Completing Quantum Mechanics within the Framework of Local Realism

1. Introduction

In Einstein’s local-realist description of the world, values of physical quantities exist objectively, and measurements performed on a system cannot immediately affect another distant system, as shown in the Einstein-Bohr debate [1,2]. The essence of the debate is the legitimacy of quantum superpositions. Einstein disagreed with Born’s probabilistic interpretation of wave-functions, which are expressed by quantum superpositions. Born’s probabilistic interpretation cannot account for randomness in quantum physics; it merely provides a way to calculate probabilities of obtaining different results by measurements. Once a measurement is performed on a system, the corresponding wave-function collapses abruptly. Because the quantum-mechanical description of the world “cannot be reconciled with the idea that physics should represent a reality in time and space, free from spooky actions at a distance” ([3], p.158), Einstein called it “the fundamental dice-game” ([3], p.149).

Based on local realism, Einstein, Podolsky, and Rosen (EPR) questioned the completeness of quantum mechanics by revealing a contradiction in its conceptual foundations: the truth of Heisenberg’s uncertainty relation implies the incompleteness of quantum mechanics; by contra-position, the completeness of quantum mechanics implies the falsity of Heisenberg’s uncertainty relation [1]. Their argument rests on the existence of correlations between distant components of the combined system ([4], p.225). Bohr defended quantum mechanics without mentioning the correlations [2]. Thus the contradiction EPR revealed cannot be explained away.

Inspired by Einstein, Bell and his followers intended to complete quantum mechanics within the framework of local realism [5]. Bell derived the first of Bell inequalities to be tested by experiments [6,7,8,9] and proved Bell’s theorem to interpret the experimental results [6]. To his surprise, his theorem shows Einstein to be wrong in the Einstein-Bohr debate [6,10], which is the opposite of what Bell intended [5,11]. Why does Bell’s approach lead him to prove the opposite of what he intended? Can quantum mechanics be completed in a way different from Bell’s approach? If yes, does the completed quantum theory imply the falsity of Heisenberg’s uncertainty relation? Concerning different understandings of Bell’s theorem [12,13,14,15,16,17], including those disproving or questioning Bell’s theorem [16,17], the questions are still open. This study aims to answer the above questions by introducing a new principle, the general principle of measurements, which is proved as a mathematical theorem. The main findings are as follows. Bell’s approach is problematic; deterministic correlations in Einstein’s local-realist description of the world are mistaken for “nonlocal-interactions” (non-locality) in the world described by Bell’s theorem. Quantum mechanics can be completed within the framework of local realism in a way consistent with the formal axiomatic definition of a general Hilbert space. Using disjunction (“or”) as the logical relation between superposed orthonormal vectors, the completed quantum theory precludes inexplicable collapses of wave-functions and is intuitively comprehensible, thus alleviating much difficulty in understanding quantum mechanics. Among various world views, Einstein’s local-realist world view is correct.

2. Bell’s Approach Revisited

When deriving Bell inequalities [6,7,8,9] by resorting to a hidden-variable theory [18,19,20], Bell and his followers merely tried to reinterpret quantum mechanics while keeping the theory in its current form intact [11]. Thus Bell’s approach presumes the legitimacy of current quantum theory. Purportedly obtained from the hypotheses underlying the EPR argument, Bell inequalities cannot capture the essence of the Einstein-Bohr debate, namely, the legitimacy of quantum superpositions. Einstein never endorsed any hidden-variable theory ([4], p.254). Nevertheless, Bell regarded Einstein as a proponent of hidden variables [6] and maintained his views (see Ref. /23/ in [21] and [22]). Influenced by Bell, some physicists believe that there exists a linkage between hidden variables and elements of physical reality [9,10]. Such a linkage is nonexistent.

Bell inequalities are not results about quantum mechanics. But the world described by Bell’s theorem is the world described by quantum mechanics, which differs substantially from Einstein’s local-realist world view. In Einstein’s local-realist description of the world, the correlations between distant components of the combined system are due to interactions occurred before the components spatially separated; after the separation, there are no longer any interaction between the distant components. But the correlations still exist, even though the components have spatially separated [1]. Regrettably, the correlations in Einstein’s local-realist description of the world [1] are mistaken for “nonlocal-interactions” in the world described by Bell’s theorem ([10], pp. 1886-1887), which leads to the questionable interpretation of the experimental results obtained by testing Bell inequalities. The questionable interpretation has led to serious consequences as shown below.

One of Bell inequalities, the CHSH inequality [7], has been intensively tested by actual experiments with single pairs of correlated photons using technologies of modern optics [8,9]. According to Bell’s theorem, the Einstein-Bohr debate seems to have been resolved “in the way which Einstein would have liked least” [20]. Nowadays people believe that Einstein’s local-realist world view conflicts with the experimental results of testing the CHSH inequality [9,10].

However, derived based on the nonexistent linkage between hidden variables and elements of physical reality, the CHSH inequality merely represents an unsuccessful attempt to reinterpret quantum mechanics by reproducing its statistical predictions while presuming the legitimacy of quantum superpositions. Thus the CHSH inequality cannot capture the essence of the Einstein-Bohr debate, and the hypotheses underlying the EPR argument are irrelevant to Bell’s theorem. But why are the correlations in Einstein’s local-realist description of the world mistaken for “nonlocal-interactions” in the world descried by Bell’s theorem? Before answering this question, it is necessary to introduce a new principle, the general principle of measurements, which can be proved as a mathematical theorem.

3. General Principle of Measurements

Physical quantities are all measured in the real world based on mathematical models of space and time. Thus the corresponding models must be identified first before the new principle is introduced and proved mathematically. The mathematical model of space is the three-dimensional Euclidean space ${\mathbb{R}}^3$ endowed with the metric given by the usual distance function between two points in space; a point $z\in {\mathbb{R}}^3$ represents a precise space coordinate. The mathematical model of time is the set of nonnegative real numbers $R_0$ equipped with the metric given by the usual distance function between two nonnegative real numbers; an element $t\in R_0$ is a precise time coordinate.

To prove the general principle of measurements as a mathematical theorem, let us recall a few definitions about “metric space” and “point-set topology”. A metric space is denoted by $(X,d)$, where X is a set, and d is a metric on X. Let $r>0$ be a real number. For $x\in X$, the open ball with center x and radius r is

$$B(x;r)=\{y\in X:d(x,y)<r\}.$$

(1)

Any open subset of X is a union of open balls. All open subsets of X constitute a metric topology ${\mathcal{T}}_X$ for X. The metric topology ${\mathcal{T}}_X$ and X form a metric topological space. Consider $x\in S$ where $S\in {\mathcal{T}}_X$. If there exists $r>0$ such that

$$B(x;r)\cap S=\left\{x\right\},$$

(2)

then x is an isolated point of S. Denote by ${\mathcal{T}}_{{\mathbb{R}}^3}$ and ${\mathcal{T}}_{R_0}$ the the metric topologies for ${\mathbb{R}}^3$ and $R_0$ associated with the metrics given by the corresponding distance functions, respectively. Apparently, measuring a point z in space perfectly precisely requires z to be an isolated point of ${\mathbb{R}}^3$. Similarly, unless time t is an isolated point of $R_0$, it is impossible to measure t perfectly precisely.

Theorem 1.

(The General Principle of Measurements): Precise space and time coordinates are practically unattainable by measurements, or equivalently, neither ${\mathbb{R}}^3$ nor $R_0$ has isolated points.

Proof. 

Consider first an arbitrarily given $z\in S$, where $S\in {\mathcal{T}}_{{\mathbb{R}}^3}$ is arbitrary. Evidently, there is no $r>0$ such that

$$B(z;r)\cap S=\left\{z\right\}.$$

(3)

Thus ${\mathbb{R}}^3$ has no isolated point. Now consider $t\in S$, where $S\in {\mathcal{T}}_{R_0}$ is arbitrary. An open “ball” now is an open interval

$$B(t;r)=(t-r,t+r).$$

(4)

There are two cases: $t=0$, and $t>0$. If $t=0$, then $B(0;r)\notin {\mathcal{T}}_{R_0}$ for any $r>0$, and there is no $S\in {\mathcal{T}}_{R_0}$ such that

$$S\cap B(0;r)=\left\{0\right\}.$$

(5)

Thus 0 is not an isolated point of $R_0$. If $t>0$, there is no $r>0$ such that

$$S\cap B(t;r)=\left\{t\right\}.$$

(6)

Thus t is not an isolated point of $R_0$ either. Consequently, $R_0$ has no isolated point. □

Proved as a mathematical theorem, the general principle of measurements does not involve issues concerning practical measurements raised by Bohr [2] or accuracy of measurement outcomes in practice. Any random phenomenon observed in physics can only be described reasonably after a large number of measurement outcomes are obtained in different repetitions of a given experiment under “the same experimental conditions” specified by precise space and time coordinates. According to the general principle of measurements, “the same experimental conditions” so specified do not exist in the real world.

Before the advent of quantum mechanics, physicists held a commonsense: the same experimental conditions always produce the same results. In other words, results produced by the same experimental conditions are deterministic. This commonsense is approximately true. Random phenomena observed in classical physics are mainly due to lack of knowledge needed to describe physical situations that typically involve a large number of single microscopic objects; their behaviors are usually assumed to be independent. Randomness in such situations is explainable using statistical mechanics. The general principle of measurements is ignorable. Quantum mechanics changed this commonsense. Nowadays physicists hold a new commonsense: the same experimental conditions do not produce the same results in quantum physics, or equivalently, the results produced by the same experimental conditions in quantum physics are random [23]. However, this new commonsense is misleading and largely responsible for erroneously interpreted experimental results.

A given physical object can, of course, have mutually exclusive properties. But the same object possesses mutually exclusive properties only at different times rather than simultaneously. In quantum physics, mutually exclusive properties of an object are observed in different repetitions of an experiment with different objects taken from an ensemble; the observed properties actually correspond to different measurement outcomes associated with different objects in the ensemble but are attached to an imaginary object. As a consequence of violating the general principle of measurements by taking precise space and time coordinates for granted to specify “the same experimental conditions”, the imaginary object does not existent in the real world. Thus the origin of randomness in quantum physics is concealed. In various experiments involving quantum superpositions, randomness actually stems from violating the general principle of measurements by taking precise space and time coordinates for granted. Caused by lack of knowledge concerning precise space and time coordinates used to specify “the same experimental conditions”, such randomness is not explainable using statistical mechanics.

Consider, again, the optical experiment designed to test the CHSH inequality [9]. Expressed as a quantum superposition, an “entangled state” describes the single pairs of correlated photons and is used to calculate the probabilities of obtaining the corresponding outcomes by measuring the polarizations of the correlated photons in the pairs, which implies the legitimacy of Born’s probabilistic interpretation of wave-functions. Thus the failure of the CHSH inequality is inevitable. The “entangled state” depends on “the same experimental conditions” specified by precise space coordinates, which are the corresponding points on a unit sphere $U\subset {\mathbb{R}}^3$. “Bloch sphere” is not contained in ${\mathbb{R}}^3$ and should not be confused with U.

The points on U correspond to (a) the polarizations as well as propagating directions of different photons detected in different repetitions of the experiment, and (b) the orientations of the polarizers for measuring the polarizations of the photons. Taking the precise space coordinates for granted [9], the “entangled state” violates the general principle of measurements and is illegitimate. The random phenomenon observed in the experiment is exactly due to lack of knowledge about the precise space coordinates used to specify “the same experimental conditions” for measuring the polarizations of different photons detected in different repetitions. Using statistical mechanics cannot explain such randomness.

Taking precise time coordinates for granted to specify “the same experimental conditions” will also conceal the origin of randomness in quantum physics and has caused much difficulty in understanding the quantum measurement problem relevant to the time development of a system described by time-dependent wave-functions [13]. Randomness due to lack of knowledge about precise time coordinates in the above situation can be analyzed similarly.

4. Hilbert Space in Quantum Mechanics

In 1927, John von Neumann provided the first formal axiomatic definition of a general Hilbert space. Based on the axiomatically defined general Hilbert space, von Neumann also provided an axiomatic formulation of quantum mechanics as its formal foundation. The name of “Hilbert space” is in honor of David Hilbert. At the beginning of the last century, Hilbert studied the classical prototype of what is known today as a Hilbert space in his work on the theory of integral equations. In functional analysis, mathematicians now denote this space by ${\ell }^2$. The general principle of measurements allows quantum mechanics to be completed within the framework of local realism in a way consistent with the formal axiomatic definition of the general Hilbert space.

Concepts used by von Neumann to define axiomatically the general Hilbert space are all highly abstract notions and have no practical meanings. Specified by an inner product, orthogonality is a purely mathematical concept. Assigning practical meanings to orthogonality is unnecessary. Moreover, the logical relation between orthogonal vectors is not needed in the formal axiomatic definition of the general Hilbert space. Elements of ${\ell }^2$, the prototypical Hilbert space, are infinite sequences of complex numbers. The logical relation between orthogonal vectors spanning ${\ell }^2$ is neither conjunction (“and”) nor disjunction (“or”); it is not necessary to assign any practical meaning to the logical relation. Only for a given application, practically meaningful concepts are necessary to define a specific Hilbert space used to describe practically meaningful objects.

However, if conjunction (“and”) is the logical relation between orthogonal vectors spanning a Hilbert space, the orthogonal vectors must not correspond to mutually exclusive properties simultaneously belonging to the same object; such an imaginary object is a consequence of violating the general principle of measurements by taking precise space and time coordinates for granted to specify “the same experimental conditions”. As shown in the last section, neither “the same experimental conditions” nor the imaginary object exist in the real world. With the inner product defined for Euclidean vectors, ${\mathbb{R}}^3$ is a Hilbert space. Orthogonal vectors spanning ${\mathbb{R}}^3$ are orthogonal only in the sense of Euclidean geometry but do not represent mutually exclusive properties simultaneously belonging to any geometric object. Thus conjunction (“and”) can serve as the logical relation between the orthogonal vectors.

The general Hilbert space axiomatically defined by von Neumann differs from any Hilbert space in quantum mechanics. The difference between the former and the latter is that the concept of orthogonality in the latter has a specific meaning: conjunction (“and”) is the logical relation between orthonormal vectors, which purportedly represent mutually exclusive properties simultaneously belonging to the same physical object. This specific meaning assigned to orthogonality makes the axiomatic formulation of quantum mechanics questionable. But von Neumann’s formal axiomatic definition of the general Hilbert space is still valid and allows disjunction (“or”) to serve as the logical relation between orthogonal vectors spanning a Hilbert space for practical applications.

For a Hilbert space in quantum mechanics completed based on the general principle of measurements, the logical relation is disjunction (“or”). Thus, represented by orthonormal vectors spanning the Hilbert space in quantum mechanics with disjunction (“or”) serving as the logical relation, different outcomes corresponding to mutually exclusive properties of a physical object are associated with different objects taken from an ensemble; the objects are measured in different repetitions of a given experiment. Obtained by measuring the corresponding object, each single outcome reveals an element of the physical reality. Consequently, a value corresponding to the single outcome can be assigned to the object, even though the precise space and time coordinates used to measure it are unknown; the value can even be taken from a continuum and cannot be obtained by measurements, such as the position or momentum of a particle moving in space.

As shown above, based on the general principle of measurements, quantum mechanics can indeed be completed within the framework of local realism, such that the completed quantum theory is consistent with the formal axiomatic definition of the general Hilbert space without changing the mathematical setting substantially. In von Neumann’s formal axiomatic definition of the general Hilbert space, the axioms concerning various calculations required by quantum mechanics, including the calculations of probabilities and expectation values, will all remain unchanged. However, some axioms in von Neumann’s axiomatic formulation of quantum mechanics, such as those implying the purported completeness of current quantum theory, the so-called inherently probabilistic nature of observations on quantum systems, and inexplicable collapses of wave-functions triggered by measurements, are questionable and should be removed; they are irrelevant to the calculations and can only make quantum mechanics difficult to understand. Removing such axioms will significantly simplify the axiomatic formulation of quantum mechanics.

With disjunction (“or”) serving as the logical relation between superposed orthonormal vectors, the notion of “quantum superposition” in the completed quantum theory will be denoted by “superposition (disjunction)”, which differs essentially from its counterpart in current quantum theory. To avoid confusion, the notion of “quantum superposition” in current quantum theory will be referred to as “superposition (conjunction)”. Violating the general principle of measurements can result in using an imaginary object described by a superpositions (conjunction) to characterize different objects measured in different repetitions. No outcome is obtained by measuring the imaginary object, which is nonexistent in the real world.

The EPR argument reveals a contradiction in the conceptual foundations of quantum mechanics, namely, the truth of Heisenberg’s uncertainty relation implies the incompleteness of current quantum theory, or equivalently, the completeness of quantum mechanics in its current form implies the falsity of Heisenberg’s uncertainty relation [1]. After quantum mechanics is completed within the framework of local realism, does the completed quantum theory imply the falsity of Heisenberg’s uncertainty relation? The answer is yes. In the real world, there is no particle described by a wave-function expressed as a superposition (conjunction), which implies an inexplicable collapse of the wave-function triggered by a measurement performed on the imaginary particle. In contrast, using a superposition (disjunction) to describe different particles taken from an ensemble and measured in different repetitions of the corresponding experiment, the completed quantum theory precludes inexplicable collapses of wave-functions and does imply the falsity of Heisenberg’s uncertainty relation.

In the optical experiment with single pairs of correlated photons taken from an ensemble [9], the pairs are described by the “entangled state” expressed as a superposition (conjunction). For a pair of correlated photons described by the “entangled state”, no polarization can be assigned to the photons if no measurement is performed, and a measurement triggers an abrupt collapse of the “entangled state” [9]. The “entangled state” cannot describe anything physically meaningful in the real world.

By contrast, corresponding to the polarizations simultaneously assigned to both correlated photons in a single pair to be detected jointly in the real world, each single outcome is obtained in one repetition of the experiment, even though the precise space coordinates used to detect the pair are unattainable by measurements and unknown. Violating the general principle of measurements brings about using an imaginary pair to characterize different pairs in the ensemble; the pairs are detected in different repetitions of the experiment. Consequently, correlations in Einstein’s local-realist world view are confused with “nonlocal-interactions” in the world described by Bell’s theorem used to prove Einstein wrong. The confusion must be clarified.

5. Correlation versus Nonlocal-Interaction

The correlations between distant components of the combined system in Einstein’s local-realist description of the world differ essentially from “nonlocal-interactions” in the world described by Bell’s theorem used to interpret the experimental results of testing Bell inequalities, where the combined system is measured in one repetition of a given experiment. The general principle of measurements can reveal the difference. Consider the experimental tests of Bell inequalities with single pairs of correlated photons taken from the corresponding ensembles. According to [24], the “locality loophole” has been closed, which is relevant to the Einstein-Bohr debate. Other loopholes, such as those concerning detections of photons and various far-fetched interpretations of Bell’s theorem, are not fundamental. Detection loopholes cannot disprove Bell’s theorem or change the world it describes. Far-fetched interpretations of Bell’s theorem are not the experiments themselves and will not be considered here.

As implied by a condition necessary to observe any random phenomenon, the correlations in Einstein’s local-realist description of the world are deterministic. The necessary condition is a banal fact: any single measurement makes no sense statistically. Clearly, jointly detecting the correlated photons in a single pair will produce one and only one outcome in one repetition of the corresponding experiment; randomness cannot manifest itself in only one repetition. Thus the correlations between the photons in the single pairs must be deterministic, and the polarizations can be simultaneously assigned to both correlated photons in each pair to be detected jointly. Therefore, using a superposition (disjunction) to describe the single pairs, the completed quantum theory precludes collapses of “entangled states” triggered by measurements. Compared to the deterministic correlations, “nonlocal-interactions” imply the legitimacy of “entangled states”, which amounts to presuming the legitimacy of current quantum theory. Without “nonlocal-interactions”, Bell’s theorem cannot interpret the experimental results of testing Bell inequalities, which are expressed in terms of statistical correlations. But why are the deterministic correlations mistaken for “nonlocal-interactions”? There are three reasons.

First, Bell and his followers did not attempt to explain random phenomena observed in outcomes obtained by experiments involving superpositions (conjunction); they merely tried to “reinterpret quantum mechanics in terms of a statistical account of an underlying hidden-variables theory” while keeping quantum mechanics in its current form intact [10]. To reproduce the statistical predictions of quantum mechanics, Bell inequalities have to be expressed by statistical correlations. Secondly, the condition necessary to observe random phenomena is ignored. According to this condition, randomness cannot manifest itself in only one repetition of the corresponding experiment. Finally, “entangled states” used in the experimental tests of Bell inequalities violate the general principle of measurements. Consequently, an imaginary pair is used to characterize different pairs detected in different repetitions, and mutually exclusive properties corresponding to different outcomes are attached to the imaginary pair. Eventually, deterministic correlations are mistaken for “nonlocal-interactions”.

6. Discussion

Inspired by Einstein, Bell and his followers intended to complete quantum mechanics within the framework of local realism [5,11]. However, misguided by the nonexistent linkage between hidden variables and elements of physical reality, they adopted a problematic approach, which cannot reveal the essential difference between deterministic correlations in Einstein’s local-realist description of the world and “nonlocal-interactions” in the world described by Bell’s theorem used to prove Einstein wrong. This regrettable situation might have been avoided, had they focused on explaining randomness in quantum physics rather than reinterpreting current quantum theory. Described himself as a follower of Einstein, Bell hoped for better theories than our current theory, insisting that current quantum theory was no more than a temporary expedient [5]; he would have been happy to see quantum mechanics completed within the framework of local realism.

By explaining randomness in quantum physics, the general principle of measurements allows quantum mechanics to be completed within the framework of local realism while keeping the formal axiomatic definition of the general Hilbert space essentially unchanged. In the axiomatic formulation of quantum mechanics, only those axioms concerning various calculations required by quantum mechanics are necessary; other axioms are questionable and can only make quantum mechanics difficult to understand. Using disjunction (“or”) as the logical relation between superposed orthonormal vectors, the completed quantum theory precludes inexplicable collapses of wave-functions and is intuitively comprehensible, thus alleviating much difficulty in understanding quantum mechanics experienced by many people, including Bell [11]. The general principle of measurements can also reveal the essential difference between deterministic correlations in Einstein’s local-realist description of the world and “nonlocal-interactions” in the world described by Bell’s theorem. Among various world views, Einstein’s local-realist world view is correct.