Abstract
This paper aims to clarify some conceptual aspects of decoherence that seem largely overlooked in the recent literature. In particular, I want to stress that decoherence theory, in the standard framework, is rather silent with respect to the description of (sub)systems and associated dynamics. Also, the selection of position basis for classical objects is more problematic than usually thought: while, on the one hand, decoherence offers a pragmatic-oriented solution to this problem, on the other hand, this can hardly be seen as a genuine ontological explanation of why the classical world is position-based. This is not to say that decoherence is not useful to the foundations of quantum mechanics; on the contrary, it is a formidable weapon, as it accounts for a realistic description of quantum systems. That powerful description, however, becomes manifest when decoherence theory itself is interpreted in a realist framework of quantum mechanics.
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Notes
- 1.
- 2.
The “environment” can be generally thought of as external or internal degrees of freedom with respect to the degrees of freedom representing our system of interest. Spatial degrees of freedom (position coordinates) may be, for example, “the environment” for spinor degrees of freedom of a spin ½ particle (spin-up, spin-down).
- 3.
Technical note: the coefficients of the entangled state superposition will generally be different from those of the initial S state superposition. However, this difference will not be relevant for the present discussion.
- 4.
Strictly speaking, this is a density operator, while the density matrix is the density operator expressed in a particular basis (generally the position basis). However, as this difference will not be relevant, I will just use the term density matrix in both cases.
- 5.
The term ε in the off-diagonal components of the matrix stands for “negligible quantity”: as the diagonalization process is mathematically described by a decreasing (quadratic) exponential, it will reach the zero value only asymptotically.
- 6.
While this is a natural interpretation of the reduced density matrix, it could be interesting to ask whether a reduced density matrix may have a more general significance independently from measurement interactions.
- 7.
As the dynamics of ρ S(x, x′,t) is linear, it cannot eliminate any state of the superposition (see, e.g., Adler (2003)).
- 8.
Note that the system-environment interactions in decoherence theory are measurement-like interactions, as there is a coupling between the system’s and environment’s degrees of freedom and the environment relative states get correlated with the system relative states. However, these interactions do not produce collapse of the wave function, as the master equations are linear and do not select one particular component. In order to have collapse of the wave function, the system has to interact—by definition–with a macroscopic measurement device.
- 9.
See e.g. Crull (2015).
- 10.
It is worth noting, however, that in some recent (more refined) interpretations of the quantum formalism this problem does not arise or could not arise, for example: Rovelli’s account of decoherence in relational quantum mechanics (Di Biagio & Rovelli, 2021); Myrvold’s ontology of quantum states in terms of distribution of values of dynamical variables (Myrvold, 2018); Chen’s density matrix realism (Chen, 2018). In all of these interpretations, the problem discussed above requires a careful and distinct analysis, which will be developed in an extended version of the present paper.
- 11.
For a presentation of the different role of decoherence in the realist interpretations of quantum mechanics (MWI, GRW and Bohm’s theory) see also Bacciagaluppi (2020).
- 12.
For a numerical estimate of the increase of the GRW collapse rate due to decoherence see, for example, M. Toroš, S. Donadi and A. Bassi (2016, pp. 10–11).
- 13.
But instantaneously well-localized for relevant timescales at the macroscopic regime and perfectly well-localized with respect to macroscopic localization.
- 14.
Separate components stand here for “components whose overlap is negligible in configuration space”, since the condition of no-overlap is not realistic: tails of Gaussian environmental particles will overlap in any region of space.
- 15.
Even though there is no rigorous formulation, this is not so different from the standard decoherence condition of “orthogonality” of states, that in most cases is reached approximately and asymptotically. For an analysis of the decoherence condition in dBB theory, see Romano (2016a). For a comparative analysis of the decoherence condition in dBB theory and MWI, see Rosaler (2015, 2016).
- 16.
- 17.
Thanks to Craig Callender and one anonymous referee for helping me elaborating this remark.
- 18.
Or Lagrangian mechanics, or Hamiltonian mechanics. However, as the ontology of classical systems (system dynamics and interaction between systems) is generally built on Newtonian mechanics, I will consider the quantum to classical dynamics transition as the transition from quantum dynamics to Newtonian mechanics.
- 19.
I consider here only the gravitational potential since, strictly speaking, there is no electromagnetic field in non-relativistic quantum mechanics. Classical electrodynamics should emerge from quantum electrodynamics, i.e. a different mathematical and physical framework. It is true that electromagnetic interactions are described also in non-relativistic quantum mechanics, e.g. the textbook presentation of the proton-electron interaction in a Hydrogen atom. However, this kind of analysis is rather phenomenological and relies on semi-classical assumption (particles described by wave functions interacting through classical electromagnetic forces). On the other hand, there is no gravitational interaction either in non-relativistic quantum mechanics, as a real quantization of the gravitational field is only done in quantum gravity. Yes, any realistic description in non-relativistic quantum mechanics is trickier than usually thought.
- 20.
We want to derive Newtonian trajectories approximately and not exactly, as any deviation of the order e.g. of the atomic scale cannot be detected at the macroscopic scale.
- 21.
- 22.
As these are notions we are already familiar from the basic course in quantum mechanics, I will not enter here in the detail of the difference between a statistical quantity and a quantity describing an individual system. For the interested reader, however, I have analyzed this issue more carefully in another paper (Sakurai, sect. 3).
- 23.
Following Zurek’s terminology, they are generally called pointer states.
- 24.
I take this specific example as the Brownian model is one of the principal models for the quantum to classical transition. Other important models, such as the collisional model, would be perfectly equivalent for the example.
- 25.
See e.g. Schlosshauer (2019, Sect. 4.2).
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Acknowledgments
I want to thank Valia Allori, Mario Hubert, Vera Matarese and Antonio Vassallo for helpful comments on earlier drafts on this paper and Craig Callender, Eddy Chen, Barry Lower and Kerry McKenzie for a nice and useful discussion of the paper in the San Diego philosophy of physics reading group. This work has been supported by the Fundação para a Ciência e a Tecnologia through the fellowship FCT Junior Researcher, hosted by the Centre of Philosophy at the University of Lisbon.
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Romano, D. (2022). The Unreasonable Effectiveness of Decoherence. In: Allori, V. (eds) Quantum Mechanics and Fundamentality . Synthese Library, vol 460. Springer, Cham. https://doi.org/10.1007/978-3-030-99642-0_1
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