## Abstract

The difficult issues related to the interpretation of quantum mechanics and, in particular, the “measurement problem” are revisited using as motivation the process of generation of structure from quantum fluctuations in inflationary cosmology. The unessential mathematical complexity of the particular problem is bypassed, facilitating the discussion of the conceptual issues, by considering, within the paradigm set up by the cosmological problem, another problem where symmetry serves as a focal point: a simplified version of Mott’s problem.

This is a preview of subscription content, access via your institution.

## Notes

We are ignoring the fact that certain interpretations are problematic.The point however is that to the extent that they are applied in a particular manner in concrete situations they do not offer predictions that differ from the text book version of Quantum Theory.

The favored version of the theory actually deals with a composite variable representing the quantum aspects of the inflaton field and a certain component of the space-time metric [6].

We refer here to the stage corresponding to several e-folds after the start of inflation, when the background corresponds to an inflating, flat, Robertson Walker space-time, and the “quantum fluctuations” are described by the Bunch-Davies vacuum, or some similarly highly symmetric state. This characterization is thought to be accurate up to exponentially small corrections in the number of e-folds, a detail that we will ignore as is customary in all inflationary analyses.

There are, apparently, some people who disagree with this view, but we will not consider their thinking any further here.

It even seems possible to construct wave packets with high

*n*in an hydrogen atom that resemble to some degree the situation above.There are apparently philosophical views inspired in Kantian ontology where this statement could be questioned.

In fact in order to do that one would need not only to define the privileged basis but also to add a postulate about actualization.

Except, of course the zero modo of the inflaton.

This point is sometimes characterized as the “transition from the quantum regime to the classical regime”, but we find this a bit misleading: most people would agree that there are no classical or quantum regimes. The fundamental description ought to be always a quantum description. However, there exist regimes in which certain quantities can be described to a sufficient accuracy by their classical counterparts represented by the corresponding expectation values. All this depends, of course, on the physical state, the underlying dynamics, the quantity of interest, and the context

**which one is considering**.On the other hand, it is worth noting that the Hamiltonian of interaction between particle and detector has a explicitly local form in the first basis but not in the second. This might be used but it would have to be explicitly formulated as part of the theory. Spontaneous localization theories, and de-Broglie Bohm approaches, for instance focus on position as playing a preferential role.

Let \(\hat{S}\) be a symmetry operator and \(|\Psi (0)\rangle \) an initial symmetric state, i.e. \(\hat{S} |\Psi (0)\rangle =|\Psi (0)\rangle \). Let \(\hat{H}(t)\) be the system’s hamiltonian, taken to be invariant under the symmetry i.e. \([\hat{H} (t),\hat{S}]=0\). Then \(\hat{S}|\Psi (t)\rangle =\hat{S}e^{i\int _{0} ^{t}H(t^{\prime })dt^{\prime }}|\Psi (0)\rangle =e^{i\int _{0}^{t}H(t^{\prime })dt^{\prime }}\hat{S}|\Psi (0)\rangle =e^{i\int _{0}^{t}H(t^{\prime })dt^{\prime } }|\Psi (0)\rangle =|\Psi (t)\rangle \) i.e. the evolved state is also symmetric.

According to [92] the posture is that one should believe both, and use the appropriate one in connection with the questions one is asking. This posture is not shared by other authors, for instance [87]. Moreover it seems the implicit views regarding the nature of science are very problematic in general (see for instance [93, 96,97,98,99]).

There exists many variants of the major themes we have considered here, and they have not been described in detail because the differences have no bearing on the issue at hand. Namely these variants fail to address the issue we face, for exactly the same reasons as the major ones they are

**closely connected**with. However, we acknowledge that there might exist some other proposal we are unaware of, and which fare better in dealing with the problem we have been considering in this work.With the possible exception of the d’ Broglie- Bohm approach, where the source of the primordial asymmetries is found in the initial conditions.

We are ignoring here the issue of how this decay became actualized into that particular direction, as the point here is to exemplify a specific technical issue.

## References

De Witt, B.S.: Quantum theory of gravity. I. The canonical theory. Phys. Rev.

**160**, 1113 (1967)Wheeler, J.A.: In: De Witt, C., Wheeler, J.A. (eds.) Battelle Reencontres 1987. Benjamin, New York (1968)

Isham, C.J.: Canonical Quantum Gravity and the Problem of Time, GIFT Semminar-0157228 (1992) qr-qc/9210011

See for instance Isham, J.: (1992) gr-qc/9210011

Guth, A.: Inflationary universe: a possible solution to the horizon and flatness problems. Phys. Rev. D

**23**, 347 (1981). For a more exhaustive discussion see for instance the relevant chapter in The Early Universe, E.W. Kolb and M.S. Turner, Frontiers in Physics Lecture Note Series (Addison Wesley Publishing Company 1990)Muckhanov, V.: Physical Foundations of Cosmology, p. 348. Cambridge University Press, Cambridge (2005)

Halliwell, J.J.: Decoherence in quantum cosmology. Phys. Rev. D

**39**, 2912 (1989)Kiefer, C.: Origin of classical structure from inflation. Nucl. Phys. Proc. Suppl.

**88**, 255 (2000). arXiv:astro-ph/0006252Polarski, D., Starobinsky, A.A.: Semiclassicality and decoherence of cosmological perturbations. Class. Quant. Grav.

**13**, 377 (1996) arXiv: gr-qc/9504030Zurek, W.H.: Environment induced superselection in cosmology. In: Cosmology in Moscow 1990, Proceedings, Quantum gravity (QC178:S4:1990), pp. 456–472. (see High Energy Physics Index 30 (1992) No. 624)

Branderberger, R., Feldman, H., Mukhavov, V.: Gauge invariant cosmological perturbations. Phys. Rep.

**215**, 203 (1992)Laflamme, R., Matacz, A.: Decoherence functional and inhomogeneities in the early universe. Int. J. Mod. Phys. D

**2**, 171 (1993) arXiv:gr-qc/9303036Castagnino, M., Lombardi, O.: The self-induced approach to decoherence in cosmology. Int. J. Theory Phys.

**42**, 1281 (2003). arXiv:quant-ph/0211163Lombardo, F.C., Lopez Nacir, D.: Decoherence during inflation: The generation of classical inhomogeneities, Phys. Rev. D

**72**, 063506 (2005). arXiv:gr-qc/0506051Martin, J.: Inflationary Cosmological Perturbations of Quantum Mechanical Origin. Lecture Notes in Physics, vol. 669, 199 (2005). arXiv:hep-th/0406011

Grishchuk, L.P., Martin, J.: Best unbiased estimates for microwave background anisotropies. Phys. Rev. D

**56**, 1924 (1997). arXiv:gr-qc/9702018Barvinsky, A.O., Kamenshchik, A.Y., Kiefer, C., Mishakov, I.V.: Decoherence in quantum cosmology at the onset of inflation. Nucl. Phys. B

**551**, 374 (1999). arXiv:gr-qc/9812043Padmanabhan, T.: Structure Formation in the Universe, p. 364. Cambridge University Press, Cambridge (1993). Section 10.4

Boucher, W., Traschen, J.: Semiclassical physics and quantum fluctuations. Phys. Rev. D

**37**, 3522–3532 (1988)Weinberg, S.: Cosmology, p. 476. Oxford University Press, Oxford (2008)

Mott, N.F.: The wave mechanics of \(\alpha \)-ray tracks. Proc. R. Soc. Lond.

**126**(800), 79 (1929)Paz, J.P., Zurek, W.H.: Environment-induced decoherence and the transition from quantum to classical. In: Heiss, D. (ed.) Lecture Notes in Physics, vol. 587. Springer, Berlin (2002)

Schlosshauer, M.: Decoherence and the Quantum-to-Classical Transition. Springer, Berlin (2007)

Joos, E., et al.: Decoherence and the Appearance of a Classical World in Quantum Theory. Springer, Berlin (2003)

Castagnino, M., Fortin, S.: Predicting decoherence in discrete models. Int. J. Theory Phys.

**50**, 2259–2267 (2011)Castagnino, M., Fortin, S., Lombardi, O.: Is the decoherence of a system the result of its interaction with the environment? Mod. Phys. Lett. A

**25**, 1431–1439 (2010)Butterfield, J., Earman, J. (eds.): Philosophy of Physics, Handbook of the Philosophy of Science. North-Holland Elsevier, Amsterdam (2007)

Harrison, E.R.: Fluctuations at the threshold of classical cosmology. Phys. Rev. D

**1**, 2726 (1970)Zel’dovich, Y.B.: A hypotesis, unifying the structure and the entropy of the universe. Mon. Not. R. Astron. Soc.

**160**, 1 (1972)Lange, A.E., et al.: Cosmological parameters from first results of Boomerang. Phys. Rev. D

**63**, 042001 (2001)Hinshaw, G., et al.: Astrophys. J. Supp.

**148**, 135 (2003)Gorski, K.M., et al.: Power spectrum of primordial inhomogeneity determined from four year COBE DMR SKY Maps. Astrophys. J.

**464**, L11 (1996)Bennett, C.L., et al.: First year wilkinson microwave anisotropy probe (WMAP) observations: preliminary results. Astrophys. J. Suppl.

**148**, 1 (2003)Bennett, C., et al.: First year Wilkinson microwave anisotropy probe (WMAP) observations: foreground emission. Astrophys. J. Suppl.

**148**, 97 (2003)Hinshaw, G. et al.: [WMAP Collaboration], Nine-year Wilkinson microwave anisotropy probe (WMAP) observations: cosmological parameter results. arXiv:1212.5226 [astro-ph.CO]

Larson, D., Dunkley, J., Hinshaw, G., Komatsu, G., Nolta, M.R., Bennett, C.L., Gold, B., Halpern, M., et al.: Seven-year Wilkinson microwave anisotropy probe (WMAP) observations: power spectra and WMAP-derived parameters. Astrophys. J. Suppl.

**192**, 16 (2011). arXiv:1001.4635 [astro-ph.CO]Ade, P.A.R.: (Planck collaboration), Planck 2013 results. XV. CMB powerspectra and likelihood (2013). arXiv:1303.5075 [astro-ph.CO]

Perez, A., Sahlmman, H., Sudarsky, D.: On the quantum mechanical origin of the seeds of cosmic structure. Class. Quantum Gravity

**23**, 2317 (2006)Diez-Tejedor, A., Sudarsky, D.: Towards a formal description of the collapse approach to the inflationary origin of the seeds of cosmic structure. JCAP

**045**, 1207 (2012). arXiv:1108.4928 [gr-qc]de Unanue, A., Sudarsky, D.: Phenomenological analysis of quantum collapse as source of the seeds of cosmic structure. Phys. Rev. D

**78**, 043510 (2008). arXiv:0801.4702 [gr-qc]León García, G., Sudarsky, D.: The slow roll condition and the amplitude of the primordial spectrum of cosmic fluctuations: contrasts and similarities of standard account and the “collapse scheme. Class. Quantum Gravity

**27**, 225017 (2010)León García, G., De Unanue, A. , Sudarsky, D.: Multiple quantum collapse of the inflaton field and its implications on the birth of cosmic structure. Class. Quantum Gravity,

**28**, 155010 (2011). arXiv:1012.2419 [gr-qc]León García, G., Sudarsky, D.: Novel possibility of observable non-Gaussianities in the inflationary spectrum of primordial inhomogeneities. Sigma

**8**, 024 (2012)Diez-Tejedor, A., León García, G., Sudarsky, D.: The collapse of the wave function in the joint metric-matter quantization for inflation. Gen. Relativ. Gravity

**44**, 2965, (2012). arXiv:1106.1176 [gr-qc]Landau, S.J., Scoccola, C.G., Sudarsky, D.: Cosmological constraints on nonstandard inflationary quantum collapse models. Phys. Rev. D

**85**, 123001 (2012). arXiv:1112.1830 [astro-ph.CO]Scully, M.O., Shea, R., Mc Cullen, J.D.: State reduction oin quantum mechanics. A calculational example. Phys. Rep.

**43**, 485–498 (1978)Zurek, W.H.: Environment-induced superselection rules. Phys. Rev. D

**26**, 1862–1880 (1982)Zurek, W.A.: Pointer basis of quantum apparatus: into what mixture does the wave packet collapse? Phys. Rev. D

**24**, 1516–1525 (1981)Barbour, J.B.: The timelessness of quantum gravity: I. The evidence from the classical theory. Class. Quantum Gravity

**11**, 2853–2873 (1994)Barbour, J.B.: The timelessness of quantum gravity: II. The apperearance of dynamics in statics configurations. Class. Quantum Gravity

**11**, 2853–2873 (1994)Earman, J.: World Enough and Space-Time. MIT Press, Cambridge, MA (1996)

Cohen, D.W.: An Introduction to Hilbert Space and Quantum Logic. Springer, London (2011)

Holik, F., Massri, C., Ciancaglini, N.: Convex quantum logic. Int. J. Theor. Phys.

**51**, 1600–1620 (2012)Holik, F., Massri, C., Plastino, A., Zuberman, L.: On the lattice structure of probability spaces in quantum mechanics. Int. J. Theory Phys.

**52**, 1836–1876 (2013)Faye, J.: Copenhagen interpretation of quantum mechanics. In: Zalta, E.N. (ed.) The Stanford Encyclopedia of Philosophy (Fall 2008 Edition). http://plato.stanford.edu/archives/fall2008/entries/qm-copenhagen/

Lombardi, O., Dieks, D.: Modal interpretations of quantum mechanics. In: Zalta, E.N. (ed.) The Stanford Encyclopedia of Philosophy (Winter 2012 Edition). http://plato.stanford.edu/archives/win2012/entries/qm-modal/

Cohen-Tannoudji, C., Diu, B., Laloë, F.: Quantum Mechanics. Wiley, New York (1978)

Ballentine, L.E.: Quantum Mechanics. Prentice Hall, New York (1990)

van Fraassen, B.C.: A formal approach to the philosophy of science. In: Colodny, R. (ed.) Paradigms and Paradoxes: The Philosophical Challenge of the Quantum Domain, pp. 303–366. University of Pittsburgh Press, Pittsburgh (1972)

Bacciagaluppi, G.: Kohen-Specker theorem in the modal interpretation of quantum mechanics. Int. J. Theor. Phys.

**34**, 1206–1215 (1995)Clifton, R.: The properties of modal interpretations of quantum mechanics. Br. J. Philos. Sci.

**47**, 371–398 (1996)Vermaas, P.E.: Two no-go theorems for modal interpretations of quantum mechanics. Stud. Hist. Philos. Mod. Phys.

**30**, 403–431 (1999)Bacciagaluppi, G., Dickson, M.: Dynamics for modal interpretations. Found. Phys.

**29**, 1165–1201 (1999)Kochen, S.: A new interpretation of quantum mechanics. In: Mittelstaedt, P., Lahti, P. (eds.) Symposium on the Foundations of Modern Physics. World Scientific, Singapore (1985)

Dieks, D.: The formalism of quantum theory: an objetive description of reality? Annalen der Physik

**7**, 174–190 (1988)Dieks, D.: Quantum mechanics without the projection postulate and its realistic interpretation. Found. Phys.

**38**, 1397–1423 (1989)Dieks, D.: Resolution of the measurement problem through decoherence of the quantum state. Phys. Lett. A

**142**, 439–446 (1989)Bene, G., Dieks, D.: A perspectival version of the modal interpretation of quantum mechanicsand the origin of macroscopic behaviour. Found. Phys.

**32**, 645–671 (2002)Lombardi, O., Fortin, S., Castagnino, M.: The problem of identifying the system and the environment in the phenomenon of decoherence. In: de Regt, H.W., Hartmann, S., Okasha, S. (eds.) European Philosophy of Science Association (EPSA). Philosophical Issues in the Sciences, vol. 3, pp. 161–174. Springer, Berlin (2012)

Castagnino, M., Fortin, S., Lombardi, O.: Suppression of decoherence in a generalization of the spin-bath model. J. Phys. A: Math. Theor.

**43**, 065304 (2010)Lombardi, O., Castagnino, M.: A modal-Hamiltonian interpretation of quantum mechanics. Stud. Hist. Philos. Mod. Phys.

**39**, 380–443 (2008)Ardenghi, J.S., Castagnino, M., Lombardi, O.: Modal Hamiltonian interpretation of quantum mechanics and Casimir operators: the road toward quantum field theory. Int. J. Theor. Phys.

**50**, 774–791 (2011)Bohm, D.: A suggested interpretation of quantum theory in terms of “hidden” variables I. Phys. Rev.

**85**, 166–179 (1952)Bohm, D.: A suggested interpretation of quantum theory in terms of “hidden” variables II. Phys. Rev.

**85**, 180–193 (1952)Valentini, A.: Inflationary cosmology as a probe of primordial quantum mechanics. Phys. Rev. D

**82**, 063513 (2010)Holland, P.R.: The Quantum Theory of Motion: An Account of the De Broglie-Bohm Causal Interpretation of Quantum Mechanics. Cambridge University Press, Cambridge (1995)

Valentini, A.: Inflationary cosmology as a probe of primordial quantum mechanics. Phys. Rev. D

**82**, 063513 (2010)Pinto-Neto, N., Santos, G., Struyve, W.: Quantum-to-classical transition of primordial cosmological perturbations in de Broglie-Bohm quantum theory. Phys. Rev. D

**85**, 083506 (2012). [arXiv:1110.1339]Bohm, D.: Proof that probability density approach \(|\psi |^{2}\) in causal interpretations of the quantum theory. Phys. Rev.

**89**, 458–466 (1953)Vaidman, L.: Many-worlds interpretation of quantum mechanics. In: Zalta, E.N. (ed.) The Stanford Encyclopedia of Philosophy (Fall 2008 Edition). http://plato.stanford.edu/archives/fall2008/entries/qm-manyworlds/

Everett, H.: “Relative state” formulation of quantum mechanics. Rev. Mod. Phys.

**29**, 454–462 (1957)Bacciagaluppi, G.: The role of decoherence in quantum mechanics. In: Zalta, E.N. (ed.) The Stanford Encyclopedia of Philosophy (Winter 2012 Edition). http://plato.stanford.edu/archives/win2012/entries/qm-decoherence/

Griffiths, R.B.: Consistent histories and the interpretation of quantum mechanics. J. Stat. Phys.

**36**, 219–272 (1984)Omnès, R.: Logical reformulation of quantum mechanics. I. Foundations. J. Stat. Phys.

**53**, 893–932 (1988)Omnès, R.: Logical reformulation of quantum mechanics. IV. Projectors in semiclassical physics. J. Stat. Phys.

**57**, 357–382 (1989)Gell-Mann, M., Hartle, J.B.: Quantum mechanics in the light of quantum cosmology. In: Zurek, W.H. (ed.) Complexity, Entropy, and the Physics of Information, pp. 425–458. Addison-Wesley, Reading, MA (1990)

Laura, R., Vanni, L.: Time translation of quantum properties. Found. Phys.

**39**, 160–173 (2009)Vanni, L., Laura, R.: The logic of quantum measurements. Int. J. Theory Phys.

**52**, 2386–2394 (2013)Losada, M., Vanni, L., Laura, R.: Probabilities for time-dependent properties in classical and quantum mechanics. Phys. Rev. A

**87**, 052128 (2013)Losada, M., Laura, R.: The formalism of generalized contexts and decay processes. Int. J. Theor. Phys.

**52**, 1289–1299 (2013)Weinberg, S.: Collapse of the State Vector. UTTG-18-11, (2011). arXiv:1109.6462

Hartle, J.B.: Quantum physics and human language. J. Phys. A

**40**, 3101 (2007)Okon, E., Sudarsky, D.: On the consistency of the consistent histories approach to quantum mechanics. Found. Phys.

**44**, 19–33 (2014). arXiv:1301.2586Hartle, J.B.: Quantum Cosmology Problems for the 21\({}^{st}\) Century (e-Print: gr-qc/9701022)

Hartle, J.B.: Generalized Quantum mechanics for Quantum Gravity (e-Print: gr-qc/0510126)

Kent, A.: Consistent sets yield contrary inferences in quantum theory. Phys. Rev. Lett.

**87**, 15 (1997)Dowker, F., Kent, A.: On the consistent histories approach to quantum mechanics. J. Statist. Phys.

**82**, 1575 (1996). arXiv:gr-qc/9412067Bassi, A., Ghirardi, G.C.: Can the decoherent histories description of reality be considered satisfactory?. Phys. Lett. A

**257**, 247 (1999). arXiv:gr-qc/9811050Bassi, A., Ghirardi, G.C.: About the notion of truth in the decoherent histories approach: a reply to Griffiths. Phys. Lett. A

**265**, 153 (2000). [arXiv:quant-ph/9912065]Diosi, L.: Gravitation and quantum mechanical localization of macro-objects. Phys. Lett. A

**105**, 199–202 (1984)Diosi, L.: A universal master equation for the gravitational violation of quantum mechanics. Phys. Lett. A

**120**, 377 (1987)Diosi, L.: Models for universal reduction of macroscopic quantum fluctuations. Phys. Lett. A

**40**, 1165 (1989)Penrose, R.: The Emperor’s New Mind. Oxford University Press, Oxford (1989)

Penrose1 Penrose, R.: On gravity’s role in quantum state reduction. Gen. Relativ. Gravity

**28**, 581 (1996)Ghirardi, G.C., Rimini, A., Weber, T.: A unified dynamics for micro and macro systems. Phys. Rev. D

**34**, 470 (1986)Pearle, P.M.: Combining stochastic dynamical state-vector reduction with spontaneous localization. Phys. Rev. A

**39**, 2277 (1989)Bassi, A., Ghirardi, G.C.: Dynamical reduction models. Phys. Rept.

**379**, 257 (2003). [arXiv:quant-ph/0302164]Pearle, P.: Reduction of the state vector by a nonlinear Schrodinger equation. Phys. Rev. D

**13**, 857 (1976)Pearle, P.: Toward explaining why events occur. Int. J. Theory Phys.

**18**, 489 (1979)Pearle, P.: Experimental tests of dynamical state-vector reduction. Phys. Rev. D

**29**, 235 (1984)Pearle, P.: Combining stochastic dynamical state vector reduction with spontaneous localization. Phys. Rev. A

**39**, 2277 (1989)Martin, J., Vennin, V., Peter, P.: Cosmological Inflation and the Quantum Measurement Problem (2012). arXiv:1207.2086

Cañate, P., Pearl, P., Sudarsky, D.: CSL Quantum Origin of the Primordial Fluctuation. Phys. Rev. D,

**87**, 104024 (2013). arXiv:1211.3463 [gr-qc]Das, S., Lochan, K., Sahu, S., Singh, T. P.: Quantum to Classical Transition of Inflationary Perturbations—Continuous Spontaneous Localization as a Possible Mechanism. arXiv:1304.5094 [astro-ph.CO]

Okon, E., Sudarsky, D.: Benefits of objective collapse models for cosmology and quantum gravity. Found. Phys.

**44**114–143 (2014). arXiv:1309.1730v1 [gr-qc]Myrvold, W.C.: On peaceful coexistence: is the collapse postulate incompatible with relativity? Stud. Hist. Philos. Mod. Phys.

**33**, 435 (2002)Tumulka, R.: On spontaneous wave function collapse and quantum field theory. Proc. Roy. Soc. Lond. A

**462**, 1897 (2006). arXiv:quant-ph/0508230Bedingham, D.J.: Relativistic state reduction dynamics. Found. Phys.

**41**, 686 (2011). arXiv:1003.2774

## Acknowledgements

This work was supported, in part, by CONACYT (México) Project 101712, a PAPPIT-UNAM (México) project IN107412 and sabbatical fellowships from CONACYT and DGAPA-UNAM (México). D.S. wants to thank the IAFE at the university of Buenos Aires for the hospitality during the sabbatical stay. This work was partially supported by Grants: of the Research Council of Argentina (CONICET), by the Endowment for Science and by Technology of Argentina (FONCYT), and by the University of Buenos Aires. We acknowledge very useful discussions with B. Kay and Elias Okon.

## Author information

### Authors and Affiliations

### Corresponding author

## Appendix

### Appendix

In this appendix, we discuss some specific issues that arise in the attempt to use decoherence related arguments in the context of the problem at hand.

The first issue is that connected to the implication of symmetry regarding the choice of a preferential basis or so called pointer states.

The simplest example exhibiting this problem is provided by a standard EPR-R setup: Consider the decay of a spin 0 particle into two spin 1/2 particles. Take the direction of the decay as being the *x* axis (the particles momenta are \(\vec P= \pm P \hat{x}\) with \(\hat{x}\) the unit vector in the \(\vec x \) direction)^{Footnote 15}. Now, we characterize the two particle states, that emerges after the decay in terms of the \(\vec {z}\) polarization states of the two Hilbert spaces of individual particles. As it is known, the conservation of the angular momentum of the system indicates that the state must be:

The state is clearly invariant under rotations around the *x* axis (simply because it is an eigen-state with zero angular momentum along that axis). The density matrix for the system is thus \(\rho = |\chi \rangle \langle \chi |\). Now assume we decide we are not interested in one of the particles (call it 1), and thus we regard it as an environment for the system of interest (particle 2). The reduced density matrix is then:

Now suppose we want to say that as the reduced density matrix is diagonal, we have found the pointer basis and that somehow the particle must be considered as having its spin along the *z* axis defined to be either \(+1/2 \) or \(-1/2\).

The problem is that the symmetry of the state \(|\chi \rangle \) regarding rotations around the *x* axis is reflected in the fact that we could have written this density matrix also as

leading, this time, to the conclusion that the particle must be considered as having its spin along the *y* axis defined to be either \(+1/2 \) or \(-1/2\).

In fact, as the density matrix is proportional to the identity ( i.e. \(\rho ^{(2)} = \frac{1}{ 2} I\)) it would have the same form in any orthogonal basis.

One might be inclined to consider that this problem occurs only in very simple situations, such as the one of the above example, and that, in general, we will not encounter such difficulty. However that consideration is mistaken as can be seen from the general result encapsulated in the following:

### 1.1 Theorem

Consider a quantum system made of a subsystem *S* and an environment *E*, with corresponding Hilbert spaces \(H_{S}\) and \(H_{E}\) so that the complete system is described by states in the product Hilbert space \(H_{S}\otimes H_{E}\). Let *G* be a symmetry group acting on the Hilbert space of the full system in a way that does not mix the system and environment. That is, the unitary representation *O* of *G* on \(H_{S}\otimes H_{E}\) is such that \(\forall g \in G\), \(\hat{O}(g) = \hat{O}^{S}(g)\otimes \hat{O}^{E}(g)\), where \(\hat{O}^{S}(g)\) and \(\hat{O}^{E}(g)\) act on \(H_{S}\) and \(H_{E}\) respectively.

Let the system be characterized by a density matrix \(\hat{\rho }\) which is invariant under *G*. Then the reduced density matrix of the subsystem is a multiple of the identity in each invariant subspace of \(H_{S}\).

### 1.2 Proof

The reduced density matrix \(\hat{\rho }_{S}= Tr_{E} ( \hat{\rho })\).The trace over the environment of any operator \(\hat{A}\) in \(H_{S}\otimes H_{E}\) is obtained by taking any orthonormal basis \(\lbrace |e_{j}\rangle \rbrace \) of \(H_{E}\) and evaluating \(\Sigma _{j} \langle e_{j}| \hat{A} |e_{j}\rangle \).

Now, by assumption, we have \(\hat{\rho }= {\hat{O}(g)}^{\dagger }\hat{\rho }\hat{O}(g)\), \(\forall g \in G\). Then, for all \(g \in G\), we have \(\hat{\rho }_{S} = \Sigma _{j} \langle e_{j}| \hat{\rho }|e_{j}\rangle = \Sigma _{j} \langle e_{j}| {\hat{O}^{S}(g)}^{\dagger }\otimes { \hat{O}^{E}(g)}^{\dagger }\hat{\rho }\hat{O}^{S}(g)\otimes \hat{O}^{E}(g) |e_{j}\rangle = \Sigma _{j} {{\hat{O}}^{S} (g)}^{\dagger }\langle e^{\prime }_{j}| \hat{\rho }|e^{\prime }_{j}\rangle {\hat{O}}^{S}(g)\), where \(|e^{\prime }_{j}\rangle \equiv O^{E}(g) |e_{j}\rangle \). However, the fact that the operator \({\hat{O}}^{E}(g)\) is unitary implies that the transformed states \(\lbrace |e^{\prime }_{j}\rangle \rbrace \) form also an orthonormal basis of \(H_{E}\).

Thus we have \(\hat{\rho }_{S} = {{\hat{O}}^{S}(g)}^{\dagger }( \Sigma _{j} \langle e^{\prime }_{j}| \hat{\rho }|e^{\prime }_{j}\rangle ) {\hat{O}}^{S}(g ) = {{\hat{O}}^{S}(g)}^{\dagger }\hat{\rho }_{S} {\hat{O}}^{S}(g )\) or equivalently \(\hat{\rho }_{S} {{\hat{O}}^{S}(g)} = {\hat{O}}^{S}(g ) \hat{\rho }_{S}\). So we have found that \([ \hat{\rho }_{S} , {\hat{O}}^{S}(g )]=0\), \(\forall g \in G\), and thus by Schur’s lemma it follows that \(\hat{\rho }_{S}\) must be a multiple of the identity in each invariant subspace of \(H_{S}\), QED.

In particular, this result indicates that, if we start with a pure state invariant under the symmetry group, the reduced density matrix must be a multiple of the identity in each invariant subspace of \(H_{S}\). This is exemplified by the well known case of a standard EPR setting, where a spinless particle decays into two photons, and where one considers the photons’ spin degrees of freedom. The reduced density matrix describing one of the photons is a multiple of the identity, and thus the decoherence that results from tracing over the first photon’s spin does not determine a preferential basis for the characterization of the spin of the second photon. Decoherence then fails under these conditions to provide a well defined preferential context for the interpretation of the reduced density matrix, as representing the various alternatives for the state of the subsystem after decoherence.

## Rights and permissions

## About this article

### Cite this article

Castagnino, M., Fortin, S., Laura, R. *et al.* Interpretations of Quantum Theory in the Light of Modern Cosmology.
*Found Phys* **47**, 1387–1422 (2017). https://doi.org/10.1007/s10701-017-0100-9

Received:

Accepted:

Published:

Issue Date:

DOI: https://doi.org/10.1007/s10701-017-0100-9

### Keywords

- Interpretation of quantum mechanics
- Measurement problem
- Foundations of quantum mechanics