Theoretical and Observational Implications of Plancks Constant as a Running Fine Structure Constant Ahmed Farag AliJonas MureikaElias C. Vagenasand Ibrahim Elmashad

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Theoretical and Observational Implications of Planck’s Constant as a Running Fine

Structure Constant

Ahmed Farag Ali ∇△ ,∗Jonas Mureika□,⃝,†Elias C. Vagenas⊕,‡and Ibrahim Elmashad △ §

△Department of Physics, Benha University, Benha 13518, Egypt

∇Essex County College, 303 University Ave, Newark, NJ, USA 07102

□Department of Physics, Loyola Marymount University, 1 LMU Drive, Los Angeles, CA, USA 90045

⃝Kavli Institute for Theoretical Physics, University of California, Santa Barbara, CA, USA 93106 and

⊕Department of Physics, College of Science, Kuwait University,

Sabah Al Salem University City, P.O. Box 2544, Safat 1320, Kuwait

This letter explores how a reinterpretation of the generalized uncertainty principle as an eﬀective

variation of Planck’s constant provides a physical explanation for a number of fundamental quantities

and couplings. In this context, a running ﬁne structure constant is naturally emergent and the

cosmological constant problem is solved, yielding a novel connection between gravitation and

quantum ﬁeld theories. The model could potentially clarify the recent experimental observations

by the DESI Collaboration that could imply a fading of dark energy over time. When applied to

quantum systems and their characteristic length scales, a simple geometric relationship between

energy and entropy is disclosed. Lastly, a mass-radius relation for both quantum and classical

systems reveals a phase transition-like behaviour similar to thermodynamical systems, which we

speculate to be a consequence of topological defects in the universe.

I. INTRODUCTION

Various approaches to quantum gravity such as string

theory, loop quantum gravity, and quantum geometry

suggest a generalized form of the uncertainty principle

(GUP) that implies the existence of a minimal

measurable length. Several forms of the GUP that

include non-relativistic and relativistic forms have been

proposed [1–8], which can be collectively written in the

form :

∆x∆p≥

ℏ

2f(p, x).(1)

Phenomenological and experimental implications of the

GUP have been investigated in low and high-energy

regimes. These include atomic systems [9, 10], quantum

optical systems [11], gravitational bar detectors [12],

gravitational decoherence [13, 14], composite particles

[15], astrophysical systems [16], condensed matter

systems [17], and macroscopic harmonic oscillators

[18]. Reviews of the GUP, its phenomenology, and its

experimental implications can be found in Refs. [19, 20].

Recently, we proposed a reinterpretation of the GUP

using an eﬀective Planck constant [21] by absorbing

the additional momentum uncertainty dependence, ℏ′=

ℏf(p, x). This implies a generic GUP of the form:

∆x∆p≥

ℏ′

2.(2)

∗email: aali29@essex.edu ; ahmed.ali@fsc.bu.edu.eg

†email: jmureika@lmu.edu

‡email: elias.vagenas@ku.edu.kw

§email: ibrahim.elmashad@fsc.bu.edu.eg

Previously, in Ref. [22], the GUP was conceptualized as

an eﬀective variation of the Planck constant by isolating

an invariant phase space with minimal length in the

context of Liouville’s theorem. Planck constant was

proposed to vary with momentum in Ref. [23], where

the authors introduced a generalized picture of the

de Broglie relation and derived a form of generalized

uncertainty principle which is similar to the one obtained

in string theory. In Ref. [21], we argued that the charge

radii of hadrons/nuclei along with their corresponding

masses support the existence of an eﬀective variation

of ℏthat suggests a universality of a minimal length

in the associated scattering process. We suggested

a relation that simulates the fundamental connection

between nature constants (ℓPMPc=ℏ,) by replacing

the Planck length ℓPby the charge radius (r) and the

Planck mass MPby the mass of the hadron/nuclei (m).

This relation reads :

r m c =ℏ′,(3)

where ris the charge radius of the particle, mis the

particle’s relativistic mass 1,cis the speed of light and

ℏ′is the eﬀective Planck constant. Here, cis a constant to

maintain consistency with the theory of relativity. It was

further shown in [21] that applying Eq. (3) to a variety of

hadronic particles, as well as to larger nuclei, a clear trend

in the eﬀective ℏ′was apparent. It was suggested this

eﬀective variation of Planck constant might be related to

the timeless state of the universe [24]. In this letter,

we propose a connection between Eq. (3) and the

universal Bekenstein bound [25] (Sec. II). Furthermore,

1The particle’s relativistic mass mis given by m=

m0/p1−v2/c2, where m0is the rest mass and vis the particle’s

speed.

arXiv:2210.06262v3 [physics.gen-ph] 6 Jun 2024

we demonstrate that in the case of the electron, a clear

connection arises with the value of ﬁne structure constant

(Sec.III). In this sense, the eﬀective variation of Planck

constant is interpreted as a running coupling. We explain

the value of the cosmological constant [26] (Sec. IV).

In addition, we investigate the conceptual connection

between the eﬀective Planck constant and the second

law of thermodynamics (Sec. V). Next, we study the

conceptual connection between our formula and both

the de Broglie and Compton wavelengths (Sec. VI).

Lastly, we present a graphical study of the mass-radius

relation, which shows phase transition behavior that may

be a consequence of topological defects in the universe.

(Sec. VII).

II. THE UNIVERSAL BEKENSTEIN BOUND

The Bekenstein bound is deﬁned as the maximal amount

of information contained in a physical system. That is,

if a physical system has ﬁnite energy and is contained in

a ﬁnite space, it must be described by a ﬁnite amount of

information [25]. Formally, the bound can be written

S≤2πkBrE

ℏc,(4)

where Sis the entropy of the physical system, kBis

the Boltzmann constant, ris the radius of a sphere

that encloses the physical system, and Eis its energy.

Replacing E=mc2 2, this becomes

S≤2π kBrmc

ℏ.(5)

One may wonder what happens for the massless particles

such as photons. In this case, we get E=cp instead of

E=mc2. It is noteworthy that Eq. (3) is naturally

included in the above inequality. Therefore, the bound

can be rewritten as

S≤2π kB

ℏ′

ℏ.(6)

Replacing the thermodynamic entropy Swith the

Shannon entropy H[27],

S=kBHln 2 .(7)

where Hsigniﬁes the Shannon entropy, calculated in

terms of the number of bits embedded in the quantum

states inside the sphere. The ln2 factor comes into play

because we interpret information as the base-2 logarithm

of the total number of quantum states [28]. We can

2The quantity mis the relativistic mass as deﬁned in footnote 1.

rewrite the bound as

H≤2π

ln 2

ℏ′

ℏ.(8)

This is the maximal amount of information required to

perfectly describe a physical object up to the quantum

level [25]. Eﬀectively, Eq. (3) introduces a novel

way to merge the universal Bekenstein bound with

quantum ﬁeld theory (QFT), through considering the

eﬀective Planck constant ℏ′for every physical object.

The relationship mrc =ℏ′corresponds intriguingly

to Bekenstein’s bound, a universal limit applicable

to any physical system for its complete description

at the quantum level. Stemming from gravitational

insights, Bekenstein’s bound serves as a natural cutoﬀ

that varies based on the speciﬁc physical system being

studied. The fact that it behaves like a natural limit

led us to propose a connection with renormalization

in QFT. Renormalization is a technique employed in

QFT to accurately describe physical systems at the

quantum level. This is commonly achieved by presuming

a cutoﬀ or employing mathematical techniques that

provide a cutoﬀ such as counter terms or dimensional

regularization that sets limits in order to get ﬁnite values

instead of inﬁnities. Having uncovered Bekenstein’s

bound and its implications, we’re now seeing a new

original meaning for our equation. This overlap provides

compelling context to the interpretation of mrc =ℏ′as

a potential gravitational explanation for renormalization

in physics. We expand on this connection in the following

sections.

III. A RUNNING FINE STRUCTURE

COUPLING

The ﬁne structure constant αdescribes the fundamental

coupling between two electrically charged particles. It

is one of the most accurately measured quantities in

physics, with a current experimental value of α=

0.0072973525693 at a precision of 8.1×10−11 [29].

Originally conceived as a coupling that gauges the basic

electromagnetic interaction between the electron and

proton, the ﬁne structure constant can be viewed as a

quantization of the energy distribution of electrons in the

atom. Fundamentally, atomic structure is an artifact of

quantum mechanics. As such, we posit that the eﬀective

Planck constant for the electron should explain the value

of α. Using the electron mass me= 9.1093837×10−31 kg

and its classical radius re= 2.8179403262 ×10−15 [29],

we ﬁnd that eﬀective Planck constant to be

ℏ′

e=merec= 0.007297352571 ℏ=α′

eℏ,(9)

where α′

e= 0.0072973525710 is the ﬁne structure

constant for the electron obtained from our model. This

agrees with the experimentally measured value to 10

decimal places (α= 0.0072973525693). We infer this

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TheoreticalandObservationalImplicationsofPlanck’sConstantasaRunningFineStructureConstantAhmedFaragAli∇△,∗JonasMureika□,⃝,†EliasC.Vagenas⊕,‡andIbrahimElmashad△§△DepartmentofPhysics,BenhaUniversity,Benha13518,Egypt∇EssexCountyCollege,303UniversityAve,Newark,NJ,USA07102□DepartmentofPhysics,LoyolaMarymo...

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Theoretical and Observational Implications of Plancks Constant as a Running Fine Structure Constant Ahmed Farag AliJonas MureikaElias C. Vagenasand Ibrahim Elmashad

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