<i>Special Functions and Their Applications</i>

Lebedev, N. N.; Silverman, Richard A.; Livhtenberg, D. B.

doi:10.1063/1.3047047

Cited by 906 publications

(861 citation statements)

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“…Starting from a knowledge of the actual entropy of the Universe σ ∼ 10 104 , one derives the radial wavefunction describing this nonmaximally entropic Universe: one simply substitutes the dimensionless eigenvalue λ = −σ = −10 104 into the eigenfunctions (24) and (25). Call the latter ψ (j) 10 104 as in Equation (27). The expectation value of R 2 in the states ψ (j) 10 104 should give back the initial entropy 10 104 .…”

Section: Discussionmentioning

confidence: 99%

“…As opposed to the free wavefunctions (10) and (11), the existence of zeroes of the confluent hypergeometric function 1 F 1 is a sure sign that the cosmological principle will be violated by the wavefunctions (27), but the extent of this violation remains to be determined. We claim that:…”

Section: Nonperturbativementioning

confidence: 99%

“…Due to the oscillatory behaviour of the confluent hypergeometric function, this renders the exact radial eigenfunctions (27) extremely cumbersome to work with, both analytically and numerically. To simplify matters, we will replace the exact vacuum-state eigenfunctions ψ (j) λ 0 of Equations (24) and (25) with a set of approximate radial eigenfunctions for the vacuum state.…”

Section: Nonperturbativementioning

confidence: 99%

“…Of course, our eigenfunctions (27) are not real. However, still assuming ψ * = ψ, the best possible ratio Q / V is attained for E = V .…”

Section: Nonperturbativementioning

confidence: 99%

“…Substituting this value of λ 0 into Equation (27) produces the wavefunctions ψ (j) λ 0 , with j = 1, 2, which are the analogues of the vacuum wavefunction of the usual oscillator. The bound (28) has been determined by a purely classical argument; although the uncertainty principle will shift the minimum energy (28) by a positive amount, this correction can be discarded for our purposes, as it will be negligible compared to (28) itself.…”

Section: Nonperturbativementioning

confidence: 99%

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Boltzmann Entropy of a Newtonian Universe

Cabrera

Córdoba

Isidro

2017

Entropy

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A dynamical estimate is given for the Boltzmann entropy of the Universe, under the simplifying assumptions provided by Newtonian cosmology. We first model the cosmological fluid as the probability fluid of a quantum-mechanical system. Next, following current ideas about the emergence of spacetime, we regard gravitational equipotentials as isoentropic surfaces. Therefore, gravitational entropy is proportional to the vacuum expectation value of the gravitational potential in a certain quantum state describing the matter contents of the Universe. The entropy of the matter sector can also be computed. While providing values of the entropy that turn out to be somewhat higher than existing estimates, our results are in perfect compliance with the upper bound set by the holographic principle.Keywords: newtonian cosmology; emergent quantum theory The Approach via Emergent Quantum TheoryIn this article, we will argue in favour of emergent quantum mechanics as providing an appropriate framework to estimate thermodynamical quantities of the Universe, such as the entropy.The notion that quantum mechanics is an emergent theory has been discussed at length in the literature [1][2][3][4][5][6][7]. Combined with the idea that spacetime is also is an emergent phenomenon [8][9][10][11][12], this paves the way for a computation of some thermodynamical properties of spacetime in quantum-mechanical terms. We would like to remark that a quantum-mechanical approach to the expansion of the Universe was called for long ago in Reference [13], where it was suggested to regard the expansion of the Universe as a scattering problem in quantum mechanics.The expansion of the Universe is a long-standing experimental observation [14] that has in recent years been refined thanks to very precise measurements [15,16]. In the Newtonian approximation, this receding behaviour of the galaxies can be easily modelled by a phenomenological potential-namely, an isotropic harmonic potential carrying a negative sign:As the angular frequency, we take the current value of Hubble's constant H 0 . Thus, U Hubble has the dimensions of energy per unit mass, or velocity squared.In the emergent approach to spacetime presented in Reference [12], gravity qualifies as an entropic force. If gravitational forces are entropy gradients, gravitational equipotential surfaces can be identified with isoentropic surfaces. Recalling the arguments of Reference [12], a classical point particle approaching a holographic screen causes the entropy of the latter to increase by one quantum k B . Here we will analyse a quantum-mechanical model in which the forces driving the galaxies away from each other can be modelled by the Hubble potential (1). We will replace the classical particle of Reference [12] with a collection of quantum particles (the matter contents of the Universe) described

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Section: Discussionmentioning

confidence: 99%

Section: Nonperturbativementioning

confidence: 99%

Section: Nonperturbativementioning

confidence: 99%

“…Of course, our eigenfunctions (27) are not real. However, still assuming ψ * = ψ, the best possible ratio Q / V is attained for E = V .…”

Section: Nonperturbativementioning

confidence: 99%

Section: Nonperturbativementioning

confidence: 99%