Abstract
What is the physical agent that causes noninteracting particles to get entangled? The theory developed in previous chapters is applied in the present one to give due response to this key question. For this purpose, the one-particle treatment presented in Chap. 5 is extendedExtended charge to systems of two particlesExtended particle that are embedded in the common zero-point fieldEntanglement!and zero-point field. The dynamical variables are shown to become correlated when the particles resonate to a common frequency of the background field. When the description is reduced to one in terms of matrices and vectors in the appropriate Hilbert space, the entangled state vectors emerge naturally. For systems of identical particles the properties of invariance of the field variables imply that entanglementEntanglement is maximal and must be described by totally (anti)symmetric states. The results thus obtained are applied to the HeliumHelium atom as a system with two electrons. As a result of entanglement, the total (orbital plus spin) stateHelium!spin states vectors turn out to be antisymmetric. States in which both particles are in the same orbital and spinorial state, are excluded because of the absence of a correlating field mode.
The best possible knowledge of a whole does not necessarily include the best possible knowledge of all its parts, even though they may be entirely separate...
E. Schrödinger (1935)Wheeler, J. A. Zurek, W. H.
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Notes
- 1.
- 2.
The postulate of symmetrization of the wave function of the composite system is normally justified on the basis of the indistinguishabilityIndistinguishability of the quantum corpuscles. It is within qft that the spin-statistics theorem is derived as a relativistic result, detached from arguments about indistinguishability. Nevertheless, also qft fails to provide elements to unveil the mechanism leading to this result —a fact that does not go without criticism (see e.g.Duck, I. Duck and Sudarshan 1997; Kaplan 2013 Sudarshan, E. C. G. Kaplan, G.; see also Tomonaga 1974).
- 3.
- 4.
In Blanco, R. Blanco and Santos (1979) the problem of a system of several particles embedded in the zpf is addressed, with a scope similar to the present one. However, and although that paper constitutes a significant and important effort to disclose the mechanism underlying entanglement, it differs essentially from the present one in that an identical background field is assumed for all particles, which leads to unconvincing results.
- 5.
- 6.
As discussed in Sect. 5.2.2, such variables are assumed to be functions of the form \(h(x_{i})+g(\dot{x}_{i})\) with both \(h,g\) power series of their argument.
- 7.
More general functions \(V(x_{1},\dot{x}_{1};x_{2},\dot{x}_{2})\) would be represented by linear combinations of elementary products \(FG.\)
- 8.
We recall that in the stationary regime, the principle of ergodicity holds and therefore time averages are equivalent to ensemble averages. Notice further that for the calculation of (7.46a, 7.46b), the system is considered to be either in state \(D\) or in state \(K\). Hence, according to the discussion in Sect. 5.2, these averages are taken over different subensembles, namely those containing all realizations of the system in state \(D\) or \(K\), respectively.
- 9.
The existence of common relevant frequencies and the condition of degeneracy are thus two faces of the same coin, yet it will become clear below that the former is more illuminating in disclosing the physical mechanism underlying entanglement. As stated after Eq. (5.135), focusing on the (relevant) frequencies rather than on the energy levels represents a shift from the Schrödinger to the Heisenberg approach. Here we find another example that shows that despite the formal equivalence of the two approaches, either one or the other is more suitable to understand certain aspects of the quantum phenomenon.
- 10.
In fact, as follows from the remark in note 8, the averages over states \(\left| D\right\rangle \) and \(\left| K\right\rangle \) Averaging!\(k\)-averagingimply an averaging over different (incompatible) subesembles, hence they cannot be taken simultaneosly over one and the same system.
- 11.
According to the results in Chap. 5 (see also Appendix B), the relevant frequencies are constructed via linear combinations of the resonance frequencies, hence the existence of common relevant frequencies goes back to the existence of common resonance frequencies. We can therefore rephrase the above statement by saying that the existence of (nontrivial) common resonance frequencies lies at the root of the emergence of correlations between the particles.
- 12.
In the above lines we treat as identical particles those having the same intrinsic properties relevant for the present description, namely \(m\) and \(e\). No explicit reference to their spin or other properties is made, since the description developed so far [rooted in Eq. (7.76)] leaves out any spinorial effect, thus treating the particles as spinless systems. However, the equality of their spins is tacitly assumed by considering that the Hilbert spaces in which the matrices \(\hat{F}\) and \(\hat{G}\) are defined coincide, hence possess the same dimension. In Sect. 7.4, we extend the analysis to particles with spin 1/2.
- 13.
As follows from the analysis in Sect. 7.1.4, \(\hat{F}\) \((\hat{G})\) stands for the matrix defined in \(\mathcal {H}_{1}\) \((\mathcal {H}_{2})\) associated with a variable \(F(G)\) proper of the particle immersed in the background (not effective!) field \(E_{1}(E_{2})\), with random amplitudes \(a_{1}(a_{2}).\) Now, in the case of identical particles subject to the same external potential, \(\mathcal {H}_{1}\) and \(\mathcal {H}_{2}\) coincide, and therefore the matrices \(\hat{F}\) and \(\hat{G}\) are invariant under \(I_{f}.\)
- 14.
If \(\hat{F}=\hat{G}\), then \(F_{\beta \beta ^{\prime }}G_{\beta ^{\prime }\beta }=\left| F_{\beta \beta ^{\prime }}\right| ^{2}\), thus \(\mathrm {Im}F_{\beta \beta ^{\prime }}G_{\beta ^{\prime }\beta }=0\) and nothing can be said about \(\lambda _{B}^{*}\,-\,\lambda _{B}.\) However, there exist \(\hat{F}\) and \(\hat{G}\) that have the same relevant frequencies and give nevertheless a complex value for \(F_{\beta \beta ^{\prime }}G_{\beta ^{\prime }\beta }.\) An obvious example is \(\hat{F}=\hat{p}\) and \(\hat{G}=\hat{x},\) for which \(\mathrm {Im}F_{\beta \beta ^{\prime }}G_{\beta ^{\prime }\beta }=m\omega _{\beta \beta ^{\prime }}\left| x_{\beta \beta ^{\prime }}\right| ^{2}.\)
- 15.
The interchange of particles thus defined is valid only for identical particles subject to the same external potential. This is so because to ensure that upon the exchange \(I_{s}\) the resulting state continues to be stationary, both \(\beta \) and \(\beta ^{\prime }\) must be accessible states for both particles, for any pair (\(\beta \), \(\beta ^{\prime }\)).
- 16.
In Chap. 6 the electron spinHarmonic oscillator!and spin was obtained as another quantum property that emerges from the particle-zpf interaction. In the transition to the Heisenberg descriptionHarmonic oscillator!Heisenberg description, this new variable appears represented by a two-dimensional matrix operator \(\varvec{\hat{S}}\), which is expressed in terms of the Pauli Pauli, W.matrices as \(\varvec{\hat{S}}=\hbar \varvec{\hat{\sigma }}/2\) and satisfies the usual angular-momentum commutation rules.
- 17.
The superselection rule of univalence forbids transtions from symmetric into antisymmetric vector states, and conversely.
References
Allahverdyan, A.E., Khrennikov, A., Nieuwenhuizen, ThM: Brownian entanglement. Phys. Rev. A 72, 032102 (2005)
Bell, J.S.: On the Einstein Podolsky Rosen paradox. Physics 1, 195 (1964)
Bell, J.S.: Speakable and Unspeakable in Quantum Mechanics. Cambridge University Press, Cambridge (1987)
Bell, M., Gottfried, K., Veltman, M.: John S. Bell on the Foundations of Quantum Mechanics. World Scientific, Singapore (2001)
Benatti, F., Floreanini, R., Piani, M.: Environment induced entanglement in Markovian dissipative dynamics. Phys. Rev. Lett. 91, 070402 (2003)
Blanco, R., Santos, E.: Coupled harmonic oscillators in stochastic electrodynamics. Lett. Nuovo Cim. 25, 360 (1979)
Braun, D.: Creation of entanglement by interaction with a common heat bath. Phys. Rev. Lett. 89, 277901 (2002)
*Casado, A., Marshall, T.W., Santos, E.: Type II parametric downconversion in the Wigner-function formalism: entanglement and Bell’s inequalities. J. Opt. Soc. Amer. B 15, 1572 (1998)
Coffman, V., Kundu, J., Wootters, W.K.: Distributed entanglement. Phys. Rev. A 61, 052306 (2000)
Cohen-Tannoudji, C., Diu, B., Laloë, F.: Quantum Mechanics. Wiley, New York (1977)
*de la Peña, L., Valdés-Hernández, A., Cetto, A.M.: Entanglement of particles as a result of their coupling through the common background zero-point radiation field. Physica E 42, 308 (2010)
*de la Peña, L., Cetto, A.M., Valdés-Hernández, A.: The emerging quantum. An invitation. In: Proceedings of the Advanced School on Quantum Foundations and open Quantum Systems, João Pessoa, Brazil, 16–28 Jul 2012
Duck, I., Sudarshan, E.C.G.: Pauli and the Spin-Statistics Theorem. World Scientific, Singapore (1997)
Einstein, A., Podolsky, B., Rosen, N.: Can quantum-mechanical description of physical reality be considered complete? Phys. Rev. 47, 777 (1935)
Ficek, Z., Tanaś, R.: Dark periods and revivals of entanglement in a two-qubit system. Phys. Rev. A 74, 024304 (2006)
Greiner, W.: Quantum Mechanics. Special Chapters. Springer, Berlin (1998)
Hor-Meyll, M., Auyuanet, A., Borges, C.V.S., Aragão, A., Huguenin, J.A.O., Khoury, A.Z., Davidovich, L.: Environment-induced entanglement with a single photon. Phys. Rev. A 80, 042327 (2009)
Horodecki, R., Horodecki, P., Horodecki, M., Horodecki, K.: Quantum entanglement. Rev. Mod. Phys. 81, 865 (2009)
Kaplan, I.G.: The Pauli exclusion principle. Can it be proved? Found. Phys. 43, 1233 (2013)
Kim, M.S., Lee, J., Ahn, D., Knight, P.L.: Entanglement induced by a single-mode heat environment. Phys. Rev. A 65, 040101 (2002)
Khrennikov, A.: Description of composite quantum systems by means of classical random fields. Found. Phys. 40, 1051 (2010)
Khrennikov, A.: Prequantum classical statistical field theory: Schrödinger dynamics of entangled systems as a classical stochastic process. Found. Phys. 41, 317 (2011)
Landau, L., Lifshitz, E.: The Classical Theory of Fields. Addison-Wesley, Cambridge (1951)
*Marshall, T.W., Santos, E.: Stochastic optics: a reaffirmation of the wave nature of light. Found. Phys. 18, 185 (1988)
*Marshall, T.W., Santos, E.: Semiclassical optics as an alternative to nonlocality. Recent Res. Devel. Opt. 2, 683 (2002)
Santos, E.: What is entanglement? arXiv:quant-phys/0204020 (2002)
*Santos, E.: Photons are fluctuations of a random (zeropoint) radiation field filling the whole space. In: Roychoudhuri, C., Kracklauer, A.F., Creath, K. (eds.) The nature of light. What is a photon? CRC Press, Florida (2008)
Schrödinger, E.: Die gegenwärtige Situation in der Quantenmechanik I, II, III, Naturwiss 23, 807, 823, 844. English translation: Quantum Theory and Measurement (Trans: Wheeler, J.A., Zurek, W.H. (eds.)) (Princeton University Press, Princeton, 1983) (1935)
Tomonaga, S.I.: The Story of the Spin. The University of Chicago Press, Chicago (1974)
Valdés-Hernández, A.: Investigación del origen del enredamiento cuántico desde la perspectiva de la Electrodinámica Estocástica Lineal. Ph. D. Thesis, Universidad Nacional Autónoma de México, México (2010)
*Valdés-Hernández, A., de la Peña y, L., Cetto, A.M.: Bipartite entanglement induced by a common background (zero-point) radiation field. Found. Phys. 41, 843 (2011)
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de la Peña, L., Cetto, A.M., Valdés Hernández, A. (2015). Disentangling Quantum Entanglement. In: The Emerging Quantum. Springer, Cham. https://doi.org/10.1007/978-3-319-07893-9_7
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