Interpretation of quantum mechanics

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Quantum mechanics
\Delta x \, \Delta p \ge \frac{\hbar}{2}
Uncertainty principle
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Mathematical formulation of...

Interpretations
Copenhagen · Ensemble
Hidden variable theory · Transactional
Many-worlds · Consistent histories
Quantum logic
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An interpretation of quantum mechanics is a statement which attempts to explain how quantum mechanics informs our understanding of nature. Although quantum mechanics has been tested extensively in very fine experiments, some believe the fundamentals of the theory are yet to be fully understood. There exist a number of contending schools of thought, differing over whether quantum mechanics can be understood to be deterministic, which elements of quantum mechanics can be considered "real", and other matters.

Although today this question is of special interest to philosophers of physics, many physicists continue to show a strong interest in the subject.

Contents

[edit] Historical background

The operational definition of the technical terms used by researchers in quantum theory (such as wavefunctions and matrix mechanics) progressed through intermediate stages. For instance Schrödinger originally viewed the wavefunction associated to the electron as the charge density of an object smeared out over an extended, possibly infinite, volume of space. Max Born interpreted it as the probability distribution in the space of the electron's position. Albert Einstein had great difficulty in accepting some of the consequences of the theory, such as quantum indeterminacy. Even if these matters could be treated as 'teething troubles', they have lent importance to the activity of interpretation.

It should not be assumed that most physicists think quantum mechanics needs interpretation, other than instrumentalist interpretations. The Copenhagen interpretation is the most popular among scientists, followed by the many worlds and consistent histories interpretations. But it is also true that most physicists consider non-instrumental questions (in particular ontological questions) to be irrelevant to physics. They fall back on David Mermin's expression: "shut up and calculate" often attributed (perhaps erroneously) to Richard Feynman (see [1]).

[edit] Obstructions to direct interpretation

The difficulties of interpretation reflect a number of points about the orthodox description of quantum mechanics, including:

  1. The abstract, mathematical nature of the description of quantum mechanics.
  2. The existence of what appear to be non-deterministic and irreversible processes in quantum mechanics.
  3. The phenomenon of entanglement, and in particular, the correlations between remote events that are not expected in classical theory.
  4. The complementarity of possible descriptions of reality.

First, the accepted mathematical structure of quantum mechanics is based on fairly abstract mathematics, such as Hilbert spaces and operators on those Hilbert spaces. In classical mechanics and electromagnetism, on the other hand, properties of a point mass or properties of a field are described by real numbers or functions defined on two or three dimensional sets. These have direct, spatial meaning, and in these theories there seems to be less need to provide a special interpretation for those numbers or functions.

Further, the process of measurement plays an apparently essential role in the theory. It relates the abstract elements of the theory, such as the wavefunction, to operationally definable values, such as probabilities. Measurement interacts with the system state, in somewhat peculiar ways, as is illustrated by the double-slit experiment.

The mathematical formalism used to describe the time evolution of a non-relativistic system proposes two somewhat different kinds of transformations:

  • Non-reversible and unpredictable transformations described by mathematically more complicated transformations (see quantum operations). Examples of these transformations are those that are undergone by a system as a result of measurement.

A restricted version of the problem of interpretation in quantum mechanics consists in providing some sort of plausible picture, just for the second kind of transformation. This problem may be addressed by purely mathematical reductions, for example by the many-worlds or the consistent histories interpretations.

In addition to the unpredictable and irreversible character of measurement processes, there are other elements of quantum physics that distinguish it sharply from classical physics and which cannot be represented by any classical picture. One of these is the phenomenon of entanglement, as illustrated in the EPR paradox, which seemingly violates principles of local causality.

Another obstruction to direct interpretation is the phenomenon of complementarity, which seems to violate basic principles of propositional logic. Complementarity says there is no logical picture (obeying classical propositional logic) that can simultaneously describe and be used to reason about all properties of a quantum system S. This is often phrased by saying that there are "complementary" sets A and B of propositions that can describe S, but not at the same time. Examples of A and B are propositions involving a wave description of S and a corpuscular description of S. The latter statement is one part of Niels Bohr's original formulation, which is often equated to the principle of complementarity itself.

Complementarity is not usually taken to mean that classical logic fails, although Hilary Putnam did take that view in his paper Is logic empirical?. Instead complementarity means that composition of physical properties for S (such as position and momentum both having values in certain ranges) using propositional connectives does not obey rules of classical propositional logic. As is now well-known (Omnès, 1999) the "origin of complementarity lies in the noncommutativity of operators" describing observables in quantum mechanics.

[edit] Problematic status of pictures and interpretations

The precise ontological status, of each one of the interpreting pictures, remains a matter of philosophical argument.

In other words, if we interpret the formal structure X of quantum mechanics by means of a structure Y (via a mathematical equivalence of the two structures), what is the status of Y? This is the old question of saving the phenomena, in a new guise.

Some physicists, for example Asher Peres and Chris Fuchs, seem to argue that an interpretation is nothing more than a formal equivalence between sets of rules for operating on experimental data. This would suggest that the whole exercise of interpretation is unnecessary.

[edit] Instrumentalist interpretation

Main article: Instrumentalist interpretation

Any modern scientific theory requires at the very least an instrumentalist description which relates the mathematical formalism to experimental practice and prediction. In the case of quantum mechanics, the most common instrumentalist description is an assertion of statistical regularity between state preparation processes and measurement processes. That is, if a measurement of a real-valued quantity is performed many times, each time starting with the same initial conditions, the outcome is a well-defined probability distribution over the real numbers; moreover, quantum mechanics provides a computational instrument to determine statistical properties of this distribution, such as its expectation value.

Calculations for measurements performed on a system S postulate a Hilbert space H over the complex numbers. When the system S is prepared in a pure state, it is associated with a vector in H. Measurable quantities are associated with Hermitian operators acting on H: these are referred to as observables.

Repeated measurement of an observable A for S prepared in state ψ yields a distribution of values. The expectation value of this distribution is given by the expression

\langle \psi \vert A \vert \psi \rangle.

This mathematical machinery gives a simple, direct way to compute a statistical property of the outcome of an experiment, once it is understood how to associate the initial state with a Hilbert space vector, and the measured quantity with an observable (that is, a specific Hermitian operator).

As an example of such a computation, the probability of finding the system in a given state \vert\phi\rangle is given by computing the expectation value of a (rank-1) projection operator

\Pi = \vert\phi\rangle \langle \phi \vert

The probability is then the non-negative real number given by

P = \langle \psi \vert \Pi \vert \psi \rangle = \vert \langle \psi \vert \phi \rangle \vert ^2.

By abuse of language, the bare instrumentalist description can be referred to as an interpretation, although this usage is somewhat misleading since instrumentalism explicitly avoids any explanatory role; that is, it does not attempt to answer the question of what quantum mechanics is talking about.

[edit] Summary of common interpretations of QM

[edit] Properties of interpretations

An interpretation can be characterized by whether it satisfies certain properties, such as:

To explain these properties, we need to be more explicit about the kind of picture an interpretation provides. To that end we will regard an interpretation as a correspondence between the elements of the mathematical formalism M and the elements of an interpreting structure I, where:

  • The mathematical formalism consists of the Hilbert space machinery of ket-vectors, self-adjoint operators acting on the space of ket-vectors, unitary time dependence of ket-vectors and measurement operations. In this context a measurement operation can be regarded as a transformation which carries a ket-vector into a probability distribution on ket-vectors. See also quantum operations for a formalization of this concept.
  • The interpreting structure includes states, transitions between states, measurement operations and possibly information about spatial extension of these elements. A measurement operation here refers to an operation which returns a value and results in a possible system state change. Spatial information, for instance would be exhibited by states represented as functions on configuration space. The transitions may be non-deterministic or probabilistic or there may be infinitely many states. However, the critical assumption of an interpretation is that the elements of I are regarded as physically real.

In this sense, an interpretation can be regarded as a semantics for the mathematical formalism.

In particular, the bare instrumentalist view of quantum mechanics outlined in the previous section is not an interpretation at all since it makes no claims about elements of physical reality.

The current use in physics of "completeness" and "realism" is often considered to have originated in the paper (Einstein et al., 1935) which proposed the EPR paradox. In that paper the authors proposed the concept "element of reality" and "completeness" of a physical theory. Though they did not define "element of reality", they did provide a sufficient characterization for it, namely a quantity whose value can be predicted with certainty before measuring it or disturbing it in any way. EPR define a "complete physical theory" as one in which every element of physical reality is accounted for by the theory. In the semantic view of interpretation, an interpretation of a theory is complete if every element of the interpreting structure is accounted for by the mathematical formalism. Realism is a property of each one of the elements of the mathematical formalism; any such element is real if it corresponds to something in the interpreting structure. For instance, in some interpretations of quantum mechanics (such as the many-worlds interpretation) the ket vector associated to the system state is assumed to correspond to an element of physical reality, while in others it does not.

Determinism is a property characterizing state changes due to the passage of time, namely that the state at an instant of time in the future is a function of the state at the present (see time evolution). It may not always be clear whether a particular interpreting structure is deterministic or not, precisely because there may not be a clear choice for a time parameter. Moreover, a given theory may have two interpretations, one of which is deterministic, and the other not.

Local realism has two parts:

  • The value returned by a measurement corresponds to the value of some function on the state space. Stated in another way, this value is an element of reality;
  • The effects of measurement have a propagation speed not exceeding some universal bound (e.g., the speed of light). In order for this to make sense, measurement operations must be spatially localized in the interpreting structure.

A precise formulation of local realism in terms of a local hidden variable theory was proposed by John Bell.

Bell's theorem, combined with experimental testing, restricts the kinds of properties a quantum theory can have. For instance, the experimental rejection of Bell's theorem implies that quantum mechanics cannot satisfy local realism.

[edit] Ensemble interpretation, or statistical interpretation

Main article: Ensemble Interpretation

The Ensemble interpretation, or statistical interpretation is an interpretation that can be viewed as a minimalist interpretation. That is, it is a quantum mechanical interpretation, that claims to make the fewest assumptions associated with the standard mathematical formalization. At its heart, 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, or for example, a single particle, but is an abstract mathematical, statistical quantity that only applies to an ensemble of similar prepared systems or particles. Probably the most notable supporter of such an interpretation was 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)

To date, probably the most prominent advocate of the ensemble interpretation is Leslie E. Ballentine, Professor at Simon Fraser University, and writer of the graduate level text book Quantum Mechanics, A Modern Development.

[edit] The Copenhagen interpretation

Main article: Copenhagen interpretation

The Copenhagen interpretation is the "standard" interpretation of quantum mechanics formulated by Niels Bohr and Werner Heisenberg while collaborating in Copenhagen around 1927. Bohr and Heisenberg extended the probabilistic interpretation of the wavefunction, proposed by Max Born. The Copenhagen interpretation rejects questions like "where was the particle before I measured its position" as meaningless. The measurement process randomly picks out exactly one of the many possibilities allowed for by the state's wave function.

[edit] Participatory Anthropic Principle (PAP)

Main article: Anthropic principle

Viewed by some as mysticism (see "consciousness causes collapse"), Wheeler's Participatory Anthropic Principle is the speculative theory that observation by a conscious observer is responsible for the wavefunction collapse. It is an attempt to solve the Wigner's friend paradox by simply stating that collapse occurs at the first "conscious" observer, whilst also selecting our universe from all possible universes, as the many-worlds interpretation also seeks to do - thus PAP can be seen as one competitor to the many-worlds interpretation. The key differences between PAP v. many-worlds are: PAP (non-deterministic and not an infinite number of alternate universes), many-worlds (deterministic and an infinite number of alternate universes) - see also the comparison table below. Supporters claim PAP is not a revival of substance dualism, since (in a ramification of this view) consciousness and objects are entangled and cannot be considered as distinct. Although such an idea could be added to most interpretations of quantum mechanics, PAP was added to the Copenhagen interpretation (Wheeler studied in Copenhagen under Niels Bohr in the 1930s). It is possible an experiment could be devised to test this theory, since it depends on an observer to collapse a wavefunction. The observer has to be conscious, but whether Schrödinger's cat or a person is necessary would be part of the experiment (hence the experiment could also define consciousness). However, the experiment would need to be carefully designed as, in Wheeler's view, it would need to ensure for an unobserved event that it remained unobserved for all time [1].

[edit] Consistent histories

Main article: Consistent histories

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

[edit] Objective collapse theories

Main article: Objective collapse theory

Objective collapse theories differ from the Copenhagen interpretation in regarding both the wavefunction and the process of collapse as ontologically objective. In objective theories, collapse occurs randomly ("spontaneous localization"), or when some physical threshold is reached, with observers having no special role. Thus, they are realistic, indeterministic, no-hidden-variables theories. The mechanism of collapse is not specified by standard quantum mechanics, which needs to be extended if this approach is correct, meaning that Objective Collapse is more of a theory than an interpretation. Examples include the Ghirardi-Rimini-Weber theory[2] and the Penrose interpretation. [3]

[edit] Many worlds

Main article: Many-worlds interpretation

The many-worlds interpretation (or MWI) is an interpretation of quantum mechanics that rejects the non-deterministic and irreversible wavefunction collapse associated with measurement in the Copenhagen interpretation in favor of a description in terms of quantum entanglement and reversible time evolution of states. The phenomena associated with measurement are explained by decoherence which occurs when states interact with the environment. As result of the decoherence the world-lines of macroscopic objects repeatedly split into mutually unobservable, branching histories -- distinct universes within a greater multiverse.

[edit] Stochastic mechanics

An entirely classical derivation and interpretation of the Schrödinger equation by analogy with Brownian motion was suggested by Princeton University professor Edward Nelson in 1966 (“Derivation of the Schrödinger Equation from Newtonian Mechanics”, Phys. Rev. 150, 1079-1085). Similar considerations were published already before, e.g. by R. Fürth (1933), I. Fényes (1952), W. Weizel (1953), and are referenced in Nelson's paper. More recent work on the subject can be found in M. Pavon, “Stochastic mechanics and the Feynman integral”, J. Math. Phys. 41, 6060-6078 (2000).

[edit] The decoherence approach

Main article: Decoherence

Decoherence occurs when a system interacts with its environment, or any complex external system, in such a thermodynamically irreversible way that ensures different elements in the quantum superposition of the system+environment's wavefunction can no longer interfere with each other. Decoherence does not provide a mechanism for the actual wavefunction collapse; rather it provides a mechanism for the appearance of wavefunction collapse. The quantum nature of the system is simply "leaked" into the environment so that a total superposition of the wavefunction still exists, but exists beyond the realm of measurement. Thus decoherence, as a philosophical interpretation, amounts to something similar to the many-worlds approach. It has the advantage of being supported by a detailed and plausible mathematical background, much of it developed by H. Dieter Zeh and Wojciech H. Zurek, which help to make it one of the more popular approaches among current physicists.

[edit] Many minds

Main article: Many-minds interpretation

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.

[edit] Quantum logic

Main article: 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.

[edit] The Bohm interpretation

Main article: Bohm interpretation

The Bohm interpretation of quantum mechanics is an interpretation postulated by David Bohm in which the existence of a non-local universal wavefunction allows distant particles to interact instantaneously. The interpretation generalizes Louis de Broglie's pilot wave theory from 1927, which posits that both wave and particle are real. The wavefunction 'guides' the motion of the particle, and evolves according to the Schrödinger equation. The interpretation assumes a single, nonsplitting universe (unlike the Everett many-worlds interpretation) and is deterministic (unlike the Copenhagen interpretation). It says the state of the universe evolves smoothly through time, without the collapsing of wavefunctions when a measurement occurs, as in the Copenhagen interpretation. However, it does this by assuming a number of hidden variables, namely the positions of all the particles in the universe, which, like probability amplitudes in other interpretations, can never be measured directly.

[edit] Transactional interpretation

Main article: Transactional interpretation

The transactional interpretation of quantum mechanics (TIQM) by John G. Cramer [2] is an interpretation of quantum mechanics inspired by the contribution Richard Feynman made to Quantum Electrodynamics. It describes quantum interactions in terms of a standing wave formed by retarded (forward-in-time) and advanced (backward-in-time) waves. The author argues that it avoids the philosophical problems with the Copenhagen interpretation and the role of the observer, and resolves various quantum paradoxes.

[edit] Relational quantum mechanics

Main article: 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 is to do not with objects themselves, but the relations between them [3]. For more information, see Rovelli (1996).

[edit] Modal interpretations of quantum theory

Modal interpretations of quantum mechanics were first conceived of in 1972 by B. 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:

  • The Copenhagen variant
  • Kochen-Dieks-Healey Interpretations
  • Motivating Early Modal Interpretations, based on the work of R. Clifton, M. Dickson and J. Bub.

[edit] Incomplete measurements

Main article: Theory of incomplete measurements

The theory of incomplete measurements (TIM) derives the main axioms of quantum mechanics from properties of the physical processes that are acceptable measurements. In that interpretation:

  • wavefunctions collapse because we require measurements to give consistent and repeatable results.
  • wavefunctions are complex-valued because they represent a field of "found/not-found" probabilities.
  • eigenvalue equations are associated with symbolic values of measurements, which we often choose to be real numbers.

The TIM is more than a simple interpretation of quantum mechanics, since in that theory, both general relativity and the traditional formalism of quantum mechanics are seen as approximations. However, it does give an interesting interpretation to quantum mechanics.

[edit] Comparison

The most common interpretations are summarized here (however, the assignment of values in the table is not without controversy, for the precise meanings of some of the concepts involved are unclear and, in fact, the subject of the very controversy itself):

No experimental evidence exists that would distinguish between the interpretations listed. To that extent, the physical theory stands, and is consistent with, itself and with reality; troubles come only when one attempts to "interpret" it. Nevertheless, there is active research in attempting to come up with experimental tests which would allow differences between the interpretations to be experimentally tested.

Interpretation Deterministic? Waveform
Real?		 Unique
History?		 Hidden
Variables?		 Collapsing
Wavefunctions?		 Observer
role?
Copenhagen interpretation
(Waveform not real)		 No No Yes No NA NA
Ensemble interpretation
(Waveform not real)		 No No Yes Agnostic No None
Copenhagen interpretation
(Waveform real)
Objective collapse theories		 No Yes Yes No Yes None
Consistent histories
(Decoherent approach)		 Agnostic1 Agnostic1 No No No Interpretational²
Quantum logic Agnostic Agnostic Yes³ No No Interpretational²
Many-worlds interpretation
(Decoherent approach)		 Yes Yes No No No None
Stochastic mechanics No No Yes No No None
Many-minds interpretation Yes Yes No No No Interpretational4
Bohm-de Broglie interpretation
("Pilot-wave" approach)		 Yes Yes5 Yes6 Yes No None
Transactional interpretation No Yes Yes No Yes7 None
Copenhagen interpretation
(Waveform real)
PAP		 No Yes Yes No Yes Causal
Relational
Quantum Mechanics		 No Yes Agnostic8 No Yes9 None
Incomplete
measurements		 No No10 Yes No Yes10 Interpretational²

1 If wavefunction is real then this becomes the many-worlds interpretation. If wavefunction less than real, but more than just information, then Zurek calls this the "existential interpretation".
2 Quantum mechanics is regarded as a way of predicting observations, or a theory of measurement..
3 But quantum logic is more limited in applicability than Coherent Histories.
4 Observers separate the universal wavefunction into orthogonal sets of experiences.
5 Both particle AND guiding wavefunction are real.
6 Unique particle history, but multiple wave histories.
7 In the TI the collapse of the state vector is interpreted as the completion of the transaction between emitter and absorber.
8 Comparing histories between systems in this interpretation has no well-defined meaning.
9 Any physical interaction is treated as a collapse event relative to the systems involved, not just macroscopic or conscious observers.
10 The nature and collapse of the wavefunction are derived, not axiomatic.

Each interpretation has many variants. It is difficult to get a precise definition of the Copenhagen interpretation. In the table above, two variants are shown: one that regards the waveform as being a tool for calculating probabilities only, and the other regards the waveform as an "element of reality".

[edit] See also

[edit] Related lists

[edit] References

  • Bub, J. and Clifton, R. 1996. “A uniqueness theorem for interpretations of quantum mechanics,” Studies in History and Philosophy of Modern Physics, 27B, 181-219
  • R. Carnap, The interpretation of physics, Foundations of Logic and Mathematics of the International Encyclopedia of Unified Science, University of Chicago Press, 1939.
  • D. Deutsch, The Fabric of Reality, Allen Lane, 1997. Though written for general audiences, in this book Deutsch argues forcefully against instrumentalism.
  • Dickson, M. 1994. Wavefunction tails in the modal interpretation, Proceedings of the PSA 1994, Hull, D., Forbes, M., and Burian, R. (eds), Vol. 1, pp. 366-376. East Lansing, Michigan: Philosophy of Science Association.
  • Dickson, M. and Clifton, R. 1998. Lorentz-invariance in modal interpretations The Modal Interpretation of Quantum Mechanics, Dieks, D. and Vermaas, P. (eds), pp. 9-48. Dordrecht: Kluwer Academic Publishers
  • A. Einstein, B. Podolsky and N. Rosen, Can quantum-mechanical description of physical reality be considered complete? Phys. Rev. 47 777, 1935.
  • C. Fuchs and A. Peres, Quantum theory needs no ‘interpretation’ , Physics Today, March 2000.
  • Christopher Fuchs, Quantum Mechanics as Quantum Information (and only a little more), arXiv:quant-ph/0205039 v1, (2002)
  • N. Herbert. Quantum Reality: Beyond the New Physics, New York: Doubleday, ISBN 0-385-23569-0, LoC QC174.12.H47 1985.
  • T. Hey and P. Walters, The New Quantum Universe, New York: Cambridge University Press, 2003, ISBN 0-5215-6457-3.
  • R. Jackiw and D. Kleppner, One Hundred Years of Quantum Physics, Science, Vol. 289 Issue 5481, p893, August 2000.
  • M. Jammer, The Conceptual Development of Quantum Mechanics. New York: McGraw-Hill, 1966.
  • M. Jammer, The Philosophy of Quantum Mechanics. New York: Wiley, 1974.
  • Al-Khalili, Quantum: A Guide for the Perplexed. London: Weidenfeld & Nicholson, 2003.
  • W. M. de Muynck, Foundations of quantum mechanics, an empiricist approach, Dordrecht: Kluwer Academic Publishers, 2002, ISBN 1-4020-0932-1
  • R. Omnès, Understanding Quantum Mechanics, Princeton, 1999.
  • K. Popper, Conjectures and Refutations, Routledge and Kegan Paul, 1963. The chapter "Three views Concerning Human Knowledge", addresses, among other things, the instrumentalist view in the physical sciences.
  • H. Reichenbach, Philosophic Foundations of Quantum Mechanics, Berkeley: University of California Press, 1944.
  • C. Rovelli, Relational Quantum Mechanics; Int. J. of Theor. Phys. 35 (1996) 1637. arXiv: quant-ph/9609002 [4]
  • M. Tegmark and J. A. Wheeler, 100 Years of Quantum Mysteries", Scientific American 284, 68, 2001.
  • van Fraassen, B. 1972. A formal approach to the philosophy of science, in Paradigms and Paradoxes: The Philosophical Challenge of the Quantum Domain, Colodny, R. (ed.), pp. 303-366. Pittsburgh: University of Pittsburgh Press.
  • John A. Wheeler and Wojciech Hubert Zurek (eds), Quantum Theory and Measurement, Princeton: Princeton University Press, ISBN 0-691-08316-9, LoC QC174.125.Q38 1983.
  1. ^ Science Show - 18 February 2006 - The anthropic universe
  2. ^ Frigg, R. GRW theory
  3. ^ Review of Penrose's Shadows of the Mind

[edit] Further reading

[edit] External links

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