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A Contextual Planck Parameter and the Classical Limit in Quantum Cosmology

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Barrow, John D 


jats:titleAbstract</jats:title>jats:pWe propose that whatever quantity controls the Heisenberg uncertainty relations (for a given complementary pair of observables) it should be identified with an effective Planck parameter. With this definition it is not difficult to find examples where the Planck parameter depends on the region under study, varies in time, and even depends on which pair of observables one focuses on. In quantum cosmology the effective Planck parameter depends on the size of the comoving region under study, and so depends on that chosen region and on time. With this criterion, the classical limit is expected, not for regions larger than the Planck length, jats:inline-formulajats:alternativesjats:tex-math$$l_{P}$$</jats:tex-math><mml:math xmlns:mml=""> mml:msub mml:mil</mml:mi> mml:miP</mml:mi> </mml:msub> </mml:math></jats:alternatives></jats:inline-formula>, but for those larger than jats:inline-formulajats:alternativesjats:tex-math$$l_{Q}=(l_{P}^{2}H^{-1})^{1/3}$$</jats:tex-math><mml:math xmlns:mml=""> mml:mrow mml:msub mml:mil</mml:mi> mml:miQ</mml:mi> </mml:msub> mml:mo=</mml:mo> mml:msup mml:mrow mml:mo(</mml:mo> mml:msubsup mml:mil</mml:mi> mml:mrow mml:miP</mml:mi> </mml:mrow> mml:mn2</mml:mn> </mml:msubsup> mml:msup mml:miH</mml:mi> mml:mrow mml:mo-</mml:mo> mml:mn1</mml:mn> </mml:mrow> </mml:msup> mml:mo)</mml:mo> </mml:mrow> mml:mrow mml:mn1</mml:mn> mml:mo/</mml:mo> mml:mn3</mml:mn> </mml:mrow> </mml:msup> </mml:mrow> </mml:math></jats:alternatives></jats:inline-formula>, where jats:italicH</jats:italic> is the Hubble parameter. In theories where the cosmological constant is dynamical, it is possible for the latter to remain quantum even in contexts where everything else is deemed classical. These results are derived from standard quantization methods, but we also include more speculative cases where ad hoc Planck parameters scale differently with the length scale under observation. Even more speculatively, we examine the possibility that similar complementary concepts affect thermodynamical variables, such as the temperature and the entropy of a black hole.</jats:p>


Funder: Science and Technology Facilities Council; doi:


5107 Particle and High Energy Physics, 51 Physical Sciences

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Foundations of Physics

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Springer Science and Business Media LLC