Abstract
This second chapter is the first interpretative one. Step by step it proceeds from the minimal- and the ensemble-interpretations to the (in) famous Copenhagen interpretation. Finally, the (non-standard) realist collapse view (GRW) will be presented.
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Notes
- 1.
- 2.
In fact, this holds for many-particle systems only with certain reservations.
- 3.
As already mentioned in Sect. 1.2.4, the measurement probability is identical to the expectation value of the corresponding projection operator. We likewise saw there how this can be generalized to mixed states.
- 4.
For a discussion, see the historical articles, collected in Baumann and Sexl (1984).
- 5.
That is, in a repeatable and irreversible manner?
- 6.
However, one should note that in Bohmian mechanics, only the positions of the particles are introduced as additional properties, and not, for example, local spin values for the particles, as one might think at this point.
- 7.
namely within the potential V(x) in \(\hat{H} = -(\frac{\hbar ^{2}}{2m})\frac{\partial ^{2}}{\partial x^{2}} + V(x)\)
- 8.
According to the ensemble interpretation, the state vector \(|\Psi \rangle \) describes a large number of similarly prepared systems. Independently of this, and distinct from it, is the question of whether \(|\Psi \rangle \) describes single- or many-particle systems. The version of the ensemble interpretation which has been most thoroughly worked out can be found in ((Ballentine 1998), Chap. 9).
- 9.
Caution: This contrast to Bohm’s theory exists only when one holds quantum mechanics to be complete and at the same time does not wish to apply it to particular systems. In a certain sense, every type of Bohm’s theory is an ensemble interpretation in its statistical part; we are, however, not referring to this when we speak of the “ensemble interpretation” here.
- 10.
See, e.g., Faye (2008).
- 11.
For this theory, see the section below on the “measurement problem”.
- 12.
“For the measurement setup is worthy of that name only when it is in intimate contact with the rest of the world, when there is a physical interaction between the measurement setup and the observer” ((Heisenberg 1959), p. 41).
- 13.
Caution in the case of multiple eigenvalues, where (possibly several) other measurable quantities must first be determined.
- 14.
While that “which happens in an atomic process”, namely the change of eigenvalues as properties, is supposed to be physical, for the state vector it is asserted that “the discontinuous change in the probability function occurs to be sure through the act of observation; for here, we are dealing with the discontinuous change of our knowledge at the moment of observation” ((Heisenberg 1959), p. 38). \(|\Psi \rangle \) is thus construed epistemologically.
- 15.
Take note, however, of Maudlin’s Lorentz-invariant realistic collapse interpretation, according to which the wavefunctions are hyperplane-dependent in four-dimensional spacetime; cf. ((Maudlin 1994), Chap. 7).
- 16.
Permitting ourselves a preview of the GRW theory, we note here that in that theory, Born’s rule does not hold: In that theory, a (modified) quantum mechanics is complete, and there is “nevertheless” no measurement process—which is excluded with Born’s rule.
- 17.
- 18.
Within the formalism of quantum mechanics, one can even demonstrate mathematically that under the assumption that a (real) collapse takes place, one can in principle not determine when it occurs, and thus where the (objective) cut must lie; see (Albert (1992), p. 91).
- 19.
Such formulations often produce abstruse misunderstandings: It is naturally not meant here that the “subject-independent”, objective world is in a superposition so long and to such an extent (that is as a whole) until a transcendental ego “views” it from “outside”.
- 20.
Everett’s many-worlds interpretation is to be sure also popular; it lies mathematically very close to the standard formalism and is, in contrast to (Heisenberg’s) Copenhagen interpretation, realistic. For Everett’s theory, see Sect. 5.2.
- 21.
That the eigenvectors—in contrast to the eigenvalues—are not given a reality status is also not really a defect of this version of the Copenhagen interpretation, since in mathematical theories, such as in particular the Hilbert space formalism, there are many more symbols and operations which have been interpreted as at most instrumental, without anyone complaining.
- 22.
Especially since the question of which properties the quantum-physical system acquires in a concrete case does not depend on the observer.
- 23.
Which in turn is a common feature with Bohr’s Copenhagen interpretation, in the sense that Bohr avoids a description of the (physical) measurement process in terms of mathematical physics.
- 24.
There is still another interpretation which also refutes assertion 1, but without assuming the existence of hidden variables. It is the modal interpretation of quantum mechanics (see (van Fraassen 1991), Chap. 9). However, its main problem—namely how to explain why repeated measurements (e.g. of the spin) lead with certainty again to the same result, although there has been no collapse nor have hidden variables guaranteed the result—has not been solved in a convincing way.
- 25.
Note that in this description of the measurement problem, we have not referred again to Born’s rule. In fact, this probability interpretation of the state vectors holds neither in GRW theory nor in Everett’s theory.
- 26.
Making this assumption means that we cannot balk here and like Bohr asserts dogmatically that macroscopic measurement apparatus is in any case only classically describable.
- 27.
Apart from the fact that real measurement apparatus also has a nonzero error rate.
- 28.
Caution: Tacitly, we have introduced here mathematically a product between vectors in different Hilbert spaces, which we had previously not discussed. In fact, the measurement apparatus and the quantum object form a composite system, which will be treated in detail only in the following chapter. Such a (pure) product, such as \(|M_{-1}\rangle |down\rangle \), in any case reflects rather classically a whole whose properties are completely determined by its parts. One says that the state of the whole is separable, in that here the particle (quantum object) and the measurement apparatus each individually indicate \(+1\) (the fact that they in the end always point in the same direction is a result of their interaction).
- 29.
In the ideal case! There are of course measurements which inevitably destroy the quantum object, even when it was already in an eigenstate of the corresponding observable.
- 30.
Otherwise, the measurement apparatus from the outset cannot accomplish what it is supposed to.
- 31.
The overall system is in no case in a state in which the measurement apparatus would indicate both 1 and \(-1\), which would be contradictory.
- 32.
The superposition is itself instead the eigenvector of a different, incommensurable operator, analogously to the situation in two-dimensional spin space, even though here— in the macroscopic world—it is not so simple to specify the operator explicitly.
- 33.
A circumstance which Bohr avoided explaining right from the start.
- 34.
More details will follow in later chapters.
- 35.
The corresponding projection operator would not be in diagonal form: \(\hat{P}_{final} = (c_{1}|up\rangle + c_{2}|down\rangle )(c_{2}\langle down| + c_{1}\langle up|) = |c_{1}|^{2}|up\rangle \langle up| + |c_{2}|^{2}|down\rangle \langle down| + c_{1}{c_{2}}^{*}|up\rangle \langle down| + {c_{1}}^{*}c_{2}|down\rangle \langle up|\). This has the effect that the expectation values of operators, \(\langle \Psi _{final}|\hat{O}|\Psi _{final}\rangle = \text {Tr}(\hat{P}_{final} \hat{O})\), in general contain interference terms (cross terms, which exhibit intuitively the wave character of particles, as for example in the two-slit experiment). These are lacking in \({\langle \hat{O}\rangle }_{\hat{\rho }} = \text {Tr}(\hat{\rho }_{quant}\hat{O})\).
- 36.
The original (superposition) vector was, like every vector of a pure state, indeed an eigenvector of a different (incommensurable) operator. Referring to the statistical operator that we have now obtained, the expectation value of every operator, however, is no longer the (former) eigenvalue and no longer free of variance.
- 37.
“Coherent”, i.e. “interrelated”, is the classical term for the condition which must be fulfilled by waves in order that they exhibit interference. Thus, one can also refer to quantum-mechanical superpositions as “coherent states”. The decoherence approach then attempts to clarify the conditions under which a classical world, in which precisely such superpositions no longer occur, can emerge on the basis of quantum mechanics.
- 38.
- 39.
The significance of decoherence theory for Everett’s interpretation is discussed in Sect. 5.2.4.
- 40.
An initial state can thus have two possible final states: Going in one direction of time, towards the future(?), the state can split apart.
- 41.
How does distinguishing this particular basis relate to the decoherence programme, one of whose steps forward consists in the fact that indeed one basis is dynamically distinguished? There is at least some tension between “decoherence” and GRW when it turns out that the special basis in certain situations is not the position representation. Concerning this problem, see (Schlosshauer 2007, pp. 349f.).
- 42.
It is perhaps even the case that GRW makes some predictions which deviate from those of standard quantum mechanics, so that future experiments might “prove” that GRW is also empirically more realistic—but they could also falsify the GRW theory!
- 43.
This is a common feature with Bohm’s mechanics (which, however, is deterministic).
- 44.
The fundamental GRW equation is better represented by referring to the density matrix; compare ((Frigg and Hoefer 2007), p. 374).
- 45.
In this approach, a continuous “material” field in concrete spacetime corresponds to the wavefunction in configuration space. It is supported by Ghirardi himself, among others.
- 46.
Physical difficulties of the GRW theory relate to the “indistinguishability” of similar particles, the treatment of counterfactual dependencies in the EPR problem, and in particular its compatibility with the theory of (special) relativity. This last topic is also problematic for Bohm’s mechanics.
- 47.
In the literature, a so-called counting anomaly has also been discussed, which, however, we need not consider here.
- 48.
A Hume interpretation of GRW is defended by Frigg and Hoefer (2007).
References
Albert, David Z. 1992. Quantum Mechanics and Experience. Cambridge: Harvard University Press.
Audretsch, Jürgen (ed.). 2002. Verschränkte Welt. Faszination der Quanten. Weinheim: Wiley-VCH.
Baumann, Kurt, and Roman U. Sexl. 1984. Die Interpretationen der Quantentheorie. Braunschweig: Vieweg.
Ballentine, Leslie E. 1998. Quantum Mechanics: A Modern Development. Singapore: World Scientific.
Bell, John S. 1987. Are there quantum jumps?. In: Bell. In Speakable and Unspeakable in Quantum Mechanics (Revised edition 2004), ed. Speakable Bell, 201–212. Cambridge: Cambridge University Press.
Dorato, Mauro, and Michael Esfeld. 2010. GRW as an ontology of dispositions. Studies in History and Philosophy of Modern Physics 41: 41–49.
Elby, Andrew, and Jeffrey Bub. 1994. Triorthogonal uniqueness theorem and its relevance to the interpretation of quantum mechanics. Physical Review A 49 (5): 4213.
Esfeld, Michael (2012). Das Messproblem der Quantenmechanik heute: Übersicht und Bewertung. In Esfeld (ed.). Philosophie der Physik, 88–109. Frankfurt/M.: Suhrkamp.
Faye, Jan (2008). Copenhagen Interpretation of Quantum Mechanics. In E.N. Zalta (ed.). The Stanford Encyclopedia of Philosophy (Fall 2008 edition). http://plato.stanford.edu/archives/fall2008/entries/qm-copenhagen/
van Fraassen, Bas C. 1991. Quantum Mechanics: An Empiricist View.Oxford: Clarendon Press.
Friederich, Simon. 2011. How to spell out the epistemic conception of quantum states. Studies in History and Philosophy of Modern Physics 42: 149–157.
Frigg, Roman, and Carl Hoefer. 2007. Probability in GRW theory. Studies in History and Philosophy of Modern Physics 38B: 371–389.
Fuchs, Christopher A., and Asher Peres. 2000. Quantum theory needs no interpretation. Physics Today 53 (3): 70–71.
Ghirardi, GianCarlo, Alberto Rimini, and Tullio Weber. 1986. Unified dynamics for microscopic and macroscopic systems. Physical Review D 34: 470–491.
Heisenberg, Werner. 1959. Die Kopenhagener Interpretation der Quantentheorie. In Physik und Philosophie, ed. Heisenberg, 27–42. Stuttgart: Hirzel.
Held, Carsten. 2012. Die Struktur der Quantenmechanik. In Philosophie der Physik, ed. M. Esfeld, 71–87. Frankfurt/M.: Suhrkamp.
Maudlin, Tim. 1994. Quantum Non-Locality and Relativity. Oxford: Blackwell.
Maudlin, Tim. 1995. Three measurement problems. Topoi 14: 7–15.
Maudlin, Tim. 2010. Can the world be only wavefunction? In Many Worlds? Everett, Quantum Theory, and Reality, ed. S. Saunders, et al., 121–143. Oxford: Oxford University Press.
Monton, Bradley. 2006. Quantum mechanics and \(3N\)-dimensional space. Philosophy of Science 73: 778–789.
von Neumann, John. 1932. Mathematische Grundlagen der Quantenmechanik. Berlin: Springer. (Reprint 1968).
Rosenthal, Jacob (2003). Wahrscheinlichkeiten als Tendenzen. Eine Untersuchung objectiver Probabilitätssbegriffe. Paderborn: mentis.
Schlosshauer, Maximilian. 2007. Decoherence and the Quantum-to-Classical Transition. Berlin/Heidelberg: Springer.
Stöckler, Manfred (2007). Philosophische Probleme der Quantentheorie. In: A. Bartels and M. Stöckler (eds.). Wissenschaftstheorie. Ein Studienbuch, 245–264. Paderborn: mentis.
Zeh, Heinz-Dieter. 1970. On the interpretation of measurement in quantum theory. Foundations of Physics 1: 69–76.
Zurek, Wojciech H. 1981. Pointer basis of quantum apparatus: Into what mixture does the wave packet collapse? Physical Review D 24: 1516–1525.
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Exercises
Exercises
- 1. :
-
Distinguish between two readings of Born’s rule, depending upon whether the reference to a measurement in it is essential or not.
- 2. :
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According to the Copenhagen interpretation (in Heisenberg’s version), there are two temporal dynamics of the state vector. Describe them in your own words. How is the second dynamics related to Born’s rule and to von Neumann’s projection postulate? What is problematic about it?
- 3. :
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The interpretation problem in quantum mechanics may be regarded as a trilemma. Explain the three statements and show that they are inconsistent when taken together. What is the advantage of this description as compared to the conventional one, which is guided by Born’s rule?
- 4. :
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The decoherence programme is an essential step forward. Highlight the ways in which all the interpretation variants could profit from it. Why, however, can the programme not solve the measurement problem in the end?
- 5. :
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Formulate in your own words what is accomplished by the GRW theory (in the opinion of its supporters). Defend the standard view of physics against it.
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Friebe, C. (2018). The Measurement Problem. Minimal and Collapse Interpretations. In: The Philosophy of Quantum Physics. Springer, Cham. https://doi.org/10.1007/978-3-319-78356-7_2
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