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Hermann eBooks

L'imaginaire et l'intuition dans les sciences

Roger Balian

Published Dec 9, 2009• 0 citations
Publisher

Abstract

Abstract. Quantum physics against intuition. Since Galileo, deeper understanding of physical reality often contradicts everyday’s intuition. This trend has become dramatic a century ago, rendering modern physics abstruse to the laymen. The scientist can exert his intuition and his imagination within an elaborate mathematical language, but it is impossible to translate his ideas faithfully into ordinary language; this difficulty may be one of the causes of the present tarnish of the image of science. A first shock was brought in by relativity, which revolutionized our conceptions of space and time, but quantum physics presents even more counter-intuitive features. One has been led to represent physical quantities pertaining to a system, such as the position or the velocity of a particle, no longer by ordinary numbers but by “observables”, mathematical objects which belong to a non-commutative algebra. Namely, the product of two such observables is not a unique new observable but may depend on their order: the products AB and BA may differ.The values that a physical quantity can take constitute a set of ordinary numbers, the eigenvalues a( n) of the observable A. This set may be discrete as a consequence of the algebra. The discreteness of the eigenvalues of the energy of an atom explains the spectrum of its radiation. Any component of the intrinsic angular momentum of an electron can take only two opposite values, baffling our geometric intuition.If two observables commute (i.e., AB = BA), they are compatible: both may take well-defined values chosen among the eigenvalues. If they do not commute (i.e., AB ≠ BA), which occurs for the position and velocity of a particle, for the electric and magnetic field, or for the components of an angular momentum, they are incompatible: their values cannot be measured nor even specified simultaneously. Each of them presents statistical fluctuations which must be larger than some lower bound (Heisenberg’s inequality). The state of a system, prepared under some definite conditions, should be identified as the set of expectation values of all observables.Thus, quantum physics, our most fundamental description of nature, is a theory involving irreducible probabilities and unavoidable uncertainties. Its predictions do not refer to a single system, but to a statistical ensemble, and the observer cannot be completely disregarded. Moreover it may involve aspects which are apparently contradictory (Bohr’s complementarity). An electron, an atom or a molecule may behave in limiting cases either as a particle or as a wave; under intermediate conditions, it behaves as both, but each of the descriptions is blurred with probabilities. As a wave, it can interfere with itself. Likewise, an electromagnetic field has the double nature of a set of photons and of a wave. At the macroscopic scale, relative fluctuations are small, so that determinism emerges.Quantum probabilities differ from standard ones. In particular, quantum correlations involving four observables pair wise violate Bell’s inequalities, so that microscopic quantities are intrinsically indeterminate and depend on the observation process.The GHZ paradox, due to Greenberger, Horne and Zeilinger, exemplifies the most troublesome conceptual feature of quantum physics. Indeed, it baffles not only intuition, but also current logics. It deals with a physical system made of three correlated particles. Various observables, A1, A2, A3, B1, B2 and B3 ≡ B1 B2, which we do not specify here, can be measured on such a composite system. Each of them may take discrete values a1 , a2, a3, b1, b2 equal either to +1 or to –1. Since the three observables A1 B1, A2 B2 and A3 B3 are compatible, they can be simultaneously controlled. Consider then a statistical ensemble of systems, all prepared identically in such a way that A1 B1 = A2 B2 = A3 B3 = +1. If we measure A1 and B1, we shall find either a1 = b1 = +1 or a1 = b1 = –1, that is always a1 = b1; likewise for a2 = b2 = ±1 and a3 = b1 b2 = ±1. We are tempted to infer that a simultaneous determination of the values a1, a2 and a3 of A1, A2 and A3 (which are compatible) should verify a3 = a1 a2, that is, a1 a2 a3 = 1. However both theory and experiment provide in all cases a1 a2 a3 = –1. In fact the four statements a1 = b1, a2 = b2, a3 = b1 b2 and a1 a2 a3 = –1 which seem contradictory are separately “true”, in the sense that each one can be verified experimentally; but such experiments cannot be performed on a single system. We are not allowed to make a logical reasoning based on a simultaneous consideration of the values of B1 and A2 because these observables do not commute. No calculation, no experiment can simultaneously provide values for the numbers b1 and a2. Hence, we cannot regard as true together the statements a1 = b1 and a2 = b2, although they can be checked by separate experiments. Each one is true, but their union is meaningless.Thus, if theory does not allow us to imagine thought experiments in which several statements on a system might simultaneously be checked, we cannot draw conclusions by putting such statements together. We are led to regard the “truth” of a quantum statement as contextual. The values of physical quantities are not attached only to the system in itself, but also refer to the context in which their relations may be verified. Quantum mechanics does not deal with objects per se, but with what we know and what we can infer about them. It does not provide a direct image of the reality of the object since it involves its observation. Although it is a perfect tool for letting us make statistical predictions, and although it appears as our most basic theory owing to its precision and generality, it violates our daily intuition, our elementary mathematical intuition and our feelings about reality. Its probabilities and even its logics obey rather shocking new rules.

DOI
10.3917/herm.buser.2009.01.0057

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•Jul 26, 2026•3 citations
Abstract. Quantum physics against intuition. Since Galileo, deeper understanding of physical reality often contradicts everyday’s intuition. This trend has become dramatic a century ago, rendering modern physics abstruse to the laymen. The scientist can exert his intuition and his imagination within an elaborate mathematical language, but it is impossible to translate his ideas faithfully into ordinary language; this difficulty may be one of the causes of the present tarnish of the image of science. A first shock was brought in by relativity, which revolutionized our conceptions of space and time, but quantum physics presents even more counter-intuitive features. One has been led to represent physical quantities pertaining to a system, such as the position or the velocity of a particle, no longer by ordinary numbers but by “observables”, mathematical objects which belong to a non-commutative algebra. Namely, the product of two such observables is not a unique new observable but may depend on their order: the products AB and BA may differ.The values that a physical quantity can take constitute a set of ordinary numbers, the eigenvalues a( n) of the observable A. This set may be discrete as a consequence of the algebra. The discreteness of the eigenvalues of the energy of an atom explains the spectrum of its radiation. Any component of the intrinsic angular momentum of an electron can take only two opposite values, baffling our geometric intuition.If two observables commute (i.e., AB = BA), they are compatible: both may take well-defined values chosen among the eigenvalues. If they do not commute (i.e., AB ≠ BA), which occurs for the position and velocity of a particle, for the electric and magnetic field, or for the components of an angular momentum, they are incompatible: their values cannot be measured nor even specified simultaneously. Each of them presents statistical fluctuations which must be larger than some lower bound (Heisenberg’s inequality). The state of a system, prepared under some definite conditions, should be identified as the set of expectation values of all observables.Thus, quantum physics, our most fundamental description of nature, is a theory involving irreducible probabilities and unavoidable uncertainties. Its predictions do not refer to a single system, but to a statistical ensemble, and the observer cannot be completely disregarded. Moreover it may involve aspects which are apparently contradictory (Bohr’s complementarity). An electron, an atom or a molecule may behave in limiting cases either as a particle or as a wave; under intermediate conditions, it behaves as both, but each of the descriptions is blurred with probabilities. As a wave, it can interfere with itself. Likewise, an electromagnetic field has the double nature of a set of photons and of a wave. At the macroscopic scale, relative fluctuations are small, so that determinism emerges.Quantum probabilities differ from standard ones. In particular, quantum correlations involving four observables pair wise violate Bell’s inequalities, so that microscopic quantities are intrinsically indeterminate and depend on the observation process.The GHZ paradox, due to Greenberger, Horne and Zeilinger, exemplifies the most troublesome conceptual feature of quantum physics. Indeed, it baffles not only intuition, but also current logics. It deals with a physical system made of three correlated particles. Various observables, A1, A2, A3, B1, B2 and B3 ≡ B1 B2, which we do not specify here, can be measured on such a composite system. Each of them may take discrete values a1 , a2, a3, b1, b2 equal either to +1 or to –1. Since the three observables A1 B1, A2 B2 and A3 B3 are compatible, they can be simultaneously controlled. Consider then a statistical ensemble of systems, all prepared identically in such a way that A1 B1 = A2 B2 = A3 B3 = +1. If we measure A1 and B1, we shall find either a1 = b1 = +1 or a1 = b1 = –1, that is always a1 = b1; likewise for a2 = b2 = ±1 and a3 = b1 b2 = ±1. We are tempted to infer that a simultaneous determination of the values a1, a2 and a3 of A1, A2 and A3 (which are compatible) should verify a3 = a1 a2, that is, a1 a2 a3 = 1. However both theory and experiment provide in all cases a1 a2 a3 = –1. In fact the four statements a1 = b1, a2 = b2, a3 = b1 b2 and a1 a2 a3 = –1 which seem contradictory are separately “true”, in the sense that each one can be verified experimentally; but such experiments cannot be performed on a single system. We are not allowed to make a logical reasoning based on a simultaneous consideration of the values of B1 and A2 because these observables do not commute. No calculation, no experiment can simultaneously provide values for the numbers b1 and a2. Hence, we cannot regard as true together the statements a1 = b1 and a2 = b2, although they can be checked by separate experiments. Each one is true, but their union is meaningless.Thus, if theory does not allow us to imagine thought experiments in which several statements on a system might simultaneously be checked, we cannot draw conclusions by putting such statements together. We are led to regard the “truth” of a quantum statement as contextual. The values of physical quantities are not attached only to the system in itself, but also refer to the context in which their relations may be verified. Quantum mechanics does not deal with objects per se, but with what we know and what we can infer about them. It does not provide a direct image of the reality of the object since it involves its observation. Although it is a perfect tool for letting us make statistical predictions, and although it appears as our most basic theory owing to its precision and generality, it violates our daily intuition, our elementary mathematical intuition and our feelings about reality. Its probabilities and even its logics obey rather shocking new rules.
The difficulties of the old quantum theory were largely dispelled by the development of quantum mechanics in the 1920s. First Heisenberg (1925) developed the rather abstract matrix mechanics and soon after Schro¨dinger (1926; see also Eckart, 1926) developed the physically more appealing wave mechanics. Subsequently he showed them to be mathematically equivalent. The physicists were greatly relieved; now they had once again a mathematical formalism that enabled the results of experiments to be calculated. It was possible to move forward. It is certainly a very elegant and indeed beautiful formalism, and does enable the results of many physical measurements to be calculated with great accuracy. It is used daily by atomic and nuclear physicists; there is no other way to make calculations. From the earliest days there has been intense debate about the interpretation of quantum mechanics, and it shows no signs of being settled; if anything it is becoming more intense over the years. As Penrose has remarked, no one understands quantum mechanics. Feynman (1967; see also Sykes, 1994) also admitted that ‘nobody understands quantum mechanics’, and Einstein once remarked: ‘I have thought a hundred times as much about the quantum problems as I have about general relativity theory’ (Pais, 1994, p. 57). It should therefore be evident that the most that can be done here is to try to show where the real difficulties lie, and how the failure to recognize that quantum mechanics is an essentially statistical theory has led to a series of confusing quantum paradoxes. It is just these paradoxes that have often been used to support theological speculations. It is useful to ask what ‘understanding quantum mechanics’ actually means. Certainly it means being able to calculate the results of many measurements. The same applies to Newton’s theory of gravitation, which we understand in the sense that we can use it as a basis of calculation, although we do not really understand the gravitational force itself. Very often, especially in modern physics, the mathematical formalism does not give us the sort of intuitive physical feeling for what is going on that we often obtain from classical mechanics. Perhaps we can never achieve more than this for quantum mechanics. In classical mechanics we can assume that a property that is measured objectively exists prior to the interaction of the measuring apparatus with the observed system. Quantum mechanics, however, ‘is incompatible with the proposition that measurement discovers some unknown but pre-existing property’ (Peres, 1997, p. 14). There is an important distinction between mysteries and absurdities. We may be forced to live with mysteries, but there is no reason why we should put up with absurdities such as the wave-particle duality, tunnelling through potential barriers, acausal events and the collapse of the wave function. These are the quantum paradoxes that will be discussed later on. They arise within the Copenhagen interpretation of quantum mechanics and are solved by other interpretations. These alternative interpretations enable us to make conjectures about what is going on, even though we cannot understand it in detail. This is acceptable and a stimulus to further research. It is certainly difficult enough to understand quantum mechanics, but there is no need to try to visualize logical absurdities in the effort to do so. Quantum mechanics is a mathematical formalism that makes it possible to calculate the results of some experiments. Anything more than this, such as questions about the interpretation of what is measured and whether it tells us what is actually going on in the system being studied, belongs to philosophy. Thus we are concerned here with what is essentially a philosophical debate; if it were just a matter of physics the debate would have been settled long ago. One of the difficulties is that it invokes the results of experiments and uses the language of quantum mechanics. Inevitably this language uses philosophical terms and then it is easy to give the impression that the philosophical propositions are in some way entailed by the physical results. These in turn can influence theological thinking. In order to be quite clear which arguments are justified and which are not, it is essential to separate the three levels of discourse: physical, philosophical and theological. Our task is to relate these together, and this immediately raises the traditional problems of the philosophy of science, including the relation of experiment to theory, and of theory to explanation. We begin by discussing experiments, distinguishing between those that are real and those that are imaginary.
Quantum mechanics is our most successful physical theory. It predicts the behavior of atoms, molecules, and light to ten decimal places. No experiment has ever contradicted it. It also contains a crack at its foundation. The theory’s central equation—the Schrodinger equation—describes a smooth, continuous, perfectly predictable evolution of the quantum state. But whenever anyone performs a measurement, they get a single, definite result. Which result they get appears to be random. The transition from smooth evolution to random outcome is called the “collapse of the wavefunction,” and after nearly a century of analysis, there is no agreed-upon explanation for what causes it, when it occurs, or whether it occurs at all. This is the measurement problem. The measurement problem is not a minor puzzle at the margins of the theory. It is a failure to explain the most basic operation in science: looking at the result of an experiment. Around it cluster other puzzles. Light behaves like a wave in some experiments and like a particle in others—two contradictory descriptions. Two particles that have interacted remain correlated across arbitrary distances, as though communicating faster than light. The probabilities that govern measurement outcomes appear as a postulate—the Born rule—inserted by hand with no deeper justification. And the theory offers no unified picture of reality: quantum mechanics, general relativity, and the standard model of particle physics speak incompatible mathematical languages. A century of effort has produced many interpretations of quantum mechanics. Copenhagen. Many-worlds. De Broglie-Bohm. Objective collapse. Quantum Bayesianism. None has achieved consensus. The measurement problem remains. This document proposes that the source of these puzzles is not quantum mechanics itself. It is a single, invisible assumption made at the very beginning of the theory in 1900—an assumption so deeply embedded that nobody noticed they were making it. Change that assumption, and the puzzles do not get solved. They dissolve. They stop being puzzles. They become predictable consequences of geometry.
We suggest a somewhat non-standard view on a set of curious, paradoxical from the standpoint of simple classical physics and everyday experience phenomena. There are the quantisation (discrete set of values) of the observables (e.g., energy, momentum, angular momentum); forbidden simultaneous measurements of the observables in the most cases (e.g., of a coordinate and momentum, of angular momentum projections on difference axis); counter-intuitive relations on the simultaneously measurable quantities (e.g., the famous expression for the square momentum l(l+1)l(l+1)l(l+1) with the maximal projection lll). These and other paradoxes are traditionally related to "purely quantum" phenomenon, i.e., having no analogue in the "classical world" ones. However, there are deep analogies between classical and "quantum" worlds, as soon as the quantum technique is applied to the classical phenomenon. We follow these analogies with the examples of relatively simple and well known models of classical physics, such as a simplified model of light transition through the media, a system of electric charges close to each other and far from the observer; the specific of motion in the Coulomb/Newtonian field. This text can be considered as a mini-course addressed to higher school and undergraduate students who are interested in basics of quantum mechanics, but are not yet ready for systematic study of standard courses. The text may be also useful to those who supervise such students.
We show that quantum entanglement and the Heisenberg uncertainty principle are inextricably connected. Toward this end, a complete set of commuting observables (CSCO) criterion for the entanglement is developed. Assuming (A1,A2,...) and (B1,B2,...) being two CSCO's for a given system, and C being the matrix, Cij = i [Bi,Aj], for each given row i (i=1,2,...) if at least one matrix element Cij (j=1,2,...) is nonzero, then for the simultaneous eigenstates |ψ)=|A1',A2',...) of (A1,A2,...), the simultaneous measurements of (B1,B2,...) are, in general,entangled. The only exception is when all the simultaneous eigenstates |ψ)= A1', A2',...), (ψ|C|ψ)=0. This CSCO criterion may be considered as an extension of the Heisenberg uncertainty principle to quantum systems with either two (or more) particles or multi-degrees of freedom (MDF).
The odd concepts of quantum physics are nearly a century old. They may seem difficult and forbidding at first sight, but the barrier to understanding is not their difficulty but their differentness . Unlike classical models, quantum theory does not attempt to simulate Nature. It gives us information about observations of Nature. But it does contain a “theoretical entity” that claims to contain all information, observation notwithstanding, namely the quantum state vector in Hilbert space. Scientists routinely use the language of Hilbert space, and nonscientists do not use it at all, which is understandable but unfortunate. A sort of baby-talk has become the lingua franca for much of the popular journalism of quantum physics, an awkward patois that combines some of the early groping language of Bohr, de Broglie, Schrödinger, and Heisenberg with more modern words about symmetries and states. Earnest amateurs still ask me about the mystery of Bohr orbits. There is no mystery because there are no orbits, except as historical curiosities. Same for “wave–particle duality.” Same for “quantum jumps.” A friend once asked me to address a philosophy seminar about the “paradoxes of quantum mechanics.” He said he had approached C. N. Yang first, who replied “What paradoxes?” There are some mathematical and logical rough spots in our current best theory of matter, and perhaps some of them can be framed as paradoxes. But the theory is actually rather straightforward, and it has a perfectly clear interpretation linked to experiments that, in their simplest form, anyone can perform (and students do routinely in college laboratories). No empirical observations are known that are inconsistent with the quantum framework.
For the tilted family Jw=w(a1+a2−b1−b2)+∑ijsijEijJ_w = w(a_1+a_2-b_1-b_2) + \sum_{ij} s_{ij}E_{ij}Jw​=w(a1​+a2​−b1​−b2​)+∑ij​sij​Eij​ containing the I3322I_{3322}I3322​ Bell functional at w=1w=1w=1, we map where and why the almost-quantum set (NPA level 1+AB1{+}AB1+AB) exceeds NPA level 2, using a subset-relaxation diagnostic: adjoin the twelve level-2 words to the 1+AB1{+}AB1+AB basis in subsets and measure each subset's share of the gap. The excess is sharply localized and structurally rigid. (i) At w=1w=1w=1, the four matched words A1A2,A2A1,B1B2,B2B1A_1A_2, A_2A_1, B_1B_2, B_2B_1A1​A2​,A2​A1​,B1​B2​,B2​B1​ on the marginal-anchored settings recover 92% of the gap, while either party's complete pair structure, all eight third-setting words, and every mismatched cross-party pairing recover exactly none; at the level of the certified separation witness T∗T^*T∗, infeasibility appears already under a single cross-party pair of anchored words, one per wing, while every single-party and every third-setting subset leaves T∗T^*T∗ feasible. (ii) The mechanism is algebraic: reversal-symmetrized ordering differences obey ⟨A1A2BjB3⟩−⟨A1A2B3Bj⟩=12⟨[A1,A2][Bj,B3]⟩\langle A_1A_2B_jB_3\rangle - \langle A_1A_2B_3B_j\rangle = \frac{1}{2}\langle[A_1,A_2][B_j,B_3]\rangle⟨A1​A2​Bj​B3​⟩−⟨A1​A2​B3​Bj​⟩=21​⟨[A1​,A2​][Bj​,B3​]⟩, an expectation of commutator products across the wings, and sharp sequential measurement reads only anticommutators, so the excess lives in a sector unreadable in principle on either wing alone. (iii) The gap requires the functional's marginal anchor to oppose and co-locate with the aligned correlator block: moving either, aligning their signs, or deleting the marginals annihilates the gap. (iv) In the anchor weight the gap opens only on a finite window, w∈(0.90,0.95)w \in (0.90, 0.95)w∈(0.90,0.95) to (2.25,2.50)(2.25, 2.50)(2.25,2.50), with a two-lobed interior peaking near w=2w=2w=2 at 4.07×10−34.07\times10^{-3}4.07×10−3 -- 7.7 times the separation of standard I3322I_{3322}I3322​, which sits just inside the lower edge. Experiments probing the almost-quantum boundary should therefore run the tilted functional near w=2w=2w=2, not the textbook one. The headline values are certified in exact rational arithmetic: the two-word infeasibility of T∗T^*T∗, the matched-four value, the peak, and one closure; remaining grid values are floating-point with two solver families agreeing to five digits. Files in this record: the paper (PDF); exact rational proof objects for the four headline certificates (window_paper_certificates.json); a standalone verifier requiring only the Python 3 standard library (verify_window_certificates.py); analysis scripts (why_level2.py, anchor_tracking.py); and certificate generators (certify_window.py, refreeze_window.py). See README_window.txt. Companion record with the certified separation and its certificates: doi:10.5281/zenodo.21534675.