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).
In the minimalist interpretation, a measurement plays two different roles:
Quantum amplitudes are not probabilities until a basis is chosen. You cannot (or at least, I've never seen it done) make sense of amplitudes as probabilities without picking a basis. It's the second role of a measurement that distinguishes measurements from other interactions.
It's really hard to discuss with people making claims without explaining them sufficiently so that a simple-minded physicist can follow. Now you claim again that the minimal statistical interpretation requires a difference, and again I ask, which difference that might be! I've really no clue, and I'm curious about the answer!
Again you simply make bold claims without explanation.I'm not making a claim—I'm pointing out that what you are claiming is just not true. The minimalist interpretation makes a distinction between a measurement and other kinds of interactions. It's right there in the definition of how the wave function is interpreted. I'm not making a claim about quantum mechanics; it's certainly possible that there could be an interpretation that doesn't make such a distinction (maybe Many-Worlds, or maybe Bohmian). But that isn't the minimalist interpretation.
To make my still unanswered question very simple: What's the (principle) difference between the interaction of a photon hitting a CCD screen (measurement device) and just some other plane like my desk? I don't see, where there should be a differenceI agree. There shouldn't be a difference. But the minimalist interpretation requires a difference. So the minimalist interpretation is unsatisfactory for that reason. It's fine as a rule of thumb, but it can't be literally true.
Again you simply make bold claims without explanation. To make my still unanswered question very simple: What's the (principle) difference between the interaction of a photon hitting a CCD screen (measurement device) and just some other plane like my desk? I don't see, where there should be a difference. It's all the good old electromagnetic interaction, isn't it? Of course, if you think photons to be too special (and they are special), just take any massive particle you like to explain clearly in physical terms the difference between interacting of the particle with a measurement device and just matter that isn't used as a measurement device.
To be honest, I think it's ridiculous to think that there are different laws for this interaction simply because once the material is used as a measurement apparatus and the other time it's not. The very design of any physical measurement device (starting from something as simple as a yardstick up to the most complicated high-accuracy devices used for high-precision measurements in (sub-)atomic physics) are based on the fundamental laws of physics, which are believed to hold true universally and do not have exception only because something is used as a measurement device. There's even no different physical law for things living or non-living. There's no "vis viva" but just the fundamental interactions of physics at work also in living organisms. This is just another example for claims in the past that physical laws might not be universal. It's one of the great achievements of science to find universal laws. Although being far from trivial to exist, all quantitative and qualitative experience shows this universality.
This seems pretty straightforward: If there is no distinction between measurement-like interactions and non-measurement interactions, then it should be possible to formulate the minimalist interpretation in which the word "measurement" is replaced by its definition—something like "an interaction between two systems such that a property of one system causes a macroscopic change in the other system". If you try to do that, you will see that the minimalist interpretation inherently involves a microscopic/macroscopic distinction.
I am not! If you read the physics content of all standard textbooks, all there is predicted are probabilities for the outcome of measurements, and these predictions are in excellent agreement with all experiments done so far. That's the core of quantum theory, and that's the physics described by it. It's called the minimal statistical interpretation, and it's within the Copenhagen class of interpretation, taken away the unnecessary problematic parts, i.e., the collapse (in contradiction with relativistic space-time structure and causality) and a quantum-classical cut, which nobody has ever been able to demonstrate experimentally. To the contrary, the more advanced (quantum) engineering gets, the larger systems can be prepared in states that behave "quantum like" not "classical like", although the common "classical-like states" of everyday matter around us is of course also a quantum state. Classical physics is a limit for classical behavior of macroscopic properties which are coarse-grained quantities that averaged over many microscopic degrees of freedom. The rest is the math of the central-limit theorem of standard probability theory.I'm not disagreeing with the claim that quantum mechanics makes good predictions, I'm just saying that it is patently wrong to say that it makes those predictions without distinguishing measurements from non-measurements.
Bringing up the central limit theorem is just not relevant to this question. It's a non-sequitur. It's possible (in principle, if not in practice) to treat a measurement interaction quantum-mechanically, but when you do so, the probabilities disappear. To recover probabilities, you need yet another system that is not treated quantum-mechanically that will measure the measuring device. There are no probabilities associated with a pure quantum-mechanical system. At least not in the minimal interpretation. That's why I say that bringing up the central limit theorem is a non-sequitur. The central limit theorem is concerned with probabilities, and the issue is whether there are any probabilities at all involved in a quantum system where you treat everything (including observers and measurement devices) quantum-mechanically. Invoking the central limit theorem is assuming your conclusion.
I don't see why this is even controversial. The basic assumptions of the "minimalist interpretation" only say what happens when a measurement is performed. That's very different from the assumptions of Newtonian mechanics, which say what happens when massive particles interact through forces. Whether or not anything is measured it doesn't make any difference.
I am not! If you read the physics content of all standard textbooks, all there is predicted are probabilities for the outcome of measurements, and these predictions are in excellent agreement with all experiments done so far. That's the core of quantum theory, and that's the physics described by it. It's called the minimal statistical interpretation, and it's within the Copenhagen class of interpretation, taken away the unnecessary problematic parts, i.e., the collapse (in contradiction with relativistic space-time structure and causality) and a quantum-classical cut, which nobody has ever been able to demonstrate experimentally. To the contrary, the more advanced (quantum) engineering gets, the larger systems can be prepared in states that behave "quantum like" not "classical like", although the common "classical-like states" of everyday matter around us is of course also a quantum state. Classical physics is a limit for classical behavior of macroscopic properties which are coarse-grained quantities that averaged over many microscopic degrees of freedom. The rest is the math of the central-limit theorem of standard probability theory.
Why aren't there situations where quantum states are naturally falling into eigenstates of some operator – without measurement – for instance on a star?
This could happen trillions of times and thus a probability distribution. Or is any time a quantum states projects onto an eigen state of an operator a measurement by definition?This is not standard quantum mechanics. This is what is proposed in attempts to solve the measurement problem such as the physical collapse theories like GRW. Although vanhees71 is an expert on quantum field theory, in these fundamental and basic points, he is in contradiction to almost all standard textbooks of quantum physics.
There is no distinction between measurements and other interactions.In the way i am sure you mean it i fully agree.
But the distinction is in its description; and the description (and the expectations) are encoded in the observer part. The "questions asked" about an subatomic system, are in a deep way "formulated" and encoded physically in the observing system. The computational inference machinery required, for constructing questions (ie. observations) live in the observer part of the cut in my view.
If we relax this (which takes us beyond the standard theory) things become very complicated. Its to avoid this we need the "classical reference". Of course my opinon is that at some point we need to face these problems, but that is exactly the questions we need to ask to go beyond QM as it stands, to understand QG and unification imo.
/Fredrik
The "wave function" is a probability amplitude by definition (within the standard minimal interpretation). Thus it's probabilistic from the very beginning, without any necessity to introduce classical concepts.A probability distribution itself is classical statistical concept, involving no uncertainties. At this level quantum mechanics is just a deterministic theory as is newtons mechanics.
The laws of quantum theory deductively infers distribution of events, given a preparation. So the heart of the predictions is at the level of distributions.
Its just the link to single outcomes that is probabilistic. But this link, depends on a definite distribution; which IMO is anchored in the observer part of the system. And the reason this is considered to be in the realm of classical mechanics is that intercommunication within the measurment device is considered trivial in comparasion. One effectively assumes that (if we forget about relativity for a second) that all classical observers are equivalent, and thus we attain objectivity. But this objectivity (observer equivalent) only is manifested in the classical realm.
Ie. without a classical context for the measuremnt device, you can not defined a definite distribution, and not even a certain probability. Then even the probability gets "undertain", in an uncontrollable way.
/Fredrik
I sketched this in another post a while back.
[..]
So the Born rule in my understanding requires a distinction between macroscopic coarse-grained descriptions (where the rule applies) and microscopic descriptions (where it does not).Thanks ! I think Sewell and some of the refs therein have something about this. I will reply ( if this thread is dead I'll start a new one).
Can you expand that ? It might help to understand what the 'split' actually is.I sketched this in another post a while back.
But let's suppose that coarse-graining can be mathematically defined in terms of projection operators. Let ##|psirangle## be the state of the complete system (environment plus measuring devices plus observers plus …). Then we want a set of projection operators ##Pi_j## such that:
Then the Born rule can be formulated as: The probability of the system being in coarse-grained state ##j## is given by:
##P(j) = langle psi|Pi_j|psi rangle##
So the Born rule applies to coarse-grained projection operators.
The usual Born rule can be derived from this one. The usual formulation says that if you measure a property of a subsystem, then you will get an eigenvalue, with probabilities given by the square of the amplitude corresponding to the decomposition of the subsystem state into eigenstates. But if you interpret "measurement" as meaning: "A process whereby the value of the microscopic quantity is amplified to make a macroscopic difference", then different values of the microscopic property will lead to different coarse-grained states of the measurement device.So the Born rule on coarse-grained states implies that you will get results with the right probabilities.
But note: To have agreement with observation, you only need the Born rule to apply to coarse-grained projections, not to arbitrary (microscopic) projections. And furthermore, I don't know of a way to consistently extend the Born rule in terms of projections to microscopic properties. I don't think there is any way.
So the Born rule in my understanding requires a distinction between macroscopic coarse-grained descriptions (where the rule applies) and microscopic descriptions (where it does not).
by naturally I just meant without measurement.But under what circumstances would a star or whatever naturally make a transition into an eigenstate of some operator?
No, coarse-graining doesn't explain anything. It's another way of formulating the split.Can you expand that ? It might help to understand what the 'split' actually is.
I wasn't giving my opinion about it—I was describing the orthodox interpretation of quantum mechanics, which is that the probabilities in quantum mechanics are probabilities of measurement results.
An alternative interpretation which I think is empirically equivalent is to forget about measurements, and instead think of QM as a stochastic theory for macroscopic configurations. What I think is nice about this approach is that it doesn't single out measurements, and it doesn't require the assumption that a measurement always gives an eigenvalue of the operator corresponding to the observable being measured. It doesn't require observers, so you can apply QM to situations like distant stars where there are no observations. On the other hand, it's got the same flaw as the orthodox interpretation, in that it requires a macroscopic/microscopic distinction.
Getting back to your specific comment, I'm not sure what you mean by "naturally falling into eigenstates". Could you elaborate?by naturally I just meant without measurement.
Ok, if you think so, I've to accept it, but then how can you explain the classical behavior of macroscopic objects from quantum theory at all, or are you really thinking, there's a cut on a fundamental level? If so, where's the empirical evidence for it?I'm saying that the minimalist interpretation of quantum mechanics makes a distinction between measurement interactions and other interactions. I'm not saying that it is impossible to come up with an interpretation of quantum mechanics that doesn't rely on such a split, only that your preferred interpretation requires it.
Let's suppose that we have a device that measures the spin of an electron along the z-axis as follows:
If you treat the pointer like a quantum-mechanical object, then you would have to conclude:
But the Born rule says something different:
That rule is unlike anything you would say about microscopic systems.
Of course, Newtonian physics is about measurements.No, it is not. Certainly not in the sense that QM is about measurements. Newtonian physics is about the motion of particles under the influence of forces. The connection with measurement requires an assumption that the forces and/or particle motions have an affect on the measuring device. So what Newtonian physics says about measurement is derivable from Newtonian physics (possibly with other assumptions). It is not cooked into Newtonian physics.
If you assume that a spring deforms in a linear way when a force is applied to one end, then the spring can be used for measurement of forces. But it would be a mistake to define force in terms of the deformation of springs.
No, coarse-graining doesn't explain anything. It's another way of formulating the split.Ok, if you think so, I've to accept it, but then how can you explain the classical behavior of macroscopic objects from quantum theory at all, or are you really thinking, there's a cut on a fundamental level? If so, where's the empirical evidence for it?
. That is not the truth. Newtonian physics is not formulated in terms of measurements. Neither is any other theory of physics besides the minimal interpretation..All theories are written to express the outcomes of measurements ( or observations). It is not stated explicitly because it is obvious. J J Gleason identifies any formula that gives the value of a classical outcome as an operator, in anaolgy with QT.
Of course, Newtonian physics is about measurements. To write down a position vector you already need to define it in terms of measurable quantities, e.g., the three Cartesian coordinates with respect to an appropriate reference frame (provided, e.g., by three rigid rods of unit length put together at a point or the edges in one corner of your lab, etc.). Physics is about measurable quantities.
Again you only stated that the minimal interpretation depends on a distinction between measurement and other interactions, but you did not tell WHAT difference this might be and why this distinction is even NECESSARY.
Sure, but the classical describability of macroscopic properties is not due to some cut, beyond which quantum theory isn't valid anymore, but it's explanable by coarse graining from quantum many-body systems.No, coarse-graining doesn't explain anything. It's another way of formulating the split.
Sigh. It is really difficult to make this simple argument. Of course, I have to mention measurments. I have to state it, because physics is about measurements. What else should it be about?That is not the truth. Newtonian physics is not formulated in terms of measurements. Neither is any other theory of physics besides the minimal interpretation. What you're saying is just not true. You're interpreting things through your personal philosophy.
What all theories of physics must have (if they are supposed to be fundamental) is a correspondence between observations and phenomena described in the theory. If you have a theory of light, then for it to have observational content, you need something along the lines of the assumption that seeing involves light entering our eyes and registering with sensors there. But the theory of light is not expressed in terms of observations. Maxwell's equations do not mention observations. Newton's laws don't mention observations. General Relativity doesn't mention observations. You don't need for a theory to be about measurements in order to have empirical content, you need to be able to describe how the phenomena described by the theory affects what is observable.
That's the point, not to avoid the word "measurement" or "observation". Again, where is, in your opinion, the necessity to invoke classical arguments here? You havent's defined, what you mean by "classically observe".I didn't mention the word "classical" either. I said that the probabilistic predictions of QM (at least in the minimal interpretation—things are different in the Bohmian interpretation and the consistent histories interpretation and the many-worlds interpretation) depend on a distinction between "measurement" and other interactions.
One attempt might be the following: We say that system ##A## (the measuring device) measures a property of a second system, ##B## if the interaction between the two systems causes an irreversible change in the state of system ##A## such that distinct values of the property of system ##B## reliably lead to macroscopically distinguishable states of system ##A##. This definition of "measurement" seems to necessarily involve distinguish macroscopic properties from microscopic properties.Sure, but the classical describability of macroscopic properties is not due to some cut, beyond which quantum theory isn't valid anymore, but it's explanable by coarse graining from quantum many-body systems.
I'm saying that the minimal interpretation already has that split. Try formulating the probabilistic predictions of the minimalist interpretation without mentioning "measurement".Sigh. It is really difficult to make this simple argument. Of course, I have to mention measurments. I have to state it, because physics is about measurements. What else should it be about? I never have to use the word "classical" in all these definitions. That's the point, not to avoid the word "measurement" or "observation". Again, where is, in your opinion, the necessity to invoke classical arguments here? You havent's defined, what you mean by "classically observe".
Let's take a photon. It's observed by letting it interact with a detector (in former days a photo plate, nowadays some electronic detector like a CCD). There's not the slightest hint that the interaction of the photon with the photo plate or CCD cam is any different from the electromagnetic interactions described by QED.
I'm saying that the minimal interpretation already has that split. Try formulating the probabilistic predictions of the minimalist interpretation without mentioning "measurement".If you want to treat a measurement as just another interaction, then you should be able to formulate the probabilistic predictions of quantum mechanics without mentioning the word "measurement".
One attempt might be the following: We say that system ##A## (the measuring device) measures a property of a second system, ##B## if the interaction between the two systems causes an irreversible change in the state of system ##A## such that distinct values of the property of system ##B## reliably lead to macroscopically distinguishable states of system ##A##. This definition of "measurement" seems to necessarily involve distinguish macroscopic properties from microscopic properties.
Of course, there are alternative interpretations, but the minimal interpretation seems to me to absolutely require such a distinction. You cannot make sense of the minimalist interpretation without this distinction (or something equivalent: macroscopic versus microscopic, irreversible versus reversible, measurement versus non-measurement).
I don't have a proof that it is impossible to make sense of Born probabilities without making such a distinction, I'm just claiming that the minimalist interpretation does not do so.
It's not philosophy, it's physics.No, it is philosophy. It is stunning to hear a experimentalist pretend that his lab is made of quantum object and quantum observation. Every single one of your observation is classic, in the only un-philosophically possible sense.
I just take what my experimental colleagues do in the lab and try to make sense of quantum mechanics.By counting classical "up" "down", not by observing some weird superposition. And you fail to recognize you have a cut of how many of those "identically prepared state" you'll have to classically observe before being content with the stochastic prediction.
Again, you always claim that you need a split, but you never tell why you think so.I'm saying that the minimal interpretation already has that split. Try formulating the probabilistic predictions of the minimalist interpretation without mentioning "measurement".
It's not philosophy, it's physics.No, it's philosophy.
There is no distinction between measurements and other interactionsThat might be your belief, but it isn't consistent with the axioms of quantum mechanics in the minimalist interpretation.
It's sort of funny that you simultaneously denigrate philosophy and take such strong philosophical positions.
But what you said doesn't change the fact that QM in the minimalist interpretation must make a distinction between measurements and other interactions. I'm just pointing out that you previously claimed that no such split is necessary.It's not philosophy, it's physics. I just take what my experimental colleagues do in the lab and try to make sense of quantum mechanics. The main difficulty in understanding quantum mechanics is that it is formulated by people who are too philosophical (Bohr, Heisenberg), and that it is very hard to get rid of their "doctrine" (as Einstein rightfully called it).
There is no distinction between measurements and other interactions. The interaction of a particle, say a pion, with a silicon chip within a detector at the LHC is just according to the interactions described by the Standard Model (usually it's of course the electromagnetic interaction for detecting particles or photons). There's not the slightest hint that there are different laws for the interaction of a pion with some semiconductor if it's used to detect the particle or with the same piece of matter if it's not used to detect the particle.
Again, you always claim that you need a split, but you never tell why you think so. Mostly this misconception comes about, because it's somehow diffused into the teaching of QT through taking Bohr et al as the authorities having the final word on the interpretation of QT, but that's not an argument at all. There is no evidence for such a "cut" by any modern experiment, as far as I know, or do you know any experimental evidence, published in a serious peer-reviewed journal, which claims to prove that there's distinction between interactions of particles with matter (i.e., many-body quantum systems) depending on whether this matter is used as a detector or whether it's not used as such? I'd be very surprised, to say the least ;-).
What else than measurement results should any physical theory describe? Physics is about objectively observables facts of nature. It's not an empty mathematical game of thought, where you solve Schrödinger's equation just for fun without needing any "meaning" of the wave function, i.e., just because for some reason you like the puzzle to solve the equation.A theory of physics does not have to be based on measurements in order to have observational content. What you need for empirical content to a theory are correspondences: Such and such phenomenon described in the theory is assumed to correspond to such and such observation. You need for the theory to show how observations are affected by the objects and fields and so forth in the theory.
If human beings and measurement devices are physical objects described by the theory, then you should be able, in principle, to predict what happens to humans or measurement devices in this or that circumstance. That gives empirical content to the theory.
In every other theory besides quantum mechanics–special relativity, general relativity, electromagnetism, Newtonian mechanics, etc.–what is described is the behavior of particles and fields. That is enough to have empirical content if we (and our measuring devices) are ourselves made up out of those particles and fields.
What else than measurement results should any physical theory describe? Physics is about objectively observables facts of nature. It's not an empty mathematical game of thought, where you solve Schrödinger's equation just for fun without needing any "meaning" of the wave function, i.e., just because for some reason you like the puzzle to solve the equation.It's sort of funny that you simultaneously denigrate philosophy and take such strong philosophical positions.
But what you said doesn't change the fact that QM in the minimalist interpretation must make a distinction between measurements and other interactions. I'm just pointing out that you previously claimed that no such split is necessary.
That's not true. The wave function gives probabilities for measurement results. Without distinguishing measurement results from other properties, there are no probabilities in QM.
To have probabilities you have to have events—the things that have associated probabilities. The events for QM are measurement results.What else than measurement results should any physical theory describe? Physics is about objectively observables facts of nature. It's not an empty mathematical game of thought, where you solve Schrödinger's equation just for fun without needing any "meaning" of the wave function, i.e., just because for some reason you like the puzzle to solve the equation.
Why aren't there situations where quantum states are naturally falling into eigenstates of some operator – without measurement – for instance on a star?
This could happen trillions of times and thus a probability distribution. Or is any time a quantum states projects onto an eigen state of an operator a measurement by definition?I wasn't giving my opinion about it—I was describing the orthodox interpretation of quantum mechanics, which is that the probabilities in quantum mechanics are probabilities of measurement results.
An alternative interpretation which I think is empirically equivalent is to forget about measurements, and instead think of QM as a stochastic theory for macroscopic configurations. What I think is nice about this approach is that it doesn't single out measurements, and it doesn't require the assumption that a measurement always gives an eigenvalue of the operator corresponding to the observable being measured. It doesn't require observers, so you can apply QM to situations like distant stars where there are no observations. On the other hand, it's got the same flaw as the orthodox interpretation, in that it requires a macroscopic/microscopic distinction.
Getting back to your specific comment, I'm not sure what you mean by "naturally falling into eigenstates". Could you elaborate?
That's not true. The wave function gives probabilities for measurement results. Without distinguishing measurement results from other properties, there are no probabilities in QM.
To have probabilities you have to have events—the things that have associated probabilities. The events for QM are measurement results.Why aren't there situations where quantum states are naturally falling into eigenstates of some operator – without measurement – for instance on a star?
This could happen trillions of times and thus a probability distribution. Or is any time a quantum states projects onto an eigen state of an operator a measurement by definition?
The "wave function" is a probability amplitude by definition (within the standard minimal interpretation). Thus it's probabilistic from the very beginning, without any necessity to introduce classical concepts.That's not true. The wave function gives probabilities for measurement results. Without distinguishing measurement results from other properties, there are no probabilities in QM.
To have probabilities you have to have events—the things that have associated probabilities. The events for QM are measurement results.
The "wave function" is a probability amplitude by definition (within the standard minimal interpretation). Thus it's probabilistic from the very beginning, without any necessity to introduce classical concepts.
In QM of course everything is probabilistic from the very beginningThe evolution of the wave function is deterministic. Probabilities come in when you make a division between a macroscopic system (the measuring device) and the system being measured. That division is necessary for there to be any probabilities at all.
In QM of course everything is probabilistic from the very beginning, but there is no cut anywhere in the formalism. Where do you need that cut?
There's nothing, however, hinting at a "quantum classical cut", i.e., there's nothing contradicting QT in favor of a classical descriptionBut the formalism doesn't actually make any predictions without such a cut. Without a distinction between measurements and other interactions, or between macroscopic and microscopic, there are no probabilities in QM, and the theories only predictions are probabilistic.
To be more precise, we need someting that behaves with good enough accuracy classically, and quantum many-body theory shows that many-body quantum systems are behaving to good accuracy classically. That's all you need to explain why quantum theory is successful in providing its probabilistic description of the outcome of measurements on quantum systems with macroscopic measurement apparati. There's nothing, however, hinting at a "quantum classical cut", i.e., there's nothing contradicting QT in favor of a classical description, but for many-body systems very often the classical description is a very accurate description for macroscopic "coarse-grained quantities", which are sufficiently accurate to describe the relevant behavior of many-body systems, including measurement apparati. Particularly there's no difference between measurement devices and any other kind of matter since indeed measurement devices are composed of the same elementary particles as anything around us.
Between Bohr's (mis)understanding of quantum theory and today are 83 years with tremendous progressAs I see it the probabilisitic foundation required for QM is anchored in the classical "certainty".
The fact that one can in principle describe classical systems as emergent from a complex many-body QM picture, does not mean we do not need the classical measurement device.
Such a fallacious conclusions sits in the same category as those that suggest solving the observer problem by removing the observer, and instead attaching things in a metaphysical or mathematical realm and claim its objective.
This is a deep necessary insight that Bohr appears to have had. You can not make certain statistical predictions, without a certain distributions, and certain symmetries. These are manifested only on the classical side of things in the infinite ensembles etc; or in the "observer" part of this, if we are to generalize beyond classical observers.
This is easy to see if you analyse this from the point of view of inference. It should also be intuitive for any experimental work as the accuracy and confidence in the statistical predictions, requires a solid control and knowhow of the classical measurement devices. But from the perspective of mathematical physics, the statistical predictions of QM is anchored in axioms, that sit in the mathematical realm and its very easy to be seduced and confused by this.
And that essense is what i read out of Bohrs original view as well is that he understood this, this is why a proper formulation of quantum theory itself REQUIRES the classical reference. I think this is a fundamental insight.
We certainly need to improve this to understand QG and unification, but can't see anyone so far has done better than Bohr. We obviously grossly improved and developed the SM for particle physics and QFT, but the foundations remain at Bohr level.
/Fredrik
This is completely wrong. There is no quantum reality in Copenhagen.I'm not sure I can agree with this statement. In the Copenhagen interpretation, as I understand it, is we take a state vector, and from this state vector, we can decompose it into a bunch of elements. We then assign a probability distribution to this set, and give weights to each element. However, until the wavefunction "collapses", nothing is "real" for the classical world. The classical world is ignorant of the underlying probability distribution. We only "see" the outcome!
So can we not consider that a quantum reality? It could be that I'm too invested in the math of the interpretation, and not the interpretation itself.
EDIT: Feel free to PM me as well, I don't want to divert the discussion from the main thread as I haven't read every post. Hopefully this isn't off-topic!
Can you explain in more detail what this means?See this paper (attached):
“Could GR Contextuality Resolve the Missing Mass Problem?” W.M. Stuckey, Timothy McDevitt, A.K. Sten, and Michael Silberstein. Honorable Mention in the Gravity Research Foundation 2018 Awards for Essays on Gravitation, May 2018.
and this one referenced therein (also attached with errata):
“The Observable Universe Inside a Black Hole,” W.M. Stuckey, American Journal of Physics 62, No. 9, 788 – 795 (1994).
The idea is simple, as I've written many times on PF. When you combine two different GR solutions (two spacetime regions with different geometries) into one new solution, the mass of the matter responsible for the combined solution can be different for observers in each of the two different spacetime regions. In the AJP paper, we have a sphere of FLRW dust surrounded by Schwarzschild vacuum. The mass of the dust as measured by co-moving observers in the FLRW dust sphere equals the mass M of the Schwarzschild metric for the flat-space FLRW model and is less/greater than that mass in the open/closed models. So per GR, mass is a geometric property of spacetime, not an intrinsic property of matter.
We have had a real breakthrough in quantizing gravity – string theory and gauge/gravity duality.Well, there's not yet a single observable predictio from string theory. AdS/CFT has some applications even in my field of relativistic heavy-ion collisions, but to call it a breakthrough is a bit too enthusiastic ;-)).
the contextuality already inherent in GR (multiple values of mass for same matter)Can you explain in more detail what this means?
We have had a real breakthrough in quantizing gravity – string theory and gauge/gravity duality.There's definitely no consensus for that approach and it's been around for decades. If that's your belief, keep at it though!
If it were my expertise and if I had some good idea somehow I'd rather try to find a way to formulate a consistent quantum theory of gravitation than tackle some vague philosophical problems with no clear scientific content. I don't believe in the scholastic idea of finding any useful science without a firm confirmation on empirical grounds. That seems to be the reason why we still have no real breakthrough in understanding the most pressing issue in the foundation of physics, i.e., to find a consistent unification of QT (so far relativistic local and microcausal QFTs) and gravity (so far GR, which is a classical relativistic field theory). I think the trouble is that we have not the slightest clue about what effects a quantization of gravity we have to expect since there are no observations hinting at such effects.We have had a real breakthrough in quantizing gravity – string theory and gauge/gravity duality.
And if you tried to tackle QG, you’d need a starting point (“some good idea somehow”), which depends on some tacit or explicit model of physical reality you’re trying to map using empiricism and mathematics (= physics). You can’t escape the need for this model, as Becker so nicely showed in his book. Given that many brilliant physicists have worked decades without finding QG suggests to me that we should consider new models. That’s what Hardy and others in QIT argue is the value of their reconstruction project. The manner by which our model bears on QG is explained in chap 6 of our book, so I do have “some good idea” on how to proceed (and I am doing so!). This is physics, not “some vague philosophical problems with no clear scientific content.”For example, here are some papers inspired by our model:
Modified Regge Calculus as an Explanation of Dark Energy,” W.M. Stuckey, Timothy McDevitt and Michael Silberstein, Classical & Quantum Gravity 29 055015 (2012). http://arxiv.org/abs/1110.3973.
“Explaining the Supernova Data without Accelerating Expansion,” W.M. Stuckey, Timothy McDevitt and Michael Silberstein. Honorable Mention in the Gravity Research Foundation 2012 Awards for Essays on Gravitation, May 2012. International Journal of Modern Physics D 21, No. 11, 1242021 (2012) DOI: 10.1142/S0218271812420217 http://users.etown.edu/s/STUCKEYM/GRFessay2012.pdf.
“End of a Dark Age?” W.M. Stuckey, Timothy McDevitt, A.K. Sten, and Michael Silberstein. Honorable Mention in the Gravity Research Foundation 2016 Awards for Essays on Gravitation, May 2016. International Journal of Modern Physics D 25, No. 12, 1644004 (2016) DOI: 10.1142/S0218271816440041 http://arxiv.org/abs/1605.09229
This first is specifically the result of our approach to QG. The resolution of DM is via the contextuality already inherent in GR (multiple values of mass for same matter). Different models of physical reality will produce different physics.
I do not believe that the minimal interpretation is really any different from the Copenhagen interpretation when it comes to requiring a classical/quantum split. In the minimal interpretation, the meaning of quantum amplitudes is that they give statistics for measurement results. That seems to me to require a distinction between "measurements" and other interactions. That's basically the same as the classical/quantum split.
If it were my expertise and if I had some good idea somehow I'd rather try to find a way to formulate a consistent quantum theory of gravitation than tackle some vague philosophical problems with no clear scientific content. I don't believe in the scholastic idea of finding any useful science without a firm confirmation on empirical grounds. That seems to be the reason why we still have no real breakthrough in understanding the most pressing issue in the foundation of physics, i.e., to find a consistent unification of QT (so far relativistic local and microcausal QFTs) and gravity (so far GR, which is a classical relativistic field theory). I think the trouble is that we have not the slightest clue about what effects a quantization of gravity we have to expect since there are no observations hinting at such effects.And if you tried to tackle QG, you’d need a starting point (“some good idea somehow”), which depends on some tacit or explicit model of physical reality you’re trying to map using empiricism and mathematics (= physics). You can’t escape the need for this model, as Becker so nicely showed in his book. Given that many brilliant physicists have worked decades without finding QG suggests to me that we should consider new models. That’s what Hardy and others in QIT argue is the value of their reconstruction project. The manner by which our model bears on QG is explained in chap 6 of our book, so I do have “some good idea” on how to proceed (and I am doing so!). This is physics, not “some vague philosophical problems with no clear scientific content.”