Interpretations of quantum mechanics

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An interpretation of quantum mechanics is an attempt to explain how the mathematical theory of quantum mechanics "corresponds" to reality. Although quantum mechanics has held up to rigorous and extremely precise tests in an extraordinarily broad range of experiments (not one prediction from quantum mechanics has been found to be contradicted by experiments), there exist a number of contending schools of thought over their interpretation. These views on interpretation differ on such fundamental questions as whether quantum mechanics is deterministic or stochastic, which elements of quantum mechanics can be considered real, and what the nature of measurement is, among other matters.


Despite nearly a century of debate and experiment, no consensus has been reached among physicists and philosophers of physics concerning which interpretation best "represents" reality. [1] [2]


Influential figures in the interpretation of quantum mechanics

The definition of quantum theorists' terms, such as wave function and matrix mechanics , progressed through many stages. For instance, Erwin Schrödinger originally viewed the electron's wave function as its charge density smeared across space, but Max Born reinterpreted the absolute square value of the wave function as the electron's probability density distributed across space.

The views of several early pioneers of quantum mechanics, such as Niels Bohr and Werner Heisenberg, are often grouped together as the "Copenhagen interpretation", though physicists and historians of physics have argued that this terminology obscures differences between the views so designated. [3] [4] Copenhagen-type ideas were never universally embraced, and challenges to a perceived Copenhagen orthodoxy gained increasing attention in the 1950s with the pilot-wave interpretation of David Bohm and the many-worlds interpretation of Hugh Everett III. [3] [5] [6]

Moreover, the strictly formalist position, shunning interpretation, has been challenged by proposals for experiments that might one day distinguish among interpretations, as by measuring an AI consciousness [7] or via quantum computing. [8] [ non-primary source needed ]

The physicist N. David Mermin once quipped, "New interpretations appear every year. None ever disappear." [9] As a rough guide to development of the mainstream view during the 1990s and 2000s, a "snapshot" of opinions was collected in a poll by Schlosshauer et al. at the "Quantum Physics and the Nature of Reality" conference of July 2011. [10] The authors reference a similarly informal poll carried out by Max Tegmark at the "Fundamental Problems in Quantum Theory" conference in August 1997. The main conclusion of the authors is that "the Copenhagen interpretation still reigns supreme", receiving the most votes in their poll (42%), besides the rise to mainstream notability of the many-worlds interpretations: "The Copenhagen interpretation still reigns supreme here, especially if we lump it together with intellectual offsprings such as information-based interpretations and the Quantum Bayesian interpretation. In Tegmark's poll, the Everett interpretation received 17% of the vote, which is similar to the number of votes (18%) in our poll."


More or less, all interpretations of quantum mechanics share two qualities:

  1. They interpret a formalism —a set of equations and principles to generate predictions via input of initial conditions
  2. They interpret a phenomenology —a set of observations, including those obtained by empirical research and those obtained informally, such as humans' experience of an unequivocal world

Two qualities vary among interpretations:

  1. Ontology—claims about what things, such as categories and entities, exist in the world
  2. Epistemology—claims about the possibility, scope, and means toward relevant knowledge of the world

In philosophy of science, the distinction of knowledge versus reality is termed epistemic versus ontic . A general law is a regularity of outcomes (epistemic), whereas a causal mechanism may regulate the outcomes (ontic). A phenomenon can receive interpretation either ontic or epistemic. For instance, indeterminism may be attributed to limitations of human observation and perception (epistemic), or may be explained as a real existing maybe encoded in the universe (ontic). Confusing the epistemic with the ontic, if for example one were to presume that a general law actually "governs" outcomes—and that the statement of a regularity has the role of a causal mechanism—is a category mistake.

In a broad sense, scientific theory can be viewed as offering scientific realism—approximately true description or explanation of the natural world—or might be perceived with antirealism. A realist stance seeks the epistemic and the ontic, whereas an antirealist stance seeks epistemic but not the ontic. In the 20th century's first half, antirealism was mainly logical positivism, which sought to exclude unobservable aspects of reality from scientific theory.

Since the 1950s, antirealism is more modest, usually instrumentalism, permitting talk of unobservable aspects, but ultimately discarding the very question of realism and posing scientific theory as a tool to help humans make predictions, not to attain metaphysical understanding of the world. The instrumentalist view is carried by the famous quote of David Mermin, "Shut up and calculate", often misattributed to Richard Feynman. [11]

Other approaches to resolve conceptual problems introduce new mathematical formalism, and so propose alternative theories with their interpretations. An example is Bohmian mechanics, whose empirical equivalence with the three standard formalisms—Schrödinger's wave mechanics, Heisenberg's matrix mechanics, and Feynman's path integral formalism—has been demonstrated.

Interpretive challenges

  1. Abstract, mathematical nature of quantum field theories: the mathematical structure of quantum mechanics is abstract without clear interpretation of its quantities.
  2. Existence of apparently indeterministic and irreversible processes: in classical field theory, a physical property at a given location in the field is readily derived. In most mathematical formulations of quantum mechanics, measurement is given a special role in the theory, as it is the sole process that can cause a nonunitary, irreversible evolution of the state.
  3. Role of the observer in determining outcomes: the Copenhagen Interpretation implies that the wavefunction is a calculational tool, and represents reality only immediately after a measurement, perhaps performed by an observer; Everettian interpretations grant that all the possibilities can be real, and that the process of measurement-type interactions cause an effective branching process. [12]
  4. Classically unexpected correlations between remote objects: entangled quantum systems, as illustrated in the EPR paradox, obey statistics that seem to violate principles of local causality. [13]
  5. Complementarity of proffered descriptions: complementarity holds that no set of classical physical concepts can simultaneously refer to all properties of a quantum system. For instance, wave description A and particulate description B can each describe quantum system S, but not simultaneously. This implies the composition of physical properties of S does not obey the rules of classical propositional logic when using propositional connectives (see "Quantum logic"). Like contextuality, the "origin of complementarity lies in the non-commutativity of operators" that describe quantum objects (Omnès 1999).
  6. Rapidly rising intricacy, far exceeding humans' present calculational capacity, as a system's size increases: since the state space of a quantum system is exponential in the number of subsystems, it is difficult to derive classical approximations.
  7. Contextual behaviour of systems locally: Quantum contextuality demonstrates that classical intuitions, in which properties of a system hold definite values independent of the manner of their measurement, fail even for local systems. Also, physical principles such as Leibniz's Principle of the identity of indiscernibles no longer apply in the quantum domain, signalling that most classical intuitions may be incorrect about the quantum world.

Influential interpretations

Copenhagen interpretation

The Copenhagen interpretation is a collection of views about the meaning of quantum mechanics principally attributed to Niels Bohr and Werner Heisenberg. It is one of the oldest of numerous proposed interpretations of quantum mechanics, as features of it date to the development of quantum mechanics during 1925–1927, and it remains one of the most commonly taught. [14] [15] There is no definitive historical statement of what is the Copenhagen interpretation. There are some fundamental agreements and disagreements between the views of Bohr and Heisenberg. [16] [17]

Hans Primas describes nine theses of the Copenhagen interpretation: quantum physics applies to individual objects, not only ensembles of objects; their description is probabilistic; their description is the result of experiments described in terms of classical (non-quantum) physics; the "frontier" that separates the classical from the quantum can be chosen arbitrarily; the act of "observation" or "measurement" is irreversible; the act of "observation" or "measurement" involves an action upon the object measured and reduces the wave packet; complementary properties cannot be observed simultaneously; no truth can be attributed to an object except according to the results of its measurement; and that quantum descriptions are objective, in that they are independent of physicists' mental arbitrariness. [18]

Heisenberg emphasized a sharp "cut" between the observer (or the instrument) and the system being observed, [19] while Bohr offered an interpretation that is independent of a subjective observer, or measurement, or collapse: there is an "irreversible" or effectively irreversible process causing the decay of quantum coherence or the wave packet which imparts the classical behavior of "observation" or "measurement". [20] [21] [22] [23]

Quantum information theories

Quantum informational approaches [24] have attracted growing support. [25] [10] They subdivide into two kinds. [26]

The state is not an objective property of an individual system but is that information, obtained from a knowledge of how a system was prepared, which can be used for making predictions about future measurements. ...A quantum mechanical state being a summary of the observer's information about an individual physical system changes both by dynamical laws, and whenever the observer acquires new information about the system through the process of measurement. The existence of two laws for the evolution of the state vector...becomes problematical only if it is believed that the state vector is an objective property of the system...The "reduction of the wavepacket" does take place in the consciousness of the observer, not because of any unique physical process which takes place there, but only because the state is a construct of the observer and not an objective property of the physical system. [29]

Relational quantum mechanics

The essential idea behind relational quantum mechanics, following the precedent of special relativity, is that different observers may give different accounts of the same series of events: for example, to one observer at a given point in time, a system may be in a single, "collapsed" eigenstate, while to another observer at the same time, it may be in a superposition of two or more states. Consequently, if quantum mechanics is to be a complete theory, relational quantum mechanics argues that the notion of "state" describes not the observed system itself, but the relationship, or correlation, between the system and its observer(s). The state vector of conventional quantum mechanics becomes a description of the correlation of some degrees of freedom in the observer, with respect to the observed system. However, it is held by relational quantum mechanics that this applies to all physical objects, whether or not they are conscious or macroscopic. Any "measurement event" is seen simply as an ordinary physical interaction, an establishment of the sort of correlation discussed above. Thus the physical content of the theory has to do not with objects themselves, but the relations between them. [30] [31]


QBism, which originally stood for "quantum Bayesianism", is an interpretation of quantum mechanics that takes an agent's actions and experiences as the central concerns of the theory. This interpretation is distinguished by its use of a subjective Bayesian account of probabilities to understand the quantum mechanical Born rule as a normative addition to good decision-making. QBism draws from the fields of quantum information and Bayesian probability and aims to eliminate the interpretational conundrums that have beset quantum theory.

QBism deals with common questions in the interpretation of quantum theory about the nature of wavefunction superposition, quantum measurement, and entanglement. [32] [33] According to QBism, many, but not all, aspects of the quantum formalism are subjective in nature. For example, in this interpretation, a quantum state is not an element of reality—instead it represents the degrees of belief an agent has about the possible outcomes of measurements. For this reason, some philosophers of science have deemed QBism a form of anti-realism. [34] [35] The originators of the interpretation disagree with this characterization, proposing instead that the theory more properly aligns with a kind of realism they call "participatory realism", wherein reality consists of more than can be captured by any putative third-person account of it. [36] [37]

Many worlds

The many-worlds interpretation is an interpretation of quantum mechanics in which a universal wavefunction obeys the same deterministic, reversible laws at all times; in particular there is no (indeterministic and irreversible) wavefunction collapse associated with measurement. The phenomena associated with measurement are claimed to be explained by decoherence, which occurs when states interact with the environment producing entanglement, repeatedly "splitting" the universe into mutually unobservable alternate histories—effectively distinct universes within a greater multiverse.

Consistent histories

The consistent histories interpretation generalizes the conventional Copenhagen interpretation and attempts to provide a natural interpretation of quantum cosmology. The theory is based on a consistency criterion that allows the history of a system to be described so that the probabilities for each history obey the additive rules of classical probability. It is claimed to be consistent with the Schrödinger equation.

According to this interpretation, the purpose of a quantum-mechanical theory is to predict the relative probabilities of various alternative histories (for example, of a particle).

Ensemble interpretation

The ensemble interpretation, also called the statistical interpretation, can be viewed as a minimalist interpretation. That is, it claims to make the fewest assumptions associated with the standard mathematics. It takes the statistical interpretation of Born to the fullest extent. The interpretation states that the wave function does not apply to an individual system for example, a single particle but is an abstract statistical quantity that only applies to an ensemble (a vast multitude) of similarly prepared systems or particles. In the words of Einstein:

The attempt to conceive the quantum-theoretical description as the complete description of the individual systems leads to unnatural theoretical interpretations, which become immediately unnecessary if one accepts the interpretation that the description refers to ensembles of systems and not to individual systems.

Einstein in Albert Einstein: Philosopher-Scientist, ed. P.A. Schilpp (Harper & Row, New York)

The most prominent current advocate of the ensemble interpretation is Leslie E. Ballentine, professor at Simon Fraser University, author of the text book Quantum Mechanics, A Modern Development.

De Broglie–Bohm theory

The de Broglie–Bohm theory of quantum mechanics (also known as the pilot wave theory) is a theory by Louis de Broglie and extended later by David Bohm to include measurements. Particles, which always have positions, are guided by the wavefunction. The wavefunction evolves according to the Schrödinger wave equation, and the wavefunction never collapses. The theory takes place in a single spacetime, is non-local, and is deterministic. The simultaneous determination of a particle's position and velocity is subject to the usual uncertainty principle constraint. The theory is considered to be a hidden-variable theory, and by embracing non-locality it satisfies Bell's inequality. The measurement problem is resolved, since the particles have definite positions at all times. [38] Collapse is explained as phenomenological. [39]

Quantum Darwinism

Quantum Darwinism is a theory meant to explain the emergence of the classical world from the quantum world as due to a process of Darwinian natural selection induced by the environment interacting with the quantum system; where the many possible quantum states are selected against in favor of a stable pointer state. It was proposed in 2003 by Wojciech Zurek and a group of collaborators including Ollivier, Poulin, Paz and Blume-Kohout. The development of the theory is due to the integration of a number of Zurek's research topics pursued over the course of twenty-five years including: pointer states, einselection and decoherence.

Transactional interpretation

The transactional interpretation of quantum mechanics (TIQM) by John G. Cramer is an interpretation of quantum mechanics inspired by the Wheeler–Feynman absorber theory. [40] It describes the collapse of the wave function as resulting from a time-symmetric transaction between a possibility wave from the source to the receiver (the wave function) and a possibility wave from the receiver to source (the complex conjugate of the wave function). This interpretation of quantum mechanics is unique in that it not only views the wave function as a real entity, but the complex conjugate of the wave function, which appears in the Born rule for calculating the expected value for an observable, as also real.

Objective collapse theories

Objective collapse theories differ from the Copenhagen interpretation by regarding both the wave function and the process of collapse as ontologically objective (meaning these exist and occur independent of the observer). In objective theories, collapse occurs either randomly ("spontaneous localization") or when some physical threshold is reached, with observers having no special role. Thus, objective-collapse theories are realistic, indeterministic, no-hidden-variables theories. Standard quantum mechanics does not specify any mechanism of collapse; QM would need to be extended if objective collapse is correct. The requirement for an extension to QM means that objective collapse is more of a theory than an interpretation. Examples include

Consciousness causes collapse (von Neumann–Wigner interpretation)

In his treatise The Mathematical Foundations of Quantum Mechanics, John von Neumann deeply analyzed the so-called measurement problem. He concluded that the entire physical universe could be made subject to the Schrödinger equation (the universal wave function). He also described how measurement could cause a collapse of the wave function. [42] This point of view was prominently expanded on by Eugene Wigner, who argued that human experimenter consciousness (or maybe even dog consciousness) was critical for the collapse, but he later abandoned this interpretation. [43] [44]

Quantum logic

Quantum logic can be regarded as a kind of propositional logic suitable for understanding the apparent anomalies regarding quantum measurement, most notably those concerning composition of measurement operations of complementary variables. This research area and its name originated in the 1936 paper by Garrett Birkhoff and John von Neumann, who attempted to reconcile some of the apparent inconsistencies of classical boolean logic with the facts related to measurement and observation in quantum mechanics.

Modal interpretations of quantum mechanics were first conceived of in 1972 by Bas van Fraassen, in his paper "A formal approach to the philosophy of science." However, this term now is used to describe a larger set of models that grew out of this approach. The Stanford Encyclopedia of Philosophy describes several versions: [45]

Time-symmetric theories

Time-symmetric interpretations of quantum mechanics were first suggested by Walter Schottky in 1921. [46] [47] Several theories have been proposed which modify the equations of quantum mechanics to be symmetric with respect to time reversal. [48] [49] [50] [51] [52] [53] (See Wheeler–Feynman time-symmetric theory.) This creates retrocausality: events in the future can affect ones in the past, exactly as events in the past can affect ones in the future. In these theories, a single measurement cannot fully determine the state of a system (making them a type of hidden-variables theory), but given two measurements performed at different times, it is possible to calculate the exact state of the system at all intermediate times. The collapse of the wavefunction is therefore not a physical change to the system, just a change in our knowledge of it due to the second measurement. Similarly, they explain entanglement as not being a true physical state but just an illusion created by ignoring retrocausality. The point where two particles appear to "become entangled" is simply a point where each particle is being influenced by events that occur to the other particle in the future.

Not all advocates of time-symmetric causality favour modifying the unitary dynamics of standard quantum mechanics. Thus a leading exponent of the two-state vector formalism, Lev Vaidman, states that the two-state vector formalism dovetails well with Hugh Everett's many-worlds interpretation. [54]

Other interpretations

As well as the mainstream interpretations discussed above, a number of other interpretations have been proposed which have not made a significant scientific impact for whatever reason. These range from proposals by mainstream physicists to the more occult ideas of quantum mysticism.


The most common interpretations are summarized in the table below. The values shown in the cells of the table are not without controversy, for the precise meanings of some of the concepts involved are unclear and, in fact, are themselves at the center of the controversy surrounding the given interpretation. For another table comparing interpretations of quantum theory, see reference. [55]

No experimental evidence exists that distinguishes among these interpretations. To that extent, the physical theory stands, and is consistent with itself and with reality; difficulties arise only when one attempts to "interpret" the theory. Nevertheless, designing experiments which would test the various interpretations is the subject of active research.

Most of these interpretations have variants. For example, it is difficult to get a precise definition of the Copenhagen interpretation as it was developed and argued about by many people.

Interpre­tationYear pub­lishedAuthor(s) Determ­inistic? Ontic wave­function?Unique
Ensemble interpretation 1926 Max Born AgnosticNoYesAgnosticNoNoNoNoNo
Copenhagen interpretation 1927– Niels Bohr, Werner Heisenberg NoSome [56] YesNoSome [57] No [58] [59] YesNoNo
de Broglie–
Bohm theory
Louis de Broglie, David Bohm YesYes [lower-alpha 1] Yes [lower-alpha 2] YesPhenomen­ologicalNoNoYesYes
Quantum logic 1936 Garrett Birkhoff AgnosticAgnosticYes [lower-alpha 3] NoNoInterpre­tational [lower-alpha 4] AgnosticNoNo
symmetric theories
1955 Satosi Watanabe YesNoYesYesNoNoNo [60] NoYes
Many-worlds interpretation 1957 Hugh Everett YesYesNoNoNoNoYesIll-posedYes
Consciousness causes collapse 1961–
John von Neumann, Eugene Wigner, Henry Stapp NoYesYesNoYesCausalNoNoYes
Many-minds interpretation 1970 H. Dieter Zeh YesYesNoNoNoInterpre­tational [lower-alpha 5] YesIll-posedYes
Consistent histories 1984 Robert B. Griffiths NoNoNoNoNo [lower-alpha 6] NoYesNoYes
Transactional interpretation 1986 John G. Cramer NoYesYesNoYes [lower-alpha 7] NoNo [lower-alpha 8] YesNo
Objective collapse theories 1986–
Penrose interpretation
Relational interpretation 1994 Carlo Rovelli No [61] NoAgnostic [lower-alpha 9] NoYes [lower-alpha 10] Intrinsic [lower-alpha 11] Possibly [lower-alpha 12] NoNo
QBism 2010Christopher Fuchs, Rüdiger SchackNoNo [lower-alpha 13] Agnostic [lower-alpha 14] NoYes [lower-alpha 15] Intrinsic [lower-alpha 16] YesNoNo
  1. Both particle AND guiding wavefunction are real.
  2. Unique particle history, but multiple wave histories.
  3. But quantum logic is more limited in applicability than Coherent Histories.
  4. Quantum mechanics is regarded as a way of predicting observations, or a theory of measurement.
  5. Observers separate the universal wavefunction into orthogonal sets of experiences.
  6. In the consistent histories interpretation the collapse is a legitimate calculational procedure when describing the preparation of a quantum system, but it amounts to nothing more than a convenient way of calculating conditional probabilities.
  7. In the TI the collapse of the state vector is interpreted as the completion of the transaction between emitter and absorber.
  8. The transactional interpretation is explicitly non-local.
  9. Comparing histories between systems in this interpretation has no well-defined meaning.
  10. Any physical interaction is treated as a collapse event relative to the systems involved, not just macroscopic or conscious observers.
  11. The state of the system is observer-dependent, i.e., the state is specific to the reference frame of the observer.
  12. The interpretation was originally presented as local, [62] but whether locality is well-posed in RQM has been disputed. [63]
  13. A wavefunction merely encodes an agent’s expectations for future experiences. It is no more real than a probability distribution is in subjective Bayesianism.
  14. Quantum theory is a tool any agent may use to help manage their expectations. The past comes into play only insofar as an agent’s individual experiences and temperament influence their priors.
  15. Although QBism would eschew this terminology. A change in the wavefunction that an agent ascribes to a system as a result of having an experience represents a change in his or her beliefs about further experiences they may have. See Doxastic logic.
  16. Observers, or more properly, participants, are as essential to the formalism as the systems they interact with.

The silent approach

Although interpretational opinions are openly and widely discussed today, that was not always the case. A notable exponent of a tendency of silence was Paul Dirac who once wrote: "The interpretation of quantum mechanics has been dealt with by many authors, and I do not want to discuss it here. I want to deal with more fundamental things." [64] This position is not uncommon among practitioners of quantum mechanics. [65] Others, like Nico van Kampen and Willis Lamb, have openly criticized non-orthodox interpretations of quantum mechanics. [66] [67]

See also

Related Research Articles

The Copenhagen interpretation is a collection of views about the meaning of quantum mechanics principally attributed to Niels Bohr and Werner Heisenberg. It is one of the oldest of numerous proposed interpretations of quantum mechanics, as features of it date to the development of quantum mechanics during 1925–1927, and it remains one of the most commonly taught.

Many-worlds interpretation Interpretation of quantum mechanics which denies the collapse of the wavefunction.

The many-worlds interpretation (MWI) is an interpretation of quantum mechanics that asserts that the universal wavefunction is objectively real, and that there is no wavefunction collapse. This implies that all possible outcomes of quantum measurements are physically realized in some "world" or universe. In contrast to some other interpretations, such as the Copenhagen interpretation, the evolution of reality as a whole in MWI is rigidly deterministic. Many-worlds is also called the relative state formulation or the Everett interpretation, after physicist Hugh Everett, who first proposed it in 1957. Bryce DeWitt popularized the formulation and named it many-worlds in the 1960s and 1970s.

EPR paradox Early and influential critique leveled against quantum mechanics

The Einstein–Podolsky–Rosen paradox is a thought experiment proposed by physicists Albert Einstein, Boris Podolsky and Nathan Rosen (EPR), with which they argued that the description of physical reality provided by quantum mechanics was incomplete. In a 1935 paper titled "Can Quantum-Mechanical Description of Physical Reality be Considered Complete?", they argued for the existence of "elements of reality" that were not part of quantum theory, and speculated that it should be possible to construct a theory containing them. Resolutions of the paradox have important implications for the interpretation of quantum mechanics.

Quantum mechanics Branch of physics describing nature on an atomic scale

Quantum mechanics is a fundamental theory in physics that provides a description of the physical properties of nature at the scale of atoms and subatomic particles. It is the foundation of all quantum physics including quantum chemistry, quantum field theory, quantum technology, and quantum information science.

Schrödingers cat Thought experiment devised by the physicist Erwin Schrödinger

In quantum mechanics, Schrödinger's cat is a thought experiment that illustrates a paradox of quantum superposition. In the thought experiment, a hypothetical cat may be considered simultaneously both alive and dead as a result of its fate being linked to a random subatomic event that may or may not occur.

Wave–particle duality is the concept in quantum mechanics that every particle or quantum entity may be described as either a particle or a wave. It expresses the inability of the classical concepts "particle" or "wave" to fully describe the behaviour of quantum-scale objects. As Albert Einstein wrote:

It seems as though we must use sometimes the one theory and sometimes the other, while at times we may use either. We are faced with a new kind of difficulty. We have two contradictory pictures of reality; separately neither of them fully explains the phenomena of light, but together they do.

The de Broglie–Bohm theory, also known as the pilot wave theory, Bohmian mechanics, Bohm's interpretation, and the causal interpretation, is an interpretation of quantum mechanics. In addition to the wavefunction, it also postulates an actual configuration of particles exists even when unobserved. The evolution over time of the configuration of all particles is defined by a guiding equation. The evolution of the wave function over time is given by the Schrödinger equation. The theory is named after Louis de Broglie (1892–1987) and David Bohm (1917–1992).

Wigner's friend is a thought experiment in theoretical quantum physics, first conceived by the physicist Eugene Wigner in 1961, and developed into a thought experiment by David Deutsch in 1985. The scenario involves an indirect observation of a quantum measurement: An observer W observes another observer F who performs a quantum measurement on a physical system. The two observers then formulate a statement about the physical system's state after the measurement according to the laws of quantum theory. However, in most of the interpretations of quantum theory, the resulting statements of the two observers contradict each other. This reflects a seeming incompatibility of two laws in quantum theory: the deterministic and continuous time evolution of the state of a closed system and the nondeterministic, discontinuous collapse of the state of a system upon measurement. Wigner's friend is therefore directly linked to the measurement problem in quantum mechanics with its famous Schrödinger's cat paradox.

In quantum mechanics, wave function collapse occurs when a wave function—initially in a superposition of several eigenstates—reduces to a single eigenstate due to interaction with the external world. This interaction is called an "observation". It is the essence of a measurement in quantum mechanics which connects the wave function with classical observables like position and momentum. Collapse is one of two processes by which quantum systems evolve in time; the other is the continuous evolution via the Schrödinger equation. Collapse is a black box for a thermodynamically irreversible interaction with a classical environment. Calculations of quantum decoherence show that when a quantum system interacts with the environment, the superpositions apparently reduce to mixtures of classical alternatives. Significantly, the combined wave function of the system and environment continue to obey the Schrödinger equation. More importantly, this is not enough to explain wave function collapse, as decoherence does not reduce it to a single eigenstate.

The many-minds interpretation of quantum mechanics extends the many-worlds interpretation by proposing that the distinction between worlds should be made at the level of the mind of an individual observer. The concept was first introduced in 1970 by H. Dieter Zeh as a variant of the Hugh Everett interpretation in connection with quantum decoherence, and later explicitly called a many or multi-consciousness interpretation. The name many-minds interpretation was first used by David Albert and Barry Loewer in 1988.

In physics, hidden-variable theories are proposals to provide explanations of quantum mechanical phenomena through the introduction of unobservable hypothetical entities. The existence of fundamental indeterminacy for some measurements is assumed as part of the mathematical formulation of quantum mechanics; moreover, bounds for indeterminacy can be expressed in a quantitative form by the Heisenberg uncertainty principle. Most hidden-variable theories are attempts at a deterministic description of quantum mechanics, to avoid quantum indeterminacy, but at the expense of requiring the existence of nonlocal interactions.

Jorge Pullin is an American theoretical physicist known for his work on black hole collisions and quantum gravity. He is the Horace Hearne Chair in theoretical Physics at the Louisiana State University.

The transactional interpretation of quantum mechanics (TIQM) takes the wave function of the standard quantum formalism, and its complex conjugate, to be retarded and advanced waves that form a quantum interaction as a Wheeler–Feynman handshake or transaction. It was first proposed in 1986 by John G. Cramer, who argues that it helps in developing intuition for quantum processes. He also suggests that it avoids the philosophical problems with the Copenhagen interpretation and the role of the observer, and also resolves various quantum paradoxes. TIQM formed a minor plot point in his science fiction novel Einstein's Bridge.

In quantum mechanics, the measurement problem considers how, or whether, wave function collapse occurs. The inability to observe such a collapse directly has given rise to different interpretations of quantum mechanics and poses a key set of questions that each interpretation must answer.

In physics, complementarity is a conceptual aspect of quantum mechanics that Niels Bohr regarded as an essential feature of the theory. The complementarity principle holds that objects have certain pairs of complementary properties which cannot all be observed or measured simultaneously. An example of such a pair is position and momentum. Bohr considered one of the foundational truths of quantum mechanics to be the fact that setting up an experiment to measure one quantity of a pair, for instance the position of an electron, excludes the possibility of measuring the other, yet understanding both experiments is necessary to characterize the object under study. In Bohr's view, the behavior of atomic and subatomic objects cannot be separated from the measuring instruments that create the context in which the measured objects behave. Consequently, there is no "single picture" that unifies the results obtained in these different experimental contexts, and only the "totality of the phenomena" together can provide a completely informative description.

There is a diversity of views that propose interpretations of quantum mechanics. They vary in how many physicists accept or reject them. An interpretation of quantum mechanics is a conceptual scheme that proposes to relate the mathematical formalism to the physical phenomena of interest. The present article is about those interpretations which, independently of their intrinsic value, remain today less known, or are simply less debated by the scientific community, for different reasons.

In physics, the observer effect is the disturbance of an observed system by the act of observation. This is often the result of instruments that, by necessity, alter the state of what they measure in some manner. A common example is checking the pressure in an automobile tire; this is difficult to do without letting out some of the air, thus changing the pressure. Similarly, it is not possible to see any object without light hitting the object, and causing it to reflect that light. While the effects of observation are often negligible, the object still experiences a change. This effect can be found in many domains of physics, but can usually be reduced to insignificance by using different instruments or observation techniques.

Some interpretations of quantum mechanics posit a central role for an observer of a quantum phenomenon. The quantum mechanical observer is tied to the issue of observer effect, where a measurement necessarily requires interacting with the physical object being measured, affecting its properties through the interaction. The term "observable" has gained a technical meaning, denoting a Hermitian operator that represents a measurement.

Quantum Bayesianism Interpretation of quantum mechanics

In physics and the philosophy of physics, quantum Bayesianism is a collection of related approaches to the interpretation of quantum mechanics, of which the most prominent is QBism. QBism is an interpretation that takes an agent's actions and experiences as the central concerns of the theory. QBism deals with common questions in the interpretation of quantum theory about the nature of wavefunction superposition, quantum measurement, and entanglement. According to QBism, many, but not all, aspects of the quantum formalism are subjective in nature. For example, in this interpretation, a quantum state is not an element of reality—instead it represents the degrees of belief an agent has about the possible outcomes of measurements. For this reason, some philosophers of science have deemed QBism a form of anti-realism. The originators of the interpretation disagree with this characterization, proposing instead that the theory more properly aligns with a kind of realism they call "participatory realism", wherein reality consists of more than can be captured by any putative third-person account of it.

The von Neumann–Wigner interpretation, also described as "consciousness causes collapse", is an interpretation of quantum mechanics in which consciousness is postulated to be necessary for the completion of the process of quantum measurement.


  1. Murray Gell-Mann - Quantum Mechanics Interpretations - Feynman Sum over Histories - EPR Bertlemann's Richard P Feynman: Quantum Mechanical View of Reality 1 (Part 1)
  2. Schlosshauer, Maximilian; Kofler, Johannes; Zeilinger, Anton (2013-08-01). "A snapshot of foundational attitudes toward quantum mechanics". Studies in History and Philosophy of Science Part B: Studies in History and Philosophy of Modern Physics. 44 (3): 222–230. arXiv: 1301.1069 . Bibcode:2013SHPMP..44..222S. doi:10.1016/j.shpsb.2013.04.004. ISSN   1355-2198. S2CID   55537196.
  3. 1 2 Jammer, Max (1974). Philosophy of Quantum Mechanics: The interpretations of quantum mechanics in historical perspective . Wiley-Interscience.
  4. Camilleri, Kristian (2009-02-01). "Constructing the Myth of the Copenhagen Interpretation". Perspectives on Science. 17 (1): 26–57. doi:10.1162/posc.2009.17.1.26. ISSN   1530-9274. S2CID   57559199.
  5. Vaidman, L. (2002, March 24). Many-Worlds Interpretation of Quantum Mechanics. Retrieved March 19, 2010, from Stanford Encyclopedia of Philosophy:
  6. Frank J. Tipler (1994). The Physics of Immortality: Modern Cosmology, God, and the Resurrection of the Dead. Anchor Books. ISBN   978-0-385-46799-5.
  7. Quantum theory as a universal physical theory, by David Deutsch, International Journal of Theoretical Physics, Vol 24 #1 (1985)
  8. Three connections between Everett's interpretation and experiment Quantum Concepts of Space and Time, by David Deutsch, Oxford University Press (1986)
  9. Mermin, N. David (2012-07-01). "Commentary: Quantum mechanics: Fixing the shifty split". Physics Today . 65 (7): 8–10. Bibcode:2012PhT....65g...8M. doi: 10.1063/PT.3.1618 . ISSN   0031-9228.
  10. 1 2 Schlosshauer, Maximilian; Kofler, Johannes; Zeilinger, Anton (2013-01-06). "A Snapshot of Foundational Attitudes Toward Quantum Mechanics". Studies in History and Philosophy of Science Part B: Studies in History and Philosophy of Modern Physics. 44 (3): 222–230. arXiv: 1301.1069 . Bibcode:2013SHPMP..44..222S. doi:10.1016/j.shpsb.2013.04.004. S2CID   55537196.
  11. For a discussion of the provenance of the phrase "shut up and calculate", see Mermin, N. David (2004). "Could Feynman have said this?". Physics Today . 57 (5): 10–11. Bibcode:2004PhT....57e..10M. doi: 10.1063/1.1768652 .
  12. Guido Bacciagaluppi, "The role of decoherence in quantum mechanics", The Stanford Encyclopedia of Philosophy (Winter 2012), Edward N Zalta, ed.
  13. La nouvelle cuisine, by John S Bell, last article of Speakable and Unspeakable in Quantum Mechanics, second edition.
  14. Siddiqui, Shabnam; Singh, Chandralekha (2017). "How diverse are physics instructors' attitudes and approaches to teaching undergraduate level quantum mechanics?". European Journal of Physics. 38 (3): 035703. Bibcode:2017EJPh...38c5703S. doi: 10.1088/1361-6404/aa6131 .
  15. Wimmel, Hermann (1992). Quantum Physics & Observed Reality: A Critical Interpretation of Quantum Mechanics. World Scientific. p. 2. ISBN   978-981-02-1010-6.
  16. Faye, Jan (2019). "Copenhagen Interpretation of Quantum Mechanics". In Zalta, Edward N. (ed.). Stanford Encyclopedia of Philosophy . Metaphysics Research Lab, Stanford University.
  17. Camilleri, K.; Schlosshauer, M. (2015). "Niels Bohr as Philosopher of Experiment: Does Decoherence Theory Challenge Bohr's Doctrine of Classical Concepts?". Studies in History and Philosophy of Modern Physics. 49: 73–83. arXiv: 1502.06547 . Bibcode:2015SHPMP..49...73C. doi:10.1016/j.shpsb.2015.01.005. S2CID   27697360.
  18. Omnès, R. (1994). The Interpretation of Quantum Mechanics. Princeton University Press. pp. 85–90. ISBN   978-0-691-03669-4. OCLC   439453957.
  19. Pauli, Wolfgang (1994) [1958]. "Albert Einstein and the development of physics". In Enz, C. P.; von Meyenn, K. (eds.). Writings on Physics and Philosophy. Berlin: Springer-Verlag. p. 133.
  20. John Bell (1990), "Against 'measurement'", Physics World, 3 (8): 33–41, doi:10.1088/2058-7058/3/8/26
  21. Niels Bohr (1985) [May 16, 1947], Jørgen Kalckar (ed.), Niels Bohr: Collected Works, Vol. 6: Foundations of Quantum Physics I (1926-1932), pp. 451–454|volume= has extra text (help)
  22. Stig Stenholm (1983), "To fathom space and time", in Pierre Meystre (ed.), Quantum Optics, Experimental Gravitation, and Measurement Theory, Plenum Press, p. 121, The role of irreversibility in the theory of measurement has been emphasized by many. Only this way can a permanent record be obtained. The fact that separate pointer positions must be of the asymptotic nature usually associated with irreversibility has been utilized in the measurement theory of Daneri, Loinger and Prosperi (1962). It has been accepted as a formal representation of Bohr's ideas by Rosenfeld (1966).
  23. Fritz Haake (April 1, 1993), "Classical motion of meter variables in the quantum theory of measurement", Physical Review A , 47 (4): 2506–2517, Bibcode:1993PhRvA..47.2506H, doi:10.1103/PhysRevA.47.2506, PMID   9909217
  24. "In the beginning was the bit". New Scientist. 2001-02-17. Retrieved 2013-01-25.
  25. Kate Becker (2013-01-25). "Quantum physics has been rankling scientists for decades". Boulder Daily Camera. Retrieved 2013-01-25.
  26. Information, Immaterialism, Instrumentalism: Old and New in Quantum Information. Christopher G. Timpson
  27. Timpson, Op. Cit.: "Let us call the thought that information might be the basic category from which all else flows informational immaterialism."
  28. "Physics concerns what we can say about nature". (Niels Bohr, quoted in Petersen, A. (1963). The philosophy of Niels Bohr. Bulletin of the Atomic Scientists, 19(7):8–14.)
  29. Hartle, J. B. (1968). "Quantum mechanics of individual systems". Am. J. Phys. 36 (8): 704–712. arXiv: 1907.02953 . Bibcode:1968AmJPh..36..704H. doi:10.1119/1.1975096. S2CID   123454773.
  30. "Relational Quantum Mechanics (Stanford Encyclopedia of Philosophy)". Retrieved 2011-01-24.
  31. For more information, see Carlo Rovelli (1996). "Relational Quantum Mechanics". International Journal of Theoretical Physics . 35 (8): 1637–1678. arXiv: quant-ph/9609002 . Bibcode:1996IJTP...35.1637R. doi:10.1007/BF02302261. S2CID   16325959.
  32. Timpson, Christopher Gordon (2008). "Quantum Bayesianism: A study" (postscript). Studies in History and Philosophy of Science Part B: Studies in History and Philosophy of Modern Physics. 39 (3): 579–609. arXiv: 0804.2047 . Bibcode:2008SHPMP..39..579T. doi:10.1016/j.shpsb.2008.03.006. S2CID   16775153.
  33. Mermin, N. David (2012-07-01). "Commentary: Quantum mechanics: Fixing the shifty split". Physics Today. 65 (7): 8–10. Bibcode:2012PhT....65g...8M. doi: 10.1063/PT.3.1618 . ISSN   0031-9228.
  34. Bub, Jeffrey (2016). Bananaworld: Quantum Mechanics for Primates. Oxford: Oxford University Press. p. 232. ISBN   978-0198718536.
  35. Ladyman, James; Ross, Don; Spurrett, David; Collier, John (2007). Every Thing Must Go: Metaphysics Naturalized . Oxford: Oxford University Press. pp.  184. ISBN   9780199573097.
  36. For "participatory realism," see, e.g.,
    Fuchs, Christopher A. (2017). "On Participatory Realism". In Durham, Ian T.; Rickles, Dean (eds.). Information and Interaction: Eddington, Wheeler, and the Limits of Knowledge. arXiv: 1601.04360 . Bibcode:2016arXiv160104360F. ISBN   9783319437606. OCLC   967844832.
    Fuchs, Christopher A.; Timpson, Christopher G. "Does Participatory Realism Make Sense? The Role of Observership in Quantum Theory". FQXi: Foundational Questions Institute. Retrieved 2017-04-18.
  37. Cabello, Adán (2017). "Interpretations of quantum theory: A map of madness". In Lombardi, Olimpia; Fortin, Sebastian; Holik, Federico; López, Cristian (eds.). What is Quantum Information?. Cambridge University Press. pp. 138–143. arXiv: 1509.04711 . Bibcode:2015arXiv150904711C. doi:10.1017/9781316494233.009. ISBN   9781107142114. S2CID   118419619.
  38. Maudlin, T. (1995). "Why Bohm's Theory Solves the Measurement Problem". Philosophy of Science. 62 (3): 479–483. doi:10.1086/289879. S2CID   122114295.
  39. Durr, D.; Zanghi, N.; Goldstein, S. (Nov 14, 1995). "Bohmian Mechanics as the Foundation of Quantum Mechanics". arXiv: quant-ph/9511016 . Also published in J.T. Cushing; Arthur Fine; S. Goldstein (17 April 2013). Bohmian Mechanics and Quantum Theory: An Appraisal. Springer Science & Business Media. pp. 21–43. ISBN   978-94-015-8715-0.
  40. "Quantum Nocality – Cramer". Archived from the original on 2010-12-29. Retrieved 2011-01-24.
  41. "Frigg, R. GRW theory" (PDF). Retrieved 2011-01-24.
  42. von Neumann, John. (1932/1955). Mathematical Foundations of Quantum Mechanics. Princeton: Princeton University Press. Translated by Robert T. Beyer.
  43. [Michael Esfeld, (1999), "Essay Review: Wigner's View of Physical Reality", published in Studies in History and Philosophy of Modern Physics, 30B, pp. 145–154, Elsevier Science Ltd.]
  44. Zvi Schreiber (1995). "The Nine Lives of Schrödinger's Cat". arXiv: quant-ph/9501014 .
  45. Lombardi, Olimpia; Dieks, Dennis (2002-11-12). "Modal Interpretations of Quantum Mechanics". Stanford Encyclopedia of Philosophy. Retrieved 2011-01-24.
  46. Schottky, Walter (1921). "Das Kausalproblem der Quantentheorie als eine Grundfrage der modernen Naturforschung überhaupt". Naturwissenschaften. 9 (25): 492–496. Bibcode:1921NW......9..492S. doi:10.1007/BF01494985. S2CID   22228793.
  47. Schottky, Walter (1921). "Das Kausalproblem der Quantentheorie als eine Grundfrage der modernen Naturforschung überhaupt". Naturwissenschaften. 9 (26): 506–511. Bibcode:1921NW......9..506S. doi:10.1007/BF01496025. S2CID   26246226.
  48. Watanabe, Satosi (1955). "Symmetry of physical laws. Part III. Prediction and retrodiction". Reviews of Modern Physics. 27 (2): 179–186. Bibcode:1955RvMP...27..179W. doi:10.1103/revmodphys.27.179. hdl:10945/47584.
  49. Aharonov, Y.; et al. (1964). "Time Symmetry in the Quantum Process of Measurement". Phys. Rev. 134 (6B): B1410–1416. Bibcode:1964PhRv..134.1410A. doi:10.1103/physrev.134.b1410.
  50. Aharonov, Y. and Vaidman, L. "On the Two-State Vector Reformulation of Quantum Mechanics." Physica Scripta, Volume T76, pp. 85-92 (1998).
  51. Wharton, K. B. (2007). "Time-Symmetric Quantum Mechanics". Foundations of Physics. 37 (1): 159–168. Bibcode:2007FoPh...37..159W. doi:10.1007/s10701-006-9089-1. S2CID   123086913.
  52. Wharton, K. B. (2010). "A Novel Interpretation of the Klein–Gordon Equation". Foundations of Physics. 40 (3): 313–332. arXiv: 0706.4075 . Bibcode:2010FoPh...40..313W. doi:10.1007/s10701-009-9398-2. S2CID   121170138.
  53. Heaney, M. B. (2013). "A Symmetrical Interpretation of the Klein–Gordon Equation". Foundations of Physics. 43 (6): 733–746. arXiv: 1211.4645 . Bibcode:2013FoPh...43..733H. doi:10.1007/s10701-013-9713-9. S2CID   118770571.
  54. Yakir Aharonov, Lev Vaidman: The Two-State Vector Formalism of Quantum Mechanics: an Updated Review. In: Juan Gonzalo Muga, Rafael Sala Mayato, Íñigo Egusquiza (eds.): Time in Quantum Mechanics, Volume 1, Lecture Notes in Physics 734, pp. 399–447, 2nd ed., Springer, 2008, ISBN   978-3-540-73472-7, doi : 10.1007/978-3-540-73473-4_13, arXiv : quant-ph/0105101, p. 443
  55. Olimpia, Lombardi; 1979-, Fortin, Sebastian; Federico, Holik; Cristian, López (2017). "Interpretations of Quantum Theory: A Map of Madness". What is quantum information?. pp. 138–144. arXiv: 1509.04711 . doi:10.1017/9781316494233.009. ISBN   9781107142114. OCLC   965759965. S2CID   118419619.CS1 maint: numeric names: authors list (link)
  56. John L. Heilbron (1988), "The Earliest Missionaries of the Copenhagen Spirit", in E. Ullmann-Margalit (ed.), Science in Reflection, pp. 201–233, This resolution of EPR, which Rosen later characterized as a stipulation that "[physical] reality is whatever quantum mechanics is capable of describing," was applauded for its clarity by Bohr's close associates. Heisenberg, Klein, and Kramers particularly liked the reduction of the EPR thought experiment to the familiar problem of the diaphragm with holes. Perhaps the most interesting responses came from Bohr's old friend, the physicist C. W. Oseen, and from his new ally, the physicist-philosopher Philipp Frank. Oseen had understood at last what he now recognized that Bohr had been saying all along: before a measurement an atom's state with respect to the quantity measured is undefined. Frank saw that Bohr had indeed transfixed EPR on an essential ambiguity. What Frank liked most was the implication that physicists should avoid the term and concept of "physical reality". He understood Bohr to mean that complementarity characterized measuring procedures, not the things measured. Bohr acknowledged that that was indeed what he had had in mind.
  57. Henrik Zinkernagel (2016), "Niels Bohr on the wave function and the classical/quantum divide", Studies in History and Philosophy of Modern Physics, 53: 9–19, arXiv: 1603.00353 , Bibcode:2016SHPMP..53....9Z, doi:10.1016/j.shpsb.2015.11.001, S2CID   18890207, For a start, discussions of the Copenhagen interpretation in the literature are ambiguous between two different views of the wave function, both of which of course accept the Born interpretation.5 Sometimes the Copenhagen (and Bohr’s) interpretation is associated with the epistemic view of the quantum state, according to which the quantum state is but a representation of our knowledge of the physical system, and thus not a real existing entity in itself. On this view the ‘collapse’ of the wave function is not a physical process, and it just reflects an update of our information about the system; see e.g. Zeilinger (1999). By contrast, the Copenhagen interpretation has also been associated with an ontological view of the quantum state, in which the wave function somehow describes a real wave, and the collapse is a real physical process – presumably induced by the observer. This ontological view is usually attributed to von Neumann in his 1932 textbook exposition of quantum mechanics; see e.g. Henderson (2010). [...] Thus, for Bohr, the wave function is a representation of a quantum system in a particular, classically described, experimental context. Three important points need to be made regarding this contextuality: 1) When a measurement is performed (that is, when an irreversible recording has been made; see below), then the context changes, and hence the wave function changes. This can formally be seen as a "collapse" of the wave function, with the square quotes indicating that we are not talking about a physical process in which a real wave collapses.
  58. W. Heisenberg (1955), "The Development of the Interpretation of the Quantum Theory", in W. Pauli (ed.), Essays dedicated to Niels Bohr on the occasion of his seventieth birthday, Pergamon Press, Of course it is entirely justified to imagine this transition, from the possible to the actual, moved to an earlier point of time, for the observer himself does not produce the transition; but it cannot be moved back to a time when the compound system was still separate from the external world, because such an assumption would not be compatible with the validity of quantum mechanics for the closed system. We see from this that a system cut off from the external world is potential but not actual in character, or, as BOHR has often expressed it, that the system cannot be described in terms of the classical concepts. We may say that the state of the closed system represented by a Hilbert vector is indeed objective, but not real, and that the classical idea of "objectively real things" must here, to this extent, be abandoned.
  59. Niels Bohr (1958), "Quantum Physics and Philosophy—Causality and Complementarity", Essays 1958-1962 on Atomic Physics and Human Knowledge, p. 3, The description of atomic phenomena has in these respects a perfectly objective character, in the sense that no explicit reference is made to any individual observer and that therefore, with proper regard to relativistic exigencies, no ambiguity is involved in the communication of information.
  60. Elitzur, Avshalom C.; Cohen, Eliahu; Okamoto, Ryo; Takeuchi, Shigeki (2018). "Nonlocal Position Changes of a Photon Revealed by Quantum Routers". Scientific Reports. 8 (1): 7730. arXiv: 1707.09483 . Bibcode:2018NatSR...8.7730E. doi:10.1038/s41598-018-26018-y. PMC   5955892 . PMID   29769645.
  61. Martin-Dussaud, P.; Rovelli, C.; Zalamea, F. (2019). "The Notion of Locality in Relational Quantum Mechanics". Foundations of Physics. 49 (2): 96–106. arXiv: 1806.08150 . Bibcode:2019FoPh...49...96M. doi:10.1007/s10701-019-00234-6. S2CID   50796079.
  62. Smerlak, Matteo; Rovelli, Carlo (2007-03-01). "Relational EPR". Foundations of Physics. 37 (3): 427–445. arXiv: quant-ph/0604064 . Bibcode:2007FoPh...37..427S. doi:10.1007/s10701-007-9105-0. ISSN   0015-9018. S2CID   11816650.
  63. Pienaar, Jacques (2019). "Comment on "The Notion of Locality in Relational Quantum Mechanics"". Foundations of Physics. 49: 1404–1414. arXiv: 1807.06457 . Bibcode:2019FoPh...49.1404P. doi:10.1007/s10701-019-00303-w.
  64. P. A. M. Dirac, The inadequacies of quantum field theory, in Paul Adrien Maurice Dirac, B. N. Kursunoglu and E. P. Wigner, Eds. (Cambridge University, Cambridge, 1987) p. 194
  65. F. J. Duarte (2014). Quantum Optics for Engineers. New York: CRC. ISBN   978-1439888537.
  66. van Kampen, N. G. (2008). "The scandal of quantum mechanics". Am. J. Phys. 76: 989.
  67. Lamb, W. E. (2001). "Super classical quantum mechanics: the best interpretation of nonrelativistic quantum mechanics." Am. J. Phys. 69: 413-421.


Further reading

Almost all authors below are professional physicists.