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Interpretation of quantum mechanics

From Academic Kids

An interpretation of quantum mechanics is an attempt to answer the question: what exactly is quantum mechanics talking about? Quantum mechanics, as a scientific theory, has been very successful in predicting experimental results. The close correspondence between the (abstract, mathematical) formalism and the observed facts is not generally in question. That such a basic question is still posed in itself requires some explanation.

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; it was Max Born who proposed its interpretation as the probability distribution in space of the electron's position. Other leading scientists, such as Albert Einstein, had great difficulty in coming to terms with the theory. Even if these matters could be treated as 'teething troubles', they have lent importance to the activity of interpretation.

It should not, however, be assumed that most physicists consider quantum mechanics as requiring interpretation, other than very minimal instrumentalist interpretations, which are discussed below. The Copenhagen interpretation, as of 2005, still appears to be the most popular one among scientists (followed by the consistent histories and many-worlds 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 Paul Dirac's point of view, later expressed in the famous dictum: "Shut up and calculate" often (perhaps erroneously) attributed to Richard Feynman (see [1] (http://www.physicstoday.org/vol-57/iss-5/p10.html)).

Contents

Obstructions to direct interpretation

The perceived difficulties of interpretation reflect a number of points about the orthodox description of quantum mechanics.

Firstly, 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 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 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.

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 non-deterministic 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.

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.

Instrumentalist interpretation

Any modern scientific theory requires at the very least an instrumentalist description which relates the mathematical formalism to experimental practice. In the case of quantum mechanics, the most common instrumentalist description 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 one 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 onto 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 descriptions 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.

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.

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 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 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.

Comparison

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.

Some of the most common interpretations are summarized here:

Interpretation Deterministic? Waveform real? One Universe? Avoids
hidden variables?
Avoids
collapsing wavefunctions?
Consistent histories
(Copenhagen "done right")
No No Yes Yes Yes
Copenhagen interpretation
(Waveform not real)
No No Yes Yes Yes
Copenhagen interpretation
(Waveform real)
No Yes Yes Yes No
Transactional interpretation Yes Yes Yes Yes No
Consciousness causes collapse No Yes Yes Yes No
Everett many-worlds interpretation Yes Yes No Yes1 Yes
Bohm interpretation Yes Yes2 Yes No Yes

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

  • R. Carnap, The interpretation of physics, Foundations of Logic and Mathematics of the International Encyclopedia of Unified Science, Univesity 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.
  • 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.
  • N. Herbert. Quantum Reality: Beyond the New Physics, New York: Doubleday, ISBN 0385235690, LoC QC174.12.H47 1985.
  • M. Jammer, The Conceptual Development of Quantum Mechanics. New York: McGraw-Hill, 1966.
  • M. Jammer, The Philosophy of Quantum Mechanics. New York: Wiley, 1974.
  • 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.
  • H. Reichenbach, Philosophic Foundations of Quantum Mechanics, Berkeley: University of California Press, 1944.
  • J. A. Wheeler and H. Z. Wojciech (eds), Quantum Theory and Measurement, Princeton: Princeton University Press, ISBN 0691083169, LoC QC174.125.Q38 1983

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