Front. Astron. Space Sci. Frontiers in Astronomy and Space Sciences Front. Astron. Space Sci. 2296-987X Frontiers Media S.A. 936352 10.3389/fspas.2022.936352 Astronomy and Space Sciences Brief Research Report Generalized uncertainty principle and burning stars Moradpour et al. 10.3389/fspas.2022.936352 Moradpour H. * Ziaie A. H. * Sadeghnezhad N. * Ghasemi A. Research Institute for Astronomy and Astrophysics of Maragha (RIAAM), University of Maragheh, Maragheh, Iran

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

 *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

 09 09 2022 2022 9 936352 05 05 2022 02 08 2022 Copyright © 2022 Moradpour, Ziaie, Sadeghnezhad and Ghasemi. 2022 Moradpour, Ziaie, Sadeghnezhad and Ghasemi

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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.

 quantum gravity minimal length generalized uncertainty principle Gamow theory stellar formation 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 gets 1 2 m v 2 = 3 2 K B T ≥ U c r 0 , where K B denotes the Boltzmann constant, the subscript c in U c (r 0) indicates the Coulomb potential, and correspondingly, U c ( r 0 ) = Z i Z j e 2 r 0 denotes the maximum of the Coulomb potential between the ith and jth particles located at a distance r 0 from each other (Prialnik, 2000). In this article, Kelvin (K) is the temperature unit. Finally, we reach T ≥ 2 Z i Z j e 2 3 K B r 0 ≃ 1 ⋅ 1 × 1 0 10 Z i Z j r 0 , for 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) T ≈ 4 × 1 0 6 M M ⊙ R ⊙ R , where M ⊙ and R ⊙ are the Sun mass and radius, respectively. Clearly, T 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 (r 0) becomes of the order of their de Broglie wavelength ( r 0 ≃ ℏ p ≡ λ Q where Q implies that we are in the purely quantum mechanical regime), then quantum tunneling happens and simple calculations lead to (Prialnik, 2000) T ≥ 2 Z i Z j e 2 3 K B λ Q ≃ 9 ⋅ 6 × 1 0 6 Z i 2 Z j 2 m 1 2 ≡ T , instead of Eq. 2 for the temperature required to launch star burning. λ Q can also be obtained by solving p 2 2 m = U c ( r 0 ) r 0 = λ Q which gives (Prialnik, 2000) λ Q = ℏ 2 2 m Z i Z j e 2 , 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 T ≃ 9 ⋅ 6 × 1 0 6 K (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 λ Q 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), Δ x Δ p ≥ ℏ 2 , where x and p are ordinary canonical coordinates satisfying [x i , p j ] = iℏδ ij . It has been proposed that, in quantum gravity scenarios, the HUP is modified such that (Kempf et al., 1995; Kempf, 1996) Δ X Δ P ≥ ℏ 2 1 + β 0 ł p 2 ℏ 2 Δ P 2 , called the GUP, where l p denotes the Planck length and β 0 is the dimensionless GUP parameter. X and P are called generalized coordinates, and we work in a framework in which X i = x i , and up to the first order, we have P i = p i ( 1 + β 0 ł p 2 3 ℏ 2 p 2 ) and [ X i , P j ] = i ℏ ( 1 + β 0 ł p 2 ℏ 2 P 2 ) δ i j (Das and Vagenas, 2008; Motlaq and Pedram, 2014). Moreover, the GUP implies that there is a non-zero minimum length ( Δ X ) min = β 0 ł p . 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 β 0 (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 T (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 as λ GUP ≡ ℏ P . It is obvious that, as β 0 → 0, one obtains P → p and thus λ GUP → λ Q , which is the quantum mechanical result. Indeed, up to first order in β 0, we have λ GUP = λ Q ( 1 − β 0 l p 2 3 λ Q 2 ) , and the thermal energy per particle with temperature T is (Motlaq and Pedram, 2014) 〈 K 〉 = 〈 P 2 2 m 〉 = 3 2 K B T − 3 β 0 ł p 2 ℏ 2 m K B 2 T 2 . Mathematically, one should find the corresponding de Broglie wavelength by solving the following equation: P 2 2 m = U c r 0 r 0 = λ GUP . Inserting the result into 3 2 K B T − 3 β 0 ł p 2 ℏ 2 m K B 2 T 2 ≥ U c r 0 r 0 = λ GUP , one can finally find the GUP corrected version of Eq. 4.

Now, inserting λ GUP into Eq. 10 and then combining the results with Eq. 11, we find T GUP ± = ℏ 2 1 ± 1 − 8 β 0 l p 2 m K B T / ℏ 2 4 β 0 K B l p 2 m . in which Eq. 4 has been used for simplification. To estimate the magnitude of l p 2 m K B T / ℏ 2 , we consider the hydrogen atom for which m ∼ 10–27 kg. Now, since l p ∼ 10–35 m, K B ∼ 1 0 − 23 m 2 k g s 2 K , ℏ ∼ 1 0 − 34 m 2 k g s , and T ∼ 1 0 6 K , one easily finds l p 2 m K B T / ℏ 2 ∼ 1 0 − 46 . Moreover, because the effects of the GUP in the quantum mechanical regime are small (Hossenfelder, 2013), a reasonable basic assumption could be that β 0 l p 2 m K B T / ℏ 2 ≪ 1 . Indeed, if β 0 ≪ 1046, then we always have β 0 l p 2 m K B T / ℏ 2 ≪ 1 meaning that i) we can Taylor expand our results and ii) 1046 is an upper bound for β 0, 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.

Some bounds on the GUP parameter β 0.

Measurement/experiment β 0 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)

Expanding the above solutions (12) and bearing in mind that the true solution should recover T at β = 0, one can easily find that T GUP − is the proper solution leading to T GUP − = T 1 + 2 β 0 l p 2 m K B T ℏ 2 . up to first order in β 0. Hence, because it seems that β 0 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 T < T GUP − .

 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 β 0 (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).

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The authors would like to appreciate the anonymous referees for their valuable comments.

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 References Aghababaei S. Moradpour H. Rezaei G. Khorshidian S. (2021). Minimal length, Berry phase and spin-orbit interactions. Phys. Scr. 96, 055303. 10.1088/1402-4896/abe5d2 Bosso P. Das S. Pikovski I. Vanner M. R. (2017). Amplified transduction of Planck-scale effects using quantum optics. Phys. Rev. A 96, 023849. 10.1103/physreva.96.023849 Bushev P. A. Bourhill J. Goryachev M. Kukharchyk N. Ivanov E. Galliou S. (2019). Testing the generalized uncertainty principle with macroscopic mechanical oscillators and pendulums. Phys. Rev. D. 100, 066020. 10.1103/physrevd.100.066020 Chang L. N. Minic D. Okamura D. Takeuchi T. (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 Chemissany W. Das S. Ali A. F. Vagenas E. 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 Das S. Mann R. B. (2011). Planck scale effects on some low energy quantum phenomena. Phys. Lett. B 704, 596–599. 10.1016/j.physletb.2011.09.056 Das S. Vagenas E. C. (2008). Universality of quantum gravity corrections. Phys. Rev. Lett. 101, 221301. 10.1103/physrevlett.101.221301 Feleppa F. Moradpour H. Corda C. Aghababaei S. (2021). Constraining the generalized uncertainty principle with neutron interferometry. EPL 135, 40003. 10.1209/0295-5075/ac1240 Feng Z. W. Yang Sh. Z. Li H. L. Zu X. T. (2017). Constraining the generalized uncertainty principle with the gravitational wave event GW150914. Phys. Lett. B 768, 81–85. 10.1016/j.physletb.2017.02.043 Fityo T. (2008). Statistical physics in deformed spaces with minimal length. Phys. Lett. A 372, 5872–5877. 10.1016/j.physleta.2008.07.047 Gamow G. (1928). Z. Phys. 51, 204–212. 10.1007/bf01343196 Gecim G. Sucu Y. (2017). The GUP effect on Hawking radiation of the 2 + 1 dimensional black hole. Phys. Lett. B 773, 391–394. 10.1016/j.physletb.2017.08.053 Ghosh S. (2014). Quantum gravity effects in geodesic motion and predictions of equivalence principle violation. Cl. Quantum Gravity 31, 025025. 10.1088/0264-9381/31/2/025025 Hossenfelder S. (2013). Minimal length scale scenarios for quantum gravity. Living Rev. Relativ. 16, 2. 10.12942/lrr-2013-2 Husain V. Seahra S. S. Webster E. J. (2013). High energy modifications of blackbody radiation and dimensional reduction. Phys. Rev. D. 88, 024014. 10.1103/physrevd.88.024014 Jalalzadeh S. Gorji M. A. Nozari K. (2014). Deviation from the standard uncertainty principle and the dark energy problem. Gen. Relativ. Gravit. 46, 1632. 10.1007/s10714-013-1632-8 Kempf A. Mangano G. Mann R. B. (1995). Hilbert space representation of the minimal length uncertainty relation. Phys. Rev. D. 52, 1108–1118. 10.1103/physrevd.52.1108 Kempf A. (1996). Noncommutative geometric regularization. Phys. Rev. D. 54, 5174–5178. 10.1103/physrevd.54.5174 Lake M. J. (2021). Quantum Rep. 3, 196–227. 10.3390/quantum3010012 Lake M. J. Miller M. Ganardi R. F. Liu Z. Liang S. D. Paterek T. (2019). Generalised uncertainty relations from superpositions of geometries. Cl. Quantum Gravity 36 (15), 155012. 10.1088/1361-6382/ab2160 Lake M. J. Miller M. Liang S. D. (2020). Generalised uncertainty relations for angular momentum and spin in quantum geometry. Universe 6, 56. 10.3390/universe6040056 Lake M. J. (2022). To appear in touring the Planck scale. Antonio Aurilia Memorial. Netherlands: Springer. Available at: https://arxiv.org/abs/2008.13183 . Luciano G. G. Petruzziello L. (2019). GUP parameter from maximal acceleration. Eur. Phys. J. C 79, 283. 10.1140/epjc/s10052-019-6805-5 Mead C. A. (1964). Possible connection between gravitation and fundamental length. Phys. Rev. 135, 849–B862. 10.1103/physrev.135.b849 Mohammadi Sabet M. Moradpour H. Bahadoran M. Ziaie A. H. (2021). Minimal length implications on the Hartree–Fock theory. Phys. Scr. 96, 125016. 10.1088/1402-4896/ac2c21 Moradpour H. Aghababaei S. Ziaie A. H. (2021). A note on effects of generalized and extended uncertainty principles on jüttner gas. Symmetry 13, 213. 10.3390/sym13020213 Moradpour H. Ziaie A. H. Ghaffari S. Feleppa F. (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 Motlaq M. A. Pedram P. (2014). J. Stat. Mech. P08002. Park D. , arXiv. arXiv:2003.13856 (2020). Pikovski I. Vanner M. R. Aspelmeyer M. Kim M. S. Brukner C. (2012). Probing Planck-scale physics with quantum optics. Nat. Phys. 8, 393–397. 10.1038/nphys2262 Prialnik D. (2000). An introduction to the theory of stellar structure and evolution. Cambridge: Cambridge University Press. Scardigli F. Casadio R. (2015). Gravitational tests of the generalized uncertainty principle. Eur. Phys. J. C 75, 425. 10.1140/epjc/s10052-015-3635-y Shababi H. Ourabah K. (2020). Non-Gaussian statistics from the generalized uncertainty principle. Eur. Phys. J. Plus 135, 697. 10.1140/epjp/s13360-020-00726-9 Sprenger M. Bleicher M. Nicolini P. (2011). Neutrino oscillations as a novel probe for a minimal length. Cl. Quantum Gravity 28, 235019. 10.1088/0264-9381/28/23/235019 Völkel S. H. Konoplya R. Kokkotas K. D. (2019). Inverse problem for Hawking radiation. Phys. Rev. D. 99, 104025. 10.1103/physrevd.99.104025 Wald R. M. (2001). The thermodynamics of black holes. Living Rev. Rel. 4, 6. 10.12942/lrr-2001-6 Wang P. Yang H. Zhang X. J. , High energy phys, 1 ( 2010). Zhu T. Ren J. R. Li M. F. (2009). Influence of generalized and extended uncertainty principle on thermodynamics of FRW universe. Phys. Lett. B 674, 204–209. 10.1016/j.physletb.2009.03.020