- Email: [email protected]
Comments on the paper “Relativistic generalized uncertainty principle”
Journal Pre-proof Comments on the paper “Relativistic generalized uncertainty principle” Yassine Chargui
PII: DOI: Reference:
S0003-4916(19)30262-3 https://doi.org/10.1016/j.aop.2019.168007 YAPHY 168007
To appear in:
Annals of Physics
Received date : 7 October 2019 Accepted date : 8 October 2019 Please cite this article as: Y. Chargui, Comments on the paper “Relativistic generalized uncertainty principle”, Annals of Physics (2019), doi: https://doi.org/10.1016/j.aop.2019.168007. This is a PDF file of an article that has undergone enhancements after acceptance, such as the addition of a cover page and metadata, and formatting for readability, but it is not yet the definitive version of record. This version will undergo additional copyediting, typesetting and review before it is published in its final form, but we are providing this version to give early visibility of the article. Please note that, during the production process, errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
© 2019 Elsevier Inc. All rights reserved.
Journal Pre-proof
Comments on the paper ”Relativistic generalized uncertainty principle” Yassine Chargui a,b,∗ Department, College of Science and Arts at ArRass, Qassim University, PO Box 53, ArRass 51921, Saudi Arabia
pro of
a Physics
b Universit´ e
de Tunis El Manar, Facult´e des Sciences de Tunis, Unit´e de Recherche de Physique Nucl´eaire et des Hautes Energies, 2092 Tunis, Tunisie
Abstract
re-
We point out some misleading results reported in the recent study made by V. Todorinov et al. [1], concerning a relativistic extension of the generalized uncertainty principle (GUP). We derive, in this frame, the correct deformed Klein-Gordon (KG) and Dirac equations, valid up to the first order in the deformation parameter, and discuss their minimal coupling to external fields.
[X µ , P ν ] = i~
n
lP
In a recent paper published in this Journal [1], V. Todorinov et al considered a Lorentz covariant deformed algebra, where the commutation relations between the four-position and the four-momentum operators are given by
1 + (ε − α)γ 2 P ρ Pρ η µν + (β + 2ε)γ 2 P µ P ν
o
(1)
urn a
where α, β and ε are dimensionless parameters and γ = 1/cMpl , with Mpl the Planck mass. η µν = diag(−, +, +, +) is the metric tensor in the (3 + 1)dimensional Minkowski space-time. It should be noted that the commutation relations in Eq.(1) are exactly the same as those studied by Quesne and Tkachuk in Refs.[2,3], who used the notations β and β 0 to designate the two independent deformation parameters, such as β = (ε−α)γ 2 and β 0 = (β+ε)γ 2 . Then the authors focused on the particular case where ε = α, and adopted the following representations of X µ and P µ :
Jo
X µ = xµ 1 − αγ 2 pρ pρ ,
P µ = pµ 1 + αγ 2 pρ pρ
(2)
where the small letters x = (x0 , x) and p = (p0 , p) are used, throughout this work, to designate the canonically conjugate operators of four-position ∗ Corresponding author Email: [email protected] Preprint submitted to Elsevier Science
7 October 2019
Journal Pre-proof and four-momentum, respectively. These operators satisfy the commutation relations [xµ , pν ] = i~η µν (3) Moreover, henceforth, we shall simply use the symbol α in order to refer to αγ 2 . Now, the deformed free KG equation, valid up to the first order in α, follows merely from the invariance of the squared physical momentum, that is from the relation P µ Pµ = −m2 c2 , with m the mass of the particle. Thus, using Eq.(2) for the realization of the operator P µ , and retaining only terms of first order in α, this relation results in the equation n
o
pro of
pµ pµ + 2α (pµ pµ )2 ψ = −m2 c2 ψ
(4)
where ψ is the KG wave function. Then, Eq.(4) is equivalent to n
o
1 − 4αp20 p2 + 2αp4 − 1 − 2αp20 p20 = −m2 c2 ψ
(5)
Furthermore, the KG equation, in the position-space representation, stems ∂ ∇ into Eq.(5). This yields and p = −i~∇ from the substitutions p0 = i~c ∂t ~2 ∂ 2 1 + 2α 2 2 c ∂t
!
!
∂2 ~2 ∂ 2 2 − c 1 + 4α ∇2 ∂t2 c2 ∂t2 ) m2 c4 2 2 4 ψ=0 +2α~ c ∇ + 2 ~
re-
(
(6)
lP
This equation differs drastically from that obtained in Ref.[1]. Therein Eq.(4) was regarded as a simple algebraic equation and then solved for pµ pµ , to the first order in α. This had led to an inadequate GUP-modified KG equation. The latter is very similar to the non-deformed one, but just with a modified mass (Eq.(20) in Ref.[1]).
urn a
Next, we discuss the derivation of the GUP-deformed KG equation in the presence of a minimally-coupled electromagnetic field, with a four-potential Aµ . This will involve the replacement P µ → P µ − qAµ into the free GUPdeformed KG equation, where q is the electric charge of the considered particle. Hence, one has to start from the equation (P µ − qAµ ) (Pµ − qAµ ) ψ = −m2 c2 ψ
(7)
Then, we use Eq.(2) and expand to the first order in α. This gives n
Jo
pµ pµ + 2α (pµ pµ )2 + q 2 Aµ Aµ − q (2Aµ pµ + [pµ , Aµ ]) (1 + αpρ pρ )
−αqpµ (pρ [pρ , Aµ ] + [pρ , Aµ ] pρ )} ψ = −m2 c2 ψ
(8)
Particularly, in the case where qAµ = (V (X) /c, 0), with V (X) a static potential energy (i.e., independent of X 0 ), Eq.(9) reduces to 2
Journal Pre-proof n
c2 pµ pµ + 2αc2 (pµ pµ )2 − V (X)2 + 2cV (X) (1 + αpρ pρ ) p0 o
+αc (pρ [pρ , V (X)] + [pρ , V (X)] pρ ) p0 ψ = −m2 c4 ψ
(9)
pro of
In addition, according to Eq.(2), the physical three-position operator X, is now a function of the non-deformed three-position operator x and four-momentum operator p. For instance, the distance R = |X|, expanded to the first order in α, reads √ α R = X2 = r 1 − αpρ pρ + i~ 2 x.p (10) r Thus, the development of the Coulomb potential V (X) = −κ/ |X|, valid to the leading order in α, turns to be
κ α V (X) = − 1 + αp pρ − i~ 2 x.p r r ρ
(11)
All the above considerations were not part of the procedure followed by V. Todorinov et al [1], when studying the effect of the GUP on the relativistic hydrogen atom using the KG equation.
re-
Furthermore, throughout their work, the authors of [1] have employed the relation E = E0 (1 + αpρ pρ ) = E0 1 − αm2 c2 (12)
D
E = P0
E
lP
in order to calculate the GUP corrections to the physical energy E from those to the non-deformed energy, denoted by E0 . However, we believe this is an incorrect reasoning. As a matter of fact, the physical energy E, corresponding to a stationary state ψ, should be worked out as the expectation value of the operator energy P 0 , in the state ψ. Then, using Eq.(2), this yields ψ
D
= E0 1 − αE02 + αE0 c2 p2
E
ψ
(13)
urn a
where E0 is the non-deformed energy associated to the state ψ. That is ψ is of the form ψ = e−iE0 t/~ φ where φ is a time-independent state. We turn now to discuss briefly the derivation of the GUP-deformed Dirac equation. For a free particle of mass m, this equation could be directly written as γ µ Pµ ψ = −mcψ (14)
Jo
where γ µ are the usual 4 × 4 Dirac matrices generating a Clifford algebra. Then, in the presence of a minimally coupled electromagnetic field Aµ , this equation becomes γ µ (Pµ − qAµ ) ψ = −mcψ (15)
In particular for qAµ = (V (X) /c, 0), where V (X) is X 0 -independent, Eq.(15) leads to n o ˜ 2ψ ˜ .p − p0 + V (X) ψ = −βmc (1 + αpρ pρ ) c˜ α (16) 3
Journal Pre-proof ˜ k . In this way, the where we have introduced the notations β˜ = γ 0 and α ˜ k = βγ GUP-deformed Dirac equation, written in the position representation, reads (
2
i~ 1 + α~
∂2 − c2∇ 2 ∂t2
!!
!
∂ ˜ .∇ ∇+ −V c˜ α ∂t
∂ x, ∇ , ∂t
!)
ψ=
˜ 2ψ βmc
(17)
pro of
Remember that the operator X, when written in the position representation, ∂ entails a function of the non-deformed position vector x, ∇ and ∂t . Eq.(17) is completely different from the version of the deformed Dirac equation used in Ref.[1] to study the Dirac hydrogen atom in the context of the GUP. Finally, we should mention that when all the above considerations are taken into account, solutions of the KG equation and the Dirac equation with the Coulomb potential turn out to be not trivial at all.
re-
References
[1] V. Todorinov, P. Bosso, S. Das, Ann. Phys. 405 (2019) 92 [2] C. Quesne, V.M. Tkachuk, Czech. J. Phys. 56, 1269 (2006)
Jo
urn a
lP
[3] C. Quesne, V.M. Tkachuk, J. Phys. A, Mat. Gen. 39, 10909 (2006)
4
PII: DOI: Reference:
S0003-4916(19)30262-3 https://doi.org/10.1016/j.aop.2019.168007 YAPHY 168007
To appear in:
Annals of Physics
Received date : 7 October 2019 Accepted date : 8 October 2019 Please cite this article as: Y. Chargui, Comments on the paper “Relativistic generalized uncertainty principle”, Annals of Physics (2019), doi: https://doi.org/10.1016/j.aop.2019.168007. This is a PDF file of an article that has undergone enhancements after acceptance, such as the addition of a cover page and metadata, and formatting for readability, but it is not yet the definitive version of record. This version will undergo additional copyediting, typesetting and review before it is published in its final form, but we are providing this version to give early visibility of the article. Please note that, during the production process, errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
© 2019 Elsevier Inc. All rights reserved.
Journal Pre-proof
Comments on the paper ”Relativistic generalized uncertainty principle” Yassine Chargui a,b,∗ Department, College of Science and Arts at ArRass, Qassim University, PO Box 53, ArRass 51921, Saudi Arabia
pro of
a Physics
b Universit´ e
de Tunis El Manar, Facult´e des Sciences de Tunis, Unit´e de Recherche de Physique Nucl´eaire et des Hautes Energies, 2092 Tunis, Tunisie
Abstract
re-
We point out some misleading results reported in the recent study made by V. Todorinov et al. [1], concerning a relativistic extension of the generalized uncertainty principle (GUP). We derive, in this frame, the correct deformed Klein-Gordon (KG) and Dirac equations, valid up to the first order in the deformation parameter, and discuss their minimal coupling to external fields.
[X µ , P ν ] = i~
n
lP
In a recent paper published in this Journal [1], V. Todorinov et al considered a Lorentz covariant deformed algebra, where the commutation relations between the four-position and the four-momentum operators are given by
1 + (ε − α)γ 2 P ρ Pρ η µν + (β + 2ε)γ 2 P µ P ν
o
(1)
urn a
where α, β and ε are dimensionless parameters and γ = 1/cMpl , with Mpl the Planck mass. η µν = diag(−, +, +, +) is the metric tensor in the (3 + 1)dimensional Minkowski space-time. It should be noted that the commutation relations in Eq.(1) are exactly the same as those studied by Quesne and Tkachuk in Refs.[2,3], who used the notations β and β 0 to designate the two independent deformation parameters, such as β = (ε−α)γ 2 and β 0 = (β+ε)γ 2 . Then the authors focused on the particular case where ε = α, and adopted the following representations of X µ and P µ :
Jo
X µ = xµ 1 − αγ 2 pρ pρ ,
P µ = pµ 1 + αγ 2 pρ pρ
(2)
where the small letters x = (x0 , x) and p = (p0 , p) are used, throughout this work, to designate the canonically conjugate operators of four-position ∗ Corresponding author Email: [email protected] Preprint submitted to Elsevier Science
7 October 2019
Journal Pre-proof and four-momentum, respectively. These operators satisfy the commutation relations [xµ , pν ] = i~η µν (3) Moreover, henceforth, we shall simply use the symbol α in order to refer to αγ 2 . Now, the deformed free KG equation, valid up to the first order in α, follows merely from the invariance of the squared physical momentum, that is from the relation P µ Pµ = −m2 c2 , with m the mass of the particle. Thus, using Eq.(2) for the realization of the operator P µ , and retaining only terms of first order in α, this relation results in the equation n
o
pro of
pµ pµ + 2α (pµ pµ )2 ψ = −m2 c2 ψ
(4)
where ψ is the KG wave function. Then, Eq.(4) is equivalent to n
o
1 − 4αp20 p2 + 2αp4 − 1 − 2αp20 p20 = −m2 c2 ψ
(5)
Furthermore, the KG equation, in the position-space representation, stems ∂ ∇ into Eq.(5). This yields and p = −i~∇ from the substitutions p0 = i~c ∂t ~2 ∂ 2 1 + 2α 2 2 c ∂t
!
!
∂2 ~2 ∂ 2 2 − c 1 + 4α ∇2 ∂t2 c2 ∂t2 ) m2 c4 2 2 4 ψ=0 +2α~ c ∇ + 2 ~
re-
(
(6)
lP
This equation differs drastically from that obtained in Ref.[1]. Therein Eq.(4) was regarded as a simple algebraic equation and then solved for pµ pµ , to the first order in α. This had led to an inadequate GUP-modified KG equation. The latter is very similar to the non-deformed one, but just with a modified mass (Eq.(20) in Ref.[1]).
urn a
Next, we discuss the derivation of the GUP-deformed KG equation in the presence of a minimally-coupled electromagnetic field, with a four-potential Aµ . This will involve the replacement P µ → P µ − qAµ into the free GUPdeformed KG equation, where q is the electric charge of the considered particle. Hence, one has to start from the equation (P µ − qAµ ) (Pµ − qAµ ) ψ = −m2 c2 ψ
(7)
Then, we use Eq.(2) and expand to the first order in α. This gives n
Jo
pµ pµ + 2α (pµ pµ )2 + q 2 Aµ Aµ − q (2Aµ pµ + [pµ , Aµ ]) (1 + αpρ pρ )
−αqpµ (pρ [pρ , Aµ ] + [pρ , Aµ ] pρ )} ψ = −m2 c2 ψ
(8)
Particularly, in the case where qAµ = (V (X) /c, 0), with V (X) a static potential energy (i.e., independent of X 0 ), Eq.(9) reduces to 2
Journal Pre-proof n
c2 pµ pµ + 2αc2 (pµ pµ )2 − V (X)2 + 2cV (X) (1 + αpρ pρ ) p0 o
+αc (pρ [pρ , V (X)] + [pρ , V (X)] pρ ) p0 ψ = −m2 c4 ψ
(9)
pro of
In addition, according to Eq.(2), the physical three-position operator X, is now a function of the non-deformed three-position operator x and four-momentum operator p. For instance, the distance R = |X|, expanded to the first order in α, reads √ α R = X2 = r 1 − αpρ pρ + i~ 2 x.p (10) r Thus, the development of the Coulomb potential V (X) = −κ/ |X|, valid to the leading order in α, turns to be
κ α V (X) = − 1 + αp pρ − i~ 2 x.p r r ρ
(11)
All the above considerations were not part of the procedure followed by V. Todorinov et al [1], when studying the effect of the GUP on the relativistic hydrogen atom using the KG equation.
re-
Furthermore, throughout their work, the authors of [1] have employed the relation E = E0 (1 + αpρ pρ ) = E0 1 − αm2 c2 (12)
D
E = P0
E
lP
in order to calculate the GUP corrections to the physical energy E from those to the non-deformed energy, denoted by E0 . However, we believe this is an incorrect reasoning. As a matter of fact, the physical energy E, corresponding to a stationary state ψ, should be worked out as the expectation value of the operator energy P 0 , in the state ψ. Then, using Eq.(2), this yields ψ
D
= E0 1 − αE02 + αE0 c2 p2
E
ψ
(13)
urn a
where E0 is the non-deformed energy associated to the state ψ. That is ψ is of the form ψ = e−iE0 t/~ φ where φ is a time-independent state. We turn now to discuss briefly the derivation of the GUP-deformed Dirac equation. For a free particle of mass m, this equation could be directly written as γ µ Pµ ψ = −mcψ (14)
Jo
where γ µ are the usual 4 × 4 Dirac matrices generating a Clifford algebra. Then, in the presence of a minimally coupled electromagnetic field Aµ , this equation becomes γ µ (Pµ − qAµ ) ψ = −mcψ (15)
In particular for qAµ = (V (X) /c, 0), where V (X) is X 0 -independent, Eq.(15) leads to n o ˜ 2ψ ˜ .p − p0 + V (X) ψ = −βmc (1 + αpρ pρ ) c˜ α (16) 3
Journal Pre-proof ˜ k . In this way, the where we have introduced the notations β˜ = γ 0 and α ˜ k = βγ GUP-deformed Dirac equation, written in the position representation, reads (
2
i~ 1 + α~
∂2 − c2∇ 2 ∂t2
!!
!
∂ ˜ .∇ ∇+ −V c˜ α ∂t
∂ x, ∇ , ∂t
!)
ψ=
˜ 2ψ βmc
(17)
pro of
Remember that the operator X, when written in the position representation, ∂ entails a function of the non-deformed position vector x, ∇ and ∂t . Eq.(17) is completely different from the version of the deformed Dirac equation used in Ref.[1] to study the Dirac hydrogen atom in the context of the GUP. Finally, we should mention that when all the above considerations are taken into account, solutions of the KG equation and the Dirac equation with the Coulomb potential turn out to be not trivial at all.
re-
References
[1] V. Todorinov, P. Bosso, S. Das, Ann. Phys. 405 (2019) 92 [2] C. Quesne, V.M. Tkachuk, Czech. J. Phys. 56, 1269 (2006)
Jo
urn a
lP
[3] C. Quesne, V.M. Tkachuk, J. Phys. A, Mat. Gen. 39, 10909 (2006)
4