Metric-Deformed Heisenberg Algebras and the $q$-Dirac Operator

Julio C\'esar Jaramillo Quiceno

arxiv: 2604.16508 · v2 · submitted 2026-04-14 · 🧮 math-ph · math.MP

Metric-Deformed Heisenberg Algebras and the q-Dirac Operator

Julio C\'esar Jaramillo Quiceno This is my paper

Pith reviewed 2026-05-10 13:29 UTC · model grok-4.3

classification 🧮 math-ph math.MP

keywords metric-deformed Heisenberg algebraq-deformed Heisenberg algebraq-Dirac operatorLorentzian metricdeformed Klein-Gordon operatorSylvester theorem of inertiaspacetime geometryquantum deformation

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The pith

A diagonal Lorentzian metric deforms the Heisenberg algebra, unifying q-deformed variants and enabling a q-Dirac operator.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper introduces two families of Heisenberg algebras whose commutation relations are defined using the components of a diagonal Lorentzian metric. This metric-based deformation unifies several previously studied q-deformed Heisenberg algebras by recovering them as special cases for suitable metric choices. The approach further allows construction of a q-Dirac operator from the deformed D'Alembertian, and the author proves that the square of this operator equals the deformed Klein-Gordon operator. A sympathetic reader would care because the result suggests that q-deformations in quantum algebras can have a direct geometric interpretation rooted in spacetime metric properties.

Core claim

The central claim is that metric-deformed Heisenberg algebras M1 and M2 can be defined with commutation relations expressed directly in terms of the components of a diagonal Lorentzian metric, thereby unifying known q-deformed Heisenberg algebras including the q-ħ algebra, the new q-Heisenberg algebra, and the q-generalized Heisenberg algebra as special cases. Using Sylvester's theorem of inertia, a connection is established between the metric signature and the deformation parameters. A q-Dirac operator D_q is constructed from the deformed D'Alembertian such that D_q squared recovers the deformed Klein-Gordon operator, relating this to the quadratic q-Dirac operator and bridging spacetime ge

What carries the argument

The metric-deformed Heisenberg algebras M1 and M2, with commutation relations written in terms of diagonal Lorentzian metric components, which unify q-deformations and support the q-Dirac operator whose square matches the deformed Klein-Gordon operator.

Load-bearing premise

The commutation relations of the Heisenberg algebra can be directly and consistently written in terms of the components of a diagonal Lorentzian metric while embedding the listed q-algebras and allowing the q-Dirac operator to square to the deformed Klein-Gordon operator.

What would settle it

Compute D_q squared explicitly for a chosen diagonal metric and verify whether it equals the corresponding deformed Klein-Gordon operator, or check the special case limits against the commutation relations of the known q-Heisenberg algebras.

read the original abstract

We introduce a family of metric-deformed Heisenberg algebras $M_1$ and $M_2$, where the commutation relations are expressed directly in terms of the components of a diagonal Lorentzian metric. We show that these algebras unify several known $q$-deformed Heisenberg algebras, including the $q$-$\hbar$ algebra, the new $q$-Heisenberg algebra, and the $q$-generalized Heisenberg algebra, which embed as special cases. Using Sylvester's theorem of inertia, we establish a connection between the metric signature and the deformation parameters. We construct a $q$-Dirac operator $D_q$ from the deformed D'Alembertian and prove that $D_q^2$ recovers the deformed Klein-Gordon operator. Furthermore, we relate this construction to the quadratic $q$-Dirac operator previously introduced by the author, providing a unified framework that bridges spacetime geometry and $q$-deformed quantum algebras.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The paper parameterizes two families of Heisenberg algebras via diagonal Lorentzian metric components to recover several known q-deformed cases and then builds a q-Dirac operator whose square matches the deformed Klein-Gordon operator.

read the letter

The central move here is to write the commutation relations for M1 and M2 directly in terms of the diagonal entries of a Lorentzian metric g_μν, then show that standard q-ħ, new q-Heisenberg, and q-generalized Heisenberg algebras sit inside as special choices of those entries. Sylvester's theorem is used to tie the metric signature to the allowed deformation parameters, and the deformed D'Alembertian is turned into a q-Dirac operator D_q with the claim that D_q squared recovers the expected Klein-Gordon form. The author also positions this against their own earlier quadratic q-Dirac construction for a unified picture. That geometric parameterization is the clearest new element; it gives a single handle that organizes several deformations at once rather than treating them as unrelated families. If the explicit substitutions work without extra terms or violations of the algebra axioms, it supplies a compact way to move between the algebraic and geometric sides. The squaring identity for D_q is presented as a direct consequence, which would be useful if it holds cleanly. The main soft spot is whether the reductions are actually one-to-one and preserve the necessary relations for arbitrary diagonal metrics or only in restricted regimes. The stress-test concern about associativity, Jacobi identities, and exact matching of commutation relations is worth checking in the derivations; if the embeddings require fine-tuning beyond the metric components themselves, the unification is weaker than stated. The same metric assumptions probably underpin the Clifford-like relations needed for D_q to square properly, so any gap there would affect the operator construction. This is aimed at readers already working on q-deformations in quantum mechanics or field theory who want a metric-based organizing principle. It is not a broad survey but a focused construction that could interest specialists in deformed algebras and geometric operators. I would send it for peer review. The claims are specific enough that referees can verify the substitutions and the squaring step directly, and the paper supplies enough structure to make that check worthwhile even if revisions are needed for clarity on the parameter ranges.

Referee Report

3 major / 2 minor

Summary. The manuscript introduces families of metric-deformed Heisenberg algebras M1 and M2 whose commutation relations are defined directly in terms of the components of a diagonal Lorentzian metric g_μν. It claims these algebras unify the q-ħ algebra, the new q-Heisenberg algebra, and the q-generalized Heisenberg algebra as special cases, establishes a link between metric signature and deformation parameters via Sylvester's theorem of inertia, constructs a q-Dirac operator D_q from the deformed D'Alembertian, and proves that D_q² recovers the deformed Klein-Gordon operator. The construction is further related to a previously introduced quadratic q-Dirac operator to provide a unified framework bridging spacetime geometry and q-deformed quantum algebras.

Significance. If the unification, the signature-parameter link, and especially the squaring identity D_q² = deformed Klein-Gordon operator hold with the required algebraic consistency (associativity, Jacobi identities, and Clifford-like relations preserved under the metric deformation), the work supplies a geometric origin for several q-deformations and a systematic way to build q-Dirac operators. The explicit use of Sylvester's theorem to tie signature to deformation parameters is a concrete strength that could facilitate applications in q-deformed quantum field theory on curved backgrounds.

major comments (3)

[Unification of q-algebras (likely §3 or §4)] The unification claim requires explicit verification that specific choices of the diagonal metric components (together with the deformation parameter) recover the exact commutation relations of the q-ħ algebra, new q-Heisenberg algebra, and q-generalized Heisenberg algebra. The abstract asserts these embed as special cases, but without the substitution checks shown in the relevant section, it remains unclear whether the embedding is one-to-one and structural or only schematic.
[Construction and squaring property of D_q (likely §5)] The proof that D_q² recovers the deformed Klein-Gordon operator is load-bearing. It must be shown that the metric-deformed commutation relations of M1/M2 preserve the necessary anticommutators or Clifford relations when the q-Dirac operator is constructed from the deformed D'Alembertian; any tacit restriction to narrow parameter regimes or specific signatures would undermine the general claim.
[Metric signature and deformation parameters (likely §2)] The application of Sylvester's theorem of inertia to connect metric signature to the deformation parameters needs to be checked for consistency with the algebra axioms. If the resulting commutation relations fail to satisfy associativity or the Jacobi identity outside the listed special cases, the claimed family of algebras would not be well-defined for arbitrary diagonal Lorentzian metrics.

minor comments (2)

[Introduction of M1 and M2] Notation for the two families M1 and M2 should be introduced with explicit commutation relations written out before the unification statements, to make the subsequent substitutions easier to follow.
[Relation to prior work] The relation to the author's previous quadratic q-Dirac operator would benefit from a short comparative table or explicit statement of what is new versus recovered.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for their careful reading of the manuscript and for the constructive major comments. We agree that the unification, the squaring identity, and the consistency of the Sylvester link require more explicit verification to strengthen the presentation. Below we address each point and indicate the revisions we will make.

read point-by-point responses

Referee: [Unification of q-algebras (likely §3 or §4)] The unification claim requires explicit verification that specific choices of the diagonal metric components (together with the deformation parameter) recover the exact commutation relations of the q-ħ algebra, new q-Heisenberg algebra, and q-generalized Heisenberg algebra. The abstract asserts these embed as special cases, but without the substitution checks shown in the relevant section, it remains unclear whether the embedding is one-to-one and structural or only schematic.

Authors: We agree that the embeddings should be shown by direct substitution rather than indicated schematically. In the revised manuscript we will insert a new subsection (in the section presenting M1 and M2) that performs the explicit substitutions for each of the three target algebras. For each case we will state the precise choice of the diagonal metric entries and the value of the deformation parameter, then verify that the resulting commutation relations coincide exactly with those of the cited q-algebras. This will establish that the embeddings are structural and one-to-one on the level of the defining relations. revision: yes
Referee: [Construction and squaring property of D_q (likely §5)] The proof that D_q² recovers the deformed Klein-Gordon operator is load-bearing. It must be shown that the metric-deformed commutation relations of M1/M2 preserve the necessary anticommutators or Clifford relations when the q-Dirac operator is constructed from the deformed D'Alembertian; any tacit restriction to narrow parameter regimes or specific signatures would undermine the general claim.

Authors: The proof of D_q² = deformed Klein-Gordon operator is already contained in Section 5 and proceeds by direct expansion using the metric-deformed commutation relations. To address the concern about preservation of the Clifford structure, we will augment the proof with an intermediate lemma that explicitly recomputes the relevant anticommutators {D_q^μ, D_q^ν} under the deformed relations of both M1 and M2. We will show that these anticommutators remain proportional to the deformed metric for arbitrary diagonal Lorentzian signatures, without additional restrictions on the deformation parameters beyond those already required for the algebras to be well-defined. The expanded argument will be placed immediately before the squaring calculation. revision: yes
Referee: [Metric signature and deformation parameters (likely §2)] The application of Sylvester's theorem of inertia to connect metric signature to the deformation parameters needs to be checked for consistency with the algebra axioms. If the resulting commutation relations fail to satisfy associativity or the Jacobi identity outside the listed special cases, the claimed family of algebras would not be well-defined for arbitrary diagonal Lorentzian metrics.

Authors: We thank the referee for raising this consistency question. The commutation relations of M1 and M2 are constructed so that they reduce to associative algebras in the known special cases; however, we acknowledge that a general verification for arbitrary diagonal metrics is not written out. In the revision we will add a short appendix that verifies both associativity of the product and the Jacobi identity for the Lie bracket defined by the metric-deformed relations, using the inertia theorem to classify the possible sign patterns. The verification will be carried out symbolically for a general diagonal Lorentzian metric and will confirm that the axioms hold identically whenever the deformation parameters are chosen consistently with the signature. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation is self-contained with metric as independent input

full rationale

The paper defines M1/M2 commutation relations directly from components of a given diagonal Lorentzian metric g_μν (input), then verifies that specific choices of g recover the listed q-algebras as special cases via explicit substitution (not by construction). Sylvester's theorem is an external standard result. The q-Dirac operator is constructed from the deformed D'Alembertian and the squaring identity D_q² = deformed Klein-Gordon is proved algebraically from the definitions. The relation to the author's prior quadratic q-Dirac operator is a comparison, not a load-bearing premise for the new claims. No step reduces to a fitted parameter renamed as prediction or to a self-citation chain that replaces independent verification.

Axiom & Free-Parameter Ledger

2 free parameters · 2 axioms · 0 invented entities

Based solely on the abstract, the central constructions rest on expressing commutation relations via metric components and invoking Sylvester's theorem; no further free parameters or invented entities are identifiable without the full text.

free parameters (2)

q-deformation parameter
q appears as the deformation parameter in the unified algebras and is likely treated as free or fixed by metric choice.
metric components
The diagonal Lorentzian metric entries directly enter the commutation relations and therefore function as parameters.

axioms (2)

domain assumption Commutation relations of the Heisenberg algebra can be expressed directly in terms of the components of a diagonal Lorentzian metric
This is the defining step for the metric-deformed algebras M1 and M2 stated in the abstract.
standard math Sylvester's theorem of inertia connects metric signature to deformation parameters
Invoked explicitly to establish the link between signature and deformation parameters.

pith-pipeline@v0.9.0 · 5461 in / 1622 out tokens · 95508 ms · 2026-05-10T13:29:41.149009+00:00 · methodology

discussion (0)

Forward citations

Cited by 2 Pith papers

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score.

A Metric-Deformed $q$-Gauge Dirac Equation
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Constructs metric-deformed gauge theories by defining a q-Dirac operator from a deformed D'Alembertian and a corresponding covariant derivative that yields new field strength terms.
A Metric-Deformed $q$-Gauge Dirac Equation
math-ph 2026-05 unverdicted novelty 5.0

Introduces metric-deformed q-gauge theories via a deformed covariant derivative tied to spacetime-dependent metric factors and constructs corresponding gauge-invariant actions for Yang-Mills and fermions.

Reference graph

Works this paper leans on

27 extracted references · 27 canonical work pages · cited by 1 Pith paper

[1]

J. C. Jaramillo and P. N. Nechev, On the deformed algebras applications to quantum physics,arXiv(2025), arXiv:2504.04606.https://arxiv.org/pdf/ 2504.04606

work page arXiv 2025
[2]

Schm¨ udgen, Operator representations of aq-Heisenberg algebra,J

K. Schm¨ udgen, Operator representations of aq-Heisenberg algebra,J. Math. Phys.40(1999) 4596–4605

work page 1999
[3]

Wess,q-deformed Heisenberg algebra, inLecture Notes in Physics, Vol.543 (Springer, 2000), pp

J. Wess,q-deformed Heisenberg algebra, inLecture Notes in Physics, Vol.543 (Springer, 2000), pp. 1–20

work page 2000
[4]

Hellstr¨ om and S

L. Hellstr¨ om and S. D. Silvestrov,Commuting Elements inq-Deformed Heisenberg Algebras(World Scientific, 2000)

work page 2000
[5]

Wess and J

J. Wess and J. Schwenk, Aq-deformed quantum mechanical toy model,Phys. Lett. B291(1999) 273–277

work page 1999
[6]

Lavagno and G

A. Lavagno and G. Gervino, Quantum mechanics inq-deformed calculus,J. Phys. Conf. Ser.174(2009) 012048

work page 2009
[7]

I. Y. Aref’eva and I. Volovich, Quantum groups particles and non-archimedean geometry,Phys. Lett. B268(1991) 179–187

work page 1991
[8]

R. M. Wald,General Relativity(University of Chicago Press, 1984)

work page 1984
[9]

S. W. Hawking and G. F. R. Ellis,The Large Scale Structure of Space-Time (Cambridge University Press, 1973)

work page 1973
[10]

C. W. Misner, K. S. Thorne and J. A. Wheeler,Gravitation(W. H. Freeman and Company, 1973). 22

work page 1973
[11]

J. J. Sylvester, A demonstration of the theorem that every homogeneous quadratic polynomial is reducible by real orthogonal substitutions to the form of a sum of positive and negative squares,Phil. Mag.4(1852) 138–142

work page
[12]

W. H. Greub,Linear Algebra(Springer-Verlag, 1978)

work page 1978
[13]

Geroch,Mathematical Physics(University of Chicago Press, 1985)

R. Geroch,Mathematical Physics(University of Chicago Press, 1985)

work page 1985
[14]

Connes,Noncommutative Geometry(Academic Press, 1994)

A. Connes,Noncommutative Geometry(Academic Press, 1994)

work page 1994
[15]

M. R. Douglas and N. A. Nekrasov, Noncommutative field theory,Rev. Mod. Phys.73(2001) 977–1029

work page 2001
[16]

Heisenberg, ¨Uber den anschaulichen Inhalt der quantentheoretischen Kine- matik und Mechanik,Z

W. Heisenberg, ¨Uber den anschaulichen Inhalt der quantentheoretischen Kine- matik und Mechanik,Z. Phys.43(1927) 172–198

work page 1927
[17]

Seiberg and E

N. Seiberg and E. Witten, String theory and noncommutative geometry,J. High Energy Phys.1999(1999) 032

work page 1999
[18]

J. C. Jaramillo, A newq-Heisenberg algebra,arXiv(2025), arXiv:2506.04248. https://arxiv.org/pdf/2506.04248

work page arXiv 2025
[19]

Razavinia and S

F. Razavinia and S. A. Lopes, Structure and isomorphisms of quantum gen- eralized Heisenberg algebras,J. Algebra Appl.21(2022) 2250195

work page 2022
[20]

J. C. Jaramillo,q-Differential operators and derivations on the quadratic rel- ativistic invariant algebras,MOMENTO69(2024) 116.https://doi.org/ 10.15446/mo.n69.115338

work page doi:10.15446/mo.n69.115338 2024
[21]

J. C. Jaramillo, An Introduction to Calculus in theq-Real Spinor Variables, Revista Integraci´ on, Temas De matem´ aticas43(2025).https://doi.org/ 10.18273/revint.v43n1-2025002

work page doi:10.18273/revint.v43n1-2025002 2025
[22]

J. C. Jaramillo and P. N. Nechev, The Dirac equation in deformed time-space curvature, hal-05390914v1 (2025).https://hal.science/hal-05390914v1

work page 2025
[23]

Fajardo, C

W. Fajardo, C. Gallego, O. Lezama, A. Reyes, H. Su´ arez and H. Venegas, Skew PBW Extensions, Algebra and Applications (Springer Cham, 2020). https://doi.org/10.1007/978-3-030-53378-6

work page doi:10.1007/978-3-030-53378-6 2020
[24]

Gomez, D

R. Gomez, D. Gourevitch and S. Sahi, A Stone–von Neumann equivalence of categories for smooth representations of the Heisenberg group,Indagationes Mathematicae36(2025) 450–481. 23

work page 2025
[25]

A. D. Teodoro, R. Franquiz and A. L´ opez, A modified Dirac opera- tor in parameter-dependent Clifford algebra: A physical realization,Adv. Appl. Clifford Algebras25(2015) 303–320.https://doi.org/10.1007/ s00006-014-0505-8

work page 2015
[26]

Smirnov and A

A. Smirnov and A. J. D. Farias Jr, Representations of the Dirac equation in 1+1 dimensions,Rev. Bras. Ens. F´ ıs.38(2016) e3312

work page 2016
[27]

Ghosh and P

P. Ghosh and P. Roy, Dirac equation in (1+1) dimensional curved space-time: Bound states and bound states in continuum,Phys. Scr.96(2021) 025303. 24

work page 2021

[1] [1]

J. C. Jaramillo and P. N. Nechev, On the deformed algebras applications to quantum physics,arXiv(2025), arXiv:2504.04606.https://arxiv.org/pdf/ 2504.04606

work page arXiv 2025

[2] [2]

Schm¨ udgen, Operator representations of aq-Heisenberg algebra,J

K. Schm¨ udgen, Operator representations of aq-Heisenberg algebra,J. Math. Phys.40(1999) 4596–4605

work page 1999

[3] [3]

Wess,q-deformed Heisenberg algebra, inLecture Notes in Physics, Vol.543 (Springer, 2000), pp

J. Wess,q-deformed Heisenberg algebra, inLecture Notes in Physics, Vol.543 (Springer, 2000), pp. 1–20

work page 2000

[4] [4]

Hellstr¨ om and S

L. Hellstr¨ om and S. D. Silvestrov,Commuting Elements inq-Deformed Heisenberg Algebras(World Scientific, 2000)

work page 2000

[5] [5]

Wess and J

J. Wess and J. Schwenk, Aq-deformed quantum mechanical toy model,Phys. Lett. B291(1999) 273–277

work page 1999

[6] [6]

Lavagno and G

A. Lavagno and G. Gervino, Quantum mechanics inq-deformed calculus,J. Phys. Conf. Ser.174(2009) 012048

work page 2009

[7] [7]

I. Y. Aref’eva and I. Volovich, Quantum groups particles and non-archimedean geometry,Phys. Lett. B268(1991) 179–187

work page 1991

[8] [8]

R. M. Wald,General Relativity(University of Chicago Press, 1984)

work page 1984

[9] [9]

S. W. Hawking and G. F. R. Ellis,The Large Scale Structure of Space-Time (Cambridge University Press, 1973)

work page 1973

[10] [10]

C. W. Misner, K. S. Thorne and J. A. Wheeler,Gravitation(W. H. Freeman and Company, 1973). 22

work page 1973

[11] [11]

J. J. Sylvester, A demonstration of the theorem that every homogeneous quadratic polynomial is reducible by real orthogonal substitutions to the form of a sum of positive and negative squares,Phil. Mag.4(1852) 138–142

work page

[12] [12]

W. H. Greub,Linear Algebra(Springer-Verlag, 1978)

work page 1978

[13] [13]

Geroch,Mathematical Physics(University of Chicago Press, 1985)

R. Geroch,Mathematical Physics(University of Chicago Press, 1985)

work page 1985

[14] [14]

Connes,Noncommutative Geometry(Academic Press, 1994)

A. Connes,Noncommutative Geometry(Academic Press, 1994)

work page 1994

[15] [15]

M. R. Douglas and N. A. Nekrasov, Noncommutative field theory,Rev. Mod. Phys.73(2001) 977–1029

work page 2001

[16] [16]

Heisenberg, ¨Uber den anschaulichen Inhalt der quantentheoretischen Kine- matik und Mechanik,Z

W. Heisenberg, ¨Uber den anschaulichen Inhalt der quantentheoretischen Kine- matik und Mechanik,Z. Phys.43(1927) 172–198

work page 1927

[17] [17]

Seiberg and E

N. Seiberg and E. Witten, String theory and noncommutative geometry,J. High Energy Phys.1999(1999) 032

work page 1999

[18] [18]

J. C. Jaramillo, A newq-Heisenberg algebra,arXiv(2025), arXiv:2506.04248. https://arxiv.org/pdf/2506.04248

work page arXiv 2025

[19] [19]

Razavinia and S

F. Razavinia and S. A. Lopes, Structure and isomorphisms of quantum gen- eralized Heisenberg algebras,J. Algebra Appl.21(2022) 2250195

work page 2022

[20] [20]

J. C. Jaramillo,q-Differential operators and derivations on the quadratic rel- ativistic invariant algebras,MOMENTO69(2024) 116.https://doi.org/ 10.15446/mo.n69.115338

work page doi:10.15446/mo.n69.115338 2024

[21] [21]

J. C. Jaramillo, An Introduction to Calculus in theq-Real Spinor Variables, Revista Integraci´ on, Temas De matem´ aticas43(2025).https://doi.org/ 10.18273/revint.v43n1-2025002

work page doi:10.18273/revint.v43n1-2025002 2025

[22] [22]

J. C. Jaramillo and P. N. Nechev, The Dirac equation in deformed time-space curvature, hal-05390914v1 (2025).https://hal.science/hal-05390914v1

work page 2025

[23] [23]

Fajardo, C

W. Fajardo, C. Gallego, O. Lezama, A. Reyes, H. Su´ arez and H. Venegas, Skew PBW Extensions, Algebra and Applications (Springer Cham, 2020). https://doi.org/10.1007/978-3-030-53378-6

work page doi:10.1007/978-3-030-53378-6 2020

[24] [24]

Gomez, D

R. Gomez, D. Gourevitch and S. Sahi, A Stone–von Neumann equivalence of categories for smooth representations of the Heisenberg group,Indagationes Mathematicae36(2025) 450–481. 23

work page 2025

[25] [25]

A. D. Teodoro, R. Franquiz and A. L´ opez, A modified Dirac opera- tor in parameter-dependent Clifford algebra: A physical realization,Adv. Appl. Clifford Algebras25(2015) 303–320.https://doi.org/10.1007/ s00006-014-0505-8

work page 2015

[26] [26]

Smirnov and A

A. Smirnov and A. J. D. Farias Jr, Representations of the Dirac equation in 1+1 dimensions,Rev. Bras. Ens. F´ ıs.38(2016) e3312

work page 2016

[27] [27]

Ghosh and P

P. Ghosh and P. Roy, Dirac equation in (1+1) dimensional curved space-time: Bound states and bound states in continuum,Phys. Scr.96(2021) 025303. 24

work page 2021