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In a nontechnical sense, an interpretation of quantum of mechanics is an attempt to answer the question: what exactly is quantum mechanics talking about. That such a question is even asked is a reflection of a number of facts:
- The mathematical structure of the theory is based on fairly abstract mathematics such as Hilbert spaces and operators on Hilbert spaces. In classical mechanics, properties of a point mass in space such as position, or velocity are described by real numbers and there does not seem to be any need to provide a special interpretion for these values. A similar comment can made in regard to the understanding of electromagnetism.
- The process of measurement plays an essential role in the theory, relating 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 understanding of the theory's mathematical structures went through various preliminary stages of development. For instance Schrödinger at first did not understand the probabilistic nature of the wavefunction associated to the electron and it was Max Born who proposed its interpretation as the probability distribution in space of the electron's position. Other leading scientists, such as Einstein had enormous difficulty in coming to terms with the theory, which may have also lent importance to the activity of interpretation.
Quantum mechanics has been very successful in predicting experimental results, and the correspondence between the abstract mathematical formalism and the observed facts is not in question. However, the mathematical formalism used to describe the time evolution of a non-relativistic system proposes two somewhat different kinds of transformations:
- Reversible transformations described by unitary operators on the state space. These transformations are determined by solutions to the Schrödinger equation.
- Non-reversible transformations described by mathematically more complicated transformations (see quantum operation). Examples of these transformations are those that are undergone by a system as a result of measurement.
In a more restricted sense the problem of interpretation in quantum mechanics consists in providing some kind of picture for the second kind of transformation that has a less exceptional character. This problem can be addressed by purely mathematical reductions, for example by the many-worlds or the consistent histories interpretations. The precise ontological status of each one of the interpreting pictures, however, remains a matter of philosophical argument. In other words, if we interpret a structure X by means of a structure Y (via a mathematical equivalence of the two structures), does Y really exist? This is analogous to the problem of actualism in possible worlds semantics.
It should not be assumed that most physicists consider quantum mechanics as requiring interpretation, other than very minimal instrumentalist interpretations which are discussed below. While the Copenhagen interpretation still appears to be the most popular one among scientists (although it is closely followed by the many-worlds interpretation), it is also true that most physicists consider non-instrumental questions (in particular ontological questions) to be irrelevant to physics, falling back on Richard Feynman's famous dictum: "Shut up and calculate". Other physicists (including possibly Asher Peres and Chris Fuchs) seem to suggest that an interpretation is nothing more than a formal equivalence between sets of rules for operating on experimental data and that the whole exercise of interpretation is mostly unnecessary.
Instrumentalist interpretation
Any modern scientific theory requires at the very least an instrumentalist interpretation which relates the mathematical formalism to experimental practice. In the case of quantum mechanics, the most common instrumentalist interpretation is an assertion of statistical regularity between state preparation processes and measurement processes. This is usually glossed over into an assertion regarding the statistical regularity of a measurement performed on a system with a given state φ
Consider for example a measurement M of a physical observable with just two possible outcomes "up" or "down" that can be performed on a system S with Hilbert space H. If this measurement is carried out on a system whose quantum state is known to be φ ∈ H, then according to the rules of quantum mechanics, measurement will cause the system state to change in the following way: immediately after the measurement the system will be in of two states φdown if outcome is "down" or φup if outcome is "up". The mathematical theory gives expressions for these states as follows:
- <math> \varphi_{\operatorname{up}} = \operatorname{E}_{\operatorname{up}}(\varphi). <math>
- <math> \varphi_{\operatorname{down}} = \operatorname{E}_{\operatorname{down}}(\varphi). <math>
where Edown and Eup are orthogonal projections, which are spaces of eigenvectors of the observable. The numbers
- <math> \operatorname{P}_{\operatorname{up}} =\langle \varphi_{\operatorname{up}} \mid \varphi \rangle <math>
- <math> \operatorname{P}_{\operatorname{down}} =\langle \varphi_{\operatorname{down}} \mid \varphi \rangle <math>
have precise instrumentalist interpretations in terms of relative frequencies. That is that on an infinite run of trials of identical measurements (in all of which the system is prepared in state φ) the proportion of values with outcome "down" is Pdown and the proportion of values with outcome "up" is Pup.
Properties of interpretations
An interpretation can be characterized by whether it satisfies certain properties, such as:
To explain these properties we must say something more about the picture provided by an interpretation. To that end we will use a metaphor in which a picture is a correspondence between (a) the elements of the mathematical formalism (that is ket-vectors, unitary time dependence of states and measurement) and (b) elements of a state machine (that is states, transitions etc.). The state machine may be non-deterministic or probabilistic or may have infinitely many states, but its elements are physically real.
Completeness says how much of the picture is accounted for by the mathematical formalism and realism says how much of the formalism is accounted for in the picture. Determinism is a property characterizing state changes due to the passage of time; see time evolution. Local realism has two parts:
- The value returned by a measurement corresponds to the value of some function on the state space
- The effects of measurement have a propagation speed not exceeding some universal bound (e.g., the speed of light).
A precise formulation of local realism in terms of a theory of local hidden variables was proposed by John Bell.
Bell's theorem and its experimental verification restrict the kinds of properties a quantum theory can have. For instance, Bell's theorem implies quantum mechanics cannot satisfy local realism.
Different interpretations can be classified by the properties it accepts and rejects.
At the moment, there is no experimental evidence that would allow us to distinguish between the various interpretations listed below. 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.
The measurement problem is the main problem that the different interpretations answer differently. Some of the most common interpretations are summarized here:
1Many Worlds has no hidden variables, except for the multiple worlds themselves.
2Bohm/de Broglie say both particle AND wavefunction ("guide wave") are real.
Each interpretation has many variants. It is difficult to get a precise definition of the Copenhagen Interpretation — in the table above, two classical variants and one new version of the Copenhagen Interpretation is 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".
See also
List of physics topics:
References
- Herbert, Nick. Quantum Reality: Beyond the New Physics, New York: Doubleday, ISBN 0385235690, LoC QC174.12.H47 1985 This is the most accessible reference.
- Jammer, Max. The Conceptual Development of Quantum Mechanics. New York: McGraw-Hill, 1966.
- Jammer, Max. The Philosophy of Quantum Mechanics. New York: Wiley, 1974.
- de Muynck, Willem M. Foundations of quantum mechanics, an empiricist approach, Dordrecht: Kluwer Academic Publishers, 2002, ISBN 1-4020-0932-1
- Reichenbach, Hans. Philosophic Foundations of Quantum Mechanics, Berkeley: University of California Press, 1944.
- Wheeler, John Archibald and Wojciech Hubert Zurek. (eds)., Quantum Theory and Measurement, Princeton: Princeton University Press, ISBN 0691083169, LoC QC174.125.Q38 1983
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