Abstract
Gamow’s theory of the implications of quantum tunneling for star burning has two cornerstones: quantum mechanics and the equipartition theorem. It has been proposed that both of these foundations are affected by the existence of a non-zero minimum length, which usually appears in quantum gravity scenarios and leads to the generalized uncertainty principle (GUP). Mathematically, in the framework of quantum mechanics, the effects of the GUP are considered as perturbation terms. Here, generalizing the de Broglie wavelength relation in the presence of a minimal length, GUP corrections to the Gamow temperature are calculated, and in parallel, an upper bound for the GUP parameter is estimated.
Introduction
In the first step of star burning, its constituents must overcome the Coulomb barrier to participate in nuclear fusion (NF). This means that when the primary gas ingredients have mass m and velocity v, then using the equipartition theorem, one getswhere KB denotes the Boltzmann constant, the subscript c in Uc(r0) indicates the Coulomb potential, and correspondingly, denotes the maximum of the Coulomb potential between the ith and jth particles located at a distance r0 from each other (Prialnik, 2000). In this article, Kelvin (K) is the temperature unit. Finally, we reachfor the temperature required to overcome the Coulomb barrier. Therefore, NF happens whenever the temperature of the primary gas is comparable to Eq. 2, which clearly shows that, for the heavier nuclei, NF happens at higher temperatures. On the contrary, for the temperature of gas with mass M and radius R, we have (Prialnik, 2000)where M⊙ and R⊙ are the Sun mass and radius, respectively. Clearly, and T are far from each other, meaning that NF cannot cause star burning (Prialnik, 2000). Therefore, NF occurs if a process reduces the required temperature (2). In fact, we need a process that decreases Eq. 2 to the values comparable to Eq. 3. Quantum tunneling lets particles pass through the Coulomb barrier, which finally triggers star burning, meaning that quantum tunneling allows NF to occur at temperatures lower than T (Prialnik, 2000). Indeed, if the distance between particles (r0) becomes of the order of their de Broglie wavelength ( where Q implies that we are in the purely quantum mechanical regime), then quantum tunneling happens and simple calculations lead to (Prialnik, 2000)instead of Eq. 2 for the temperature required to launch star burning. λQ can also be obtained by solving which gives (Prialnik, 2000)meaning that quantum tunneling provides a platform for NF in stars (Prialnik, 2000). As an example, for hydrogen atoms, one can see that quantum tunneling leads to (comparable to (3)) as the Gamow temperature at which NF is underway. Based on the above argument, it is expected that any change in p affects λQ and, thus, these results.
It is also useful to mention here that the quantum tunneling theory allows the above process because the tunneling probability is not zero. Indeed, quantum tunneling is also the backbone of Gamow’s theory of the α decay process (Gamow, 1928). Relying on the inversion of the Gamow formula for α decay, which gives the transmission coefficient, a method has also been proposed for studying the inverse problem of Hawking radiation (Völkel et al., 2019).
The backbone of quantum mechanics is the Heisenberg uncertainty principle (HUP),where x and p are ordinary canonical coordinates satisfying [xi, pj] = iℏδij. It has been proposed that, in quantum gravity scenarios, the HUP is modified such that (Kempf et al., 1995; Kempf, 1996)called the GUP, where lp denotes the Planck length and β0 is the dimensionless GUP parameter. X and P are called generalized coordinates, and we work in a framework in which Xi = xi, and up to the first order, we have and (Das and Vagenas, 2008; Motlaq and Pedram, 2014). Moreover, the GUP implies that there is a non-zero minimum length . Indeed, the existence of a non-zero minimum length also emerges even when the gravitational regime is Newtonian (Mead, 1964), a common result with quantum gravity scenarios (Hossenfelder, 2013). More studies on quantum gravity can be traced to earlier studies (Lake et al., 2019; Lake et al., 2020; Lake, 2021; Lake, 2022). There have been various attempts to estimate the maximum possible upper bound on β0 (Zhu et al., 2009; Chemissany et al., 2011; Das and Mann, 2011; Sprenger et al., 2011; Pikovski et al., 2012; Husain et al., 2013; Ghosh, 2014; Jalalzadeh et al., 2014; Scardigli and Casadio, 2015; Bosso et al., 2017; Feng et al., 2017; Gecim and Sucu, 2017; Bushev et al., 2019; Luciano and Petruzziello, 2019; Park, 2020; Aghababaei et al., 2021; Feleppa et al., 2021; Mohammadi Sabet et al., 2021), and among them, it seems that the maximum estimation for the upper bound is of the order of 1078 (Scardigli and Casadio, 2015). The implications of GUP on stellar evolution (Moradpour et al., 2019; Shababi and Ourabah, 2020) and the thermodynamics of various gases (Chang et al., 2002; Fityo, 2008; Wang et al., 2010; Hossenfelder, 2013; Motlaq and Pedram, 2014; Moradpour et al., 2021) have also been studied.
Indeed, the existence of a minimal length leads to the emergence of the GUP (Hossenfelder, 2013), and it affects thermodynamics (Chang et al., 2002; Fityo, 2008; Wang et al., 2010; Hossenfelder, 2013; Motlaq and Pedram, 2014; Moradpour et al., 2021) and quantum mechanics (Kempf et al., 1995; Kempf, 1996), as P can be expanded as a function of p. This letter deals with the GUP effects on star burning facilitated by quantum tunneling. Loosely speaking, we investigate the effects of a minimal length on (the Gamow temperature).
GUP corrections to the tunneling temperature
To proceed further and in the presence of the quantum features of gravity, we introduce the generalized de Broglie wavelength asIt is obvious that, as β0 → 0, one obtains P → p and thus λGUP → λQ, which is the quantum mechanical result. Indeed, up to first order in β0, we have , and the thermal energy per particle with temperature T is (Motlaq and Pedram, 2014)Mathematically, one should find the corresponding de Broglie wavelength by solving the following equation:Inserting the result intoone can finally find the GUP corrected version of Eq. 4.
Now, inserting λGUP into Eq. 10 and then combining the results with Eq. 11, we findin which Eq. 4 has been used for simplification. To estimate the magnitude of , we consider the hydrogen atom for which m ∼ 10–27kg. Now, since lp ∼ 10–35m, , , and , one easily finds . Moreover, because the effects of the GUP in the quantum mechanical regime are small (Hossenfelder, 2013), a reasonable basic assumption could be that . Indeed, if β0 ≪ 1046, then we always have meaning that i) we can Taylor expand our results and ii) 1046 is an upper bound for β0, which is comparable to those found in previous works (Das and Vagenas, 2008; Scardigli and Casadio, 2015; Feng et al., 2017; Aghababaei et al., 2021; Feleppa et al., 2021) summarized in Table 1.
TABLE 1
| Measurement/experiment | β0 | Refs. |
|---|---|---|
| Modified mass-temperature relation | 1078 | Scardigli and Casadio (2015) |
| Light deflection | 1078 | Scardigli and Casadio (2015) |
| Pulsar PRS B 1913 + 16 data | 1071 | Scardigli and Casadio (2015) |
| Solar system data | 1069 | Scardigli and Casadio (2015) |
| GW150914 | 1060 | Feng et al. (2017) |
| Dresselhaus interaction | 1051 | Aghababaei et al. (2021) |
| Landau levels | 1050 | Das and Vagenas (2008) |
| Sagnac effect | 1049 | Feleppa et al. (2021) |
| Rashba effect | 1046 | Aghababaei et al. (2021) |
Some bounds on the GUP parameter β0.
Expanding the above solutions (12) and bearing in mind that the true solution should recover at β = 0, one can easily find that is the proper solution leading toup to first order in β0. Hence, because it seems that β0 is positive (Das and Vagenas, 2008; Scardigli and Casadio, 2015; Feng et al., 2017; Aghababaei et al., 2021; Feleppa et al., 2021), one can conclude that .
Conclusion
Motivated by the GUP proposal and the vital role of the HUP in quantum mechanics and, thus, the quantum tunneling process that facilitates star burning, we studied the effects of the GUP on the Gamow temperature. In order to determine this, the GUP modification to the de Broglie wavelength was addressed, which finally helped us to find the GUP correction to the Gamow temperature and also estimate an upper bound for β0 (1046), which agrees well with those found in previous works (Das and Vagenas, 2008; Scardigli and Casadio, 2015; Feng et al., 2017; Aghababaei et al., 2021; Feleppa et al., 2021).
Finally, based on the obtained results, it may be expected that the GUP also affects the transmission coefficients (Gamow’s formula) (Gamow, 1928; Hossenfelder, 2013; Völkel et al., 2019), meaning that the method of Völkel et al. (2019) will also be affected. This is an interesting topic for future study because Hawking radiation is a fascinating issue in black hole physics (Wald, 2001).
Statements
Data availability statement
The original contributions presented in the study are included in the article/Supplementary material. Further inquiries can be directed to the corresponding author.
Author contributions
All authors listed have made a substantial, direct, and intellectual contribution to the work and approved it for publication.
Acknowledgments
The authors would like to appreciate the anonymous referees for their valuable comments.
Conflict of interest
The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.
Publisher’s note
All claims expressed in this article are solely those of the authors and do not necessarily represent those of their affiliated organizations or those of the publisher, the editors, and the reviewers. Any product that may be evaluated in this article, or claim that may be made by its manufacturer, is not guaranteed or endorsed by the publisher.
References
1
AghababaeiS.MoradpourH.RezaeiG.KhorshidianS. (2021). Minimal length, Berry phase and spin-orbit interactions. Phys. Scr.96, 055303. 10.1088/1402-4896/abe5d2
2
BossoP.DasS.PikovskiI.VannerM. R. (2017). Amplified transduction of Planck-scale effects using quantum optics. Phys. Rev. A96, 023849. 10.1103/physreva.96.023849
3
BushevP. A.BourhillJ.GoryachevM.KukharchykN.IvanovE.GalliouS.et al (2019). Testing the generalized uncertainty principle with macroscopic mechanical oscillators and pendulums. Phys. Rev. D.100, 066020. 10.1103/physrevd.100.066020
4
ChangL. N.MinicD.OkamuraD.TakeuchiT. (2002). Exact solution of the harmonic oscillator in arbitrary dimensions with minimal length uncertainty relations. Phys. Rev. D.65, 125027. 10.1103/physrevd.65.125027
5
ChemissanyW.DasS.AliA. F.VagenasE. C. (2011). Effect of the generalized uncertainty principle on post-inflation preheating. J. Cosmol. Astropart. Phys.1112, 017. 10.1088/1475-7516/2011/12/017
6
DasS.MannR. B. (2011). Planck scale effects on some low energy quantum phenomena. Phys. Lett. B704, 596–599. 10.1016/j.physletb.2011.09.056
7
DasS.VagenasE. C. (2008). Universality of quantum gravity corrections. Phys. Rev. Lett.101, 221301. 10.1103/physrevlett.101.221301
8
FeleppaF.MoradpourH.CordaC.AghababaeiS. (2021). Constraining the generalized uncertainty principle with neutron interferometry. EPL135, 40003. 10.1209/0295-5075/ac1240
9
FengZ. W.YangSh. Z.LiH. L.ZuX. T. (2017). Constraining the generalized uncertainty principle with the gravitational wave event GW150914. Phys. Lett. B768, 81–85. 10.1016/j.physletb.2017.02.043
10
FityoT. (2008). Statistical physics in deformed spaces with minimal length. Phys. Lett. A372, 5872–5877. 10.1016/j.physleta.2008.07.047
11
GamowG. (1928). Z. Phys.51, 204–212. 10.1007/bf01343196
12
GecimG.SucuY. (2017). The GUP effect on Hawking radiation of the 2 + 1 dimensional black hole. Phys. Lett. B773, 391–394. 10.1016/j.physletb.2017.08.053
13
GhoshS. (2014). Quantum gravity effects in geodesic motion and predictions of equivalence principle violation. Cl. Quantum Gravity31, 025025. 10.1088/0264-9381/31/2/025025
14
HossenfelderS. (2013). Minimal length scale scenarios for quantum gravity. Living Rev. Relativ.16, 2. 10.12942/lrr-2013-2
15
HusainV.SeahraS. S.WebsterE. J. (2013). High energy modifications of blackbody radiation and dimensional reduction. Phys. Rev. D.88, 024014. 10.1103/physrevd.88.024014
16
JalalzadehS.GorjiM. A.NozariK. (2014). Deviation from the standard uncertainty principle and the dark energy problem. Gen. Relativ. Gravit.46, 1632. 10.1007/s10714-013-1632-8
17
KempfA.ManganoG.MannR. B. (1995). Hilbert space representation of the minimal length uncertainty relation. Phys. Rev. D.52, 1108–1118. 10.1103/physrevd.52.1108
18
KempfA. (1996). Noncommutative geometric regularization. Phys. Rev. D.54, 5174–5178. 10.1103/physrevd.54.5174
19
LakeM. J. (2021). Quantum Rep.3, 196–227. 10.3390/quantum3010012
20
LakeM. J.MillerM.GanardiR. F.LiuZ.LiangS. D.PaterekT. (2019). Generalised uncertainty relations from superpositions of geometries. Cl. Quantum Gravity36 (15), 155012. 10.1088/1361-6382/ab2160
21
LakeM. J.MillerM.LiangS. D. (2020). Generalised uncertainty relations for angular momentum and spin in quantum geometry. Universe6, 56. 10.3390/universe6040056
22
LakeM. J. (2022). To appear in touring the Planck scale. Antonio Aurilia Memorial. Netherlands: Springer. Available at: https://arxiv.org/abs/2008.13183.
23
LucianoG. G.PetruzzielloL. (2019). GUP parameter from maximal acceleration. Eur. Phys. J. C79, 283. 10.1140/epjc/s10052-019-6805-5
24
MeadC. A. (1964). Possible connection between gravitation and fundamental length. Phys. Rev.135, 849–B862. 10.1103/physrev.135.b849
25
Mohammadi SabetM.MoradpourH.BahadoranM.ZiaieA. H. (2021). Minimal length implications on the Hartree–Fock theory. Phys. Scr.96, 125016. 10.1088/1402-4896/ac2c21
26
MoradpourH.AghababaeiS.ZiaieA. H. (2021). A note on effects of generalized and extended uncertainty principles on jüttner gas. Symmetry13, 213. 10.3390/sym13020213
27
MoradpourH.ZiaieA. H.GhaffariS.FeleppaF. (2019). The generalized and extended uncertainty principles and their implications on the Jeans mass. Mon. Notices R. Astronomical Soc. Lett.488 (1), L69–L74. 10.1093/mnrasl/slz098
28
MotlaqM. A.PedramP. (2014). J. Stat. Mech.P08002.
29
ParkD., arXiv. arXiv:2003.13856 (2020).
30
PikovskiI.VannerM. R.AspelmeyerM.KimM. S.BruknerC. (2012). Probing Planck-scale physics with quantum optics. Nat. Phys.8, 393–397. 10.1038/nphys2262
31
PrialnikD. (2000). An introduction to the theory of stellar structure and evolution. Cambridge: Cambridge University Press.
32
ScardigliF.CasadioR. (2015). Gravitational tests of the generalized uncertainty principle. Eur. Phys. J. C75, 425. 10.1140/epjc/s10052-015-3635-y
33
ShababiH.OurabahK. (2020). Non-Gaussian statistics from the generalized uncertainty principle. Eur. Phys. J. Plus135, 697. 10.1140/epjp/s13360-020-00726-9
34
SprengerM.BleicherM.NicoliniP. (2011). Neutrino oscillations as a novel probe for a minimal length. Cl. Quantum Gravity28, 235019. 10.1088/0264-9381/28/23/235019
35
VölkelS. H.KonoplyaR.KokkotasK. D. (2019). Inverse problem for Hawking radiation. Phys. Rev. D.99, 104025. 10.1103/physrevd.99.104025
36
WaldR. M. (2001). The thermodynamics of black holes. Living Rev. Rel.4, 6. 10.12942/lrr-2001-6
37
WangP.YangH.ZhangX. J., High energy phys, 1 ( 2010).
38
ZhuT.RenJ. R.LiM. F. (2009). Influence of generalized and extended uncertainty principle on thermodynamics of FRW universe. Phys. Lett. B674, 204–209. 10.1016/j.physletb.2009.03.020
Summary
Keywords
quantum gravity, minimal length, generalized uncertainty principle, Gamow theory, stellar formation
Citation
Moradpour H, Ziaie AH, Sadeghnezhad N and Ghasemi A (2022) Generalized uncertainty principle and burning stars. Front. Astron. Space Sci. 9:936352. doi: 10.3389/fspas.2022.936352
Received
05 May 2022
Accepted
02 August 2022
Published
09 September 2022
Volume
9 - 2022
Edited by
Matthew J. Lake, National Astronomical Research Institute of Thailand, Thailand
Reviewed by
Christian Corda, B. M. Birla Science Centre, India
Izzet Sakalli, Eastern Mediterranean University, Turkey
Updates
Copyright
© 2022 Moradpour, Ziaie, Sadeghnezhad and Ghasemi.
This is an open-access article distributed under the terms of the Creative Commons Attribution License (CC BY). The use, distribution or reproduction in other forums is permitted, provided the original author(s) and the copyright owner(s) are credited and that the original publication in this journal is cited, in accordance with accepted academic practice. No use, distribution or reproduction is permitted which does not comply with these terms.
*Correspondence: H. Moradpour, h.moradpour@riaam.ac.ir; A. H. Ziaie, ah.ziaie@maragheh.ac.ir; N. Sadeghnezhad, nsadegh@maragheh.ac.ir
This article was submitted to High-Energy and Astroparticle Physics, a section of the journal Frontiers in Astronomy and Space Sciences
Disclaimer
All claims expressed in this article are solely those of the authors and do not necessarily represent those of their affiliated organizations, or those of the publisher, the editors and the reviewers. Any product that may be evaluated in this article or claim that may be made by its manufacturer is not guaranteed or endorsed by the publisher.