Comment on ''Non-Hermitian Quantum Mechanics with Minimal Length Uncertainty''
We demonstrate that the recent paper by Jana and Roy entitled ''Non-Hermitian quantum mechanics with minimal length uncertainty'' [SIGMA 5 (2009), 083, 7 pages, arXiv:0908.1755] contains various misconceptions. We compare with an analysis on the same topic carried out previously...
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| Published in: | Symmetry, Integrability and Geometry: Methods and Applications |
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| Date: | 2009 |
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Інститут математики НАН України
2009
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| Cite this: | Comment on ''Non-Hermitian Quantum Mechanics with Minimal Length Uncertainty'' / B. Bagchi, A. Fring // Symmetry, Integrability and Geometry: Methods and Applications. — 2009. — Т. 5. — Бібліогр.: 2 назв. — англ. |
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| author | Bagchi, B. Fring, A. |
| author_facet | Bagchi, B. Fring, A. |
| citation_txt | Comment on ''Non-Hermitian Quantum Mechanics with Minimal Length Uncertainty'' / B. Bagchi, A. Fring // Symmetry, Integrability and Geometry: Methods and Applications. — 2009. — Т. 5. — Бібліогр.: 2 назв. — англ. |
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| container_title | Symmetry, Integrability and Geometry: Methods and Applications |
| description | We demonstrate that the recent paper by Jana and Roy entitled ''Non-Hermitian quantum mechanics with minimal length uncertainty'' [SIGMA 5 (2009), 083, 7 pages, arXiv:0908.1755] contains various misconceptions. We compare with an analysis on the same topic carried out previously in our manuscript [arXiv:0907.5354]. In particular, we show that the metric operators computed for the deformed non-Hermitian Swanson models differs in both cases and is inconsistent in the former.
|
| first_indexed | 2025-12-01T02:22:33Z |
| format | Article |
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Symmetry, Integrability and Geometry: Methods and Applications SIGMA 5 (2009), 089, 2 pages
Comment on “Non-Hermitian Quantum Mechanics
with Minimal Length Uncertainty”?
Bijan BAGCHI † and Andreas FRING ‡
† Department of Applied Mathematics, University of Calcutta,
92 Acharya Prafulla Chandra Road, Kolkata 700 009, India
E-mail: BBagchi123@rediffmail.com
‡ Centre for Mathematical Science, City University London,
Northampton Square, London EC1V 0HB, UK
E-mail: A.Fring@city.ac.uk
Received August 18, 2009; Published online September 17, 2009
doi:10.3842/SIGMA.2009.089
Abstract. We demonstrate that the recent paper by Jana and Roy entitled “Non-Hermitian
quantum mechanics with minimal length uncertainty” [SIGMA 5 (2009), 083, 7 pages,
arXiv:0908.1755] contains various misconceptions. We compare with an analysis on the
same topic carried out previously in our manuscript [arXiv:0907.5354]. In particular, we
show that the metric operators computed for the deformed non-Hermitian Swanson models
differs in both cases and is inconsistent in the former.
Key words: non-Hermitian Hamiltonians; deformed canonical commutation relations; mini-
mal length
2000 Mathematics Subject Classification: 81Q10; 46C15; 81Q12
It is known for some time that the deformations of the standard canonical commutation relations
between the position operator P and the momentum operator X will inevitably lead to a minimal
length, that is a bound beyond which the localization of space-time events are no longer possible.
In a recent manuscript [1] we investigated various limits of the q-deformationed relations
[X, P ] = i~qf(N)(αδ + βγ) +
i~(q2 − 1)
αδ + βγ
(
δγX2 + αβ P 2 + iαδXP − iβγPX
)
,
in conjunction with the constraint 4αγ = (q2 + 1), with α, β, γ, δ ∈ R and f being an arbitrary
function of the number operator N . One may consider various types of Hamiltonian systems,
either Hermitian or non-Hermitian, and replace the original standard canonical variables (x0, p0),
obeying [x0, p0] = i~, by (X, P ). It is crucial to note that even when the undeformed Hamiltonian
is Hermitian H(x0, p0) = H†(x0, p0) the deformed Hamiltonian is inevitably non-Hermitian
H(X, P ) 6= H†(X, P ) as a consequence of the fact that X and/or P are no longer Hermitian. Of
course one may also deform Hamiltonians, which are already non-Hermitian when undeformed
H(x0, p0) 6= H†(x0, p0). In both cases a proper quantum mechanical description requires the
re-definition of the metric to compensate for the introduction of non-Hermitian variables and in
the latter an additional change due to the fact that the Hamiltonian was non-Hermitian in the
first place.
In a certain limit, as specified in [1], X and P allow for a well-known representation of the
form X = (1+τp2
0)x0 and P = p0, which in momentum space, i.e. x0 = i~∂p0 , corresponds to the
one used by Jana and Roy [2], up to an irrelevant additional term i~γ̃P . (Whenever constants
with the same name but different meanings occur in [2] and [1] we dress the former with a tilde.)
?This paper is a contribution to the Proceedings of the 5-th Microconference “Analytic and Algebraic Me-
thods V”. The full collection is available at http://www.emis.de/journals/SIGMA/Prague2009.html
mailto:BBagchi123@rediffmail.com
mailto:A.Fring@city.ac.uk
http://dx.doi.org/10.3842/SIGMA.2009.089
http://dx.doi.org/10.3842/SIGMA.2009.083
http://arxiv.org/abs/0908.1755
http://arxiv.org/abs/0907.5354
http://www.emis.de/journals/SIGMA/Prague2009.html
2 B. Bagchi and A. Fring
The additional term can simply be gauged away and has no physical significance. Jana and Roy
have studied the non-Hermitian displaced harmonic oscillator and the Swanson model. As we
have previously also investigated the latter in [1], we shall comment on the differences. The
conventions in [2] are
HJR(a, a†) = ωa†a + λa2 + δ̃(a†)2 +
ω
2
with λ 6= δ̃ ∈ R and a = (P − iωX)/
√
2m~ω, a† = (P + iωX)/
√
2m~ω, whereas in [1] we used
HBF(X, P ) =
P 2
2m
+
mω2
2
X2 + iµ{X, P}
with µ ∈ R as a starting point. Setting ~ = m = 1 it is easy to see that the models coincide when
λ = −δ̃ and µ = δ̃ − λ. The Hamiltonians exhibit a “twofold” non-Hermiticity, one resulting
from the fact that when λ 6= δ̃ even the undeformed Hamiltonian is non-Hermitian and the other
resulting from the replacement of the Hermitian variables (x0, p0) by (X, P ). The factor of the
metric operator to compensate for the non-Hermiticity of X coincides in both cases, but the
factor which is required due to the non-Hermitian nature of the undeformed case differs in both
cases
ρBF = e2µP 2
and ρJR = (1 + τP 2)
µ
ω2τ .
We have made the above identifications such that HJR(a, a†) = HBF(X, P ) and replaced the
deformation parameter β used in [2] by τ employed in [1]. It is well known that when given only
a non-Hermitian Hamiltonian, the metric operator can not be uniquely determined. However,
as argued in [1] with the specification of the observable X, which coincides in [2] and [1], the
outcome is unique and we can therefore directly compare ρBF and ρJR. The limit τ → 0 reduces
the deformed Hamiltonian HJR = HBF to the standard Swanson Hamiltonian, such that ρJR
and ρBF should acquire the form of a previously constructed metric operator. This is indeed the
case for ρBF, but not for ρJR. In fact it is unclear how to carry out this limit for ρJR and we
therefore conclude that the metric ρJR is incorrect.
References
[1] Bagchi B., Fring A., Minimal length in quantum mechanics and non-Hermitian Hamiltonian systems,
arXiv:0907.5354.
[2] Jana T.K., Roy P., Non-Hermitian quantum mechanics with minimal length uncertainty, SIGMA 5 (2009),
083, 7 pages, arXiv:0908.1755.
http://arxiv.org/abs/0907.5354
http://arxiv.org/abs/0908.1755
References
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| id | nasplib_isofts_kiev_ua-123456789-149130 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1815-0659 |
| language | English |
| last_indexed | 2025-12-01T02:22:33Z |
| publishDate | 2009 |
| publisher | Інститут математики НАН України |
| record_format | dspace |
| spelling | Bagchi, B. Fring, A. 2019-02-19T17:34:55Z 2019-02-19T17:34:55Z 2009 Comment on ''Non-Hermitian Quantum Mechanics with Minimal Length Uncertainty'' / B. Bagchi, A. Fring // Symmetry, Integrability and Geometry: Methods and Applications. — 2009. — Т. 5. — Бібліогр.: 2 назв. — англ. 1815-0659 2000 Mathematics Subject Classification: 81Q10; 46C15; 81Q12 https://nasplib.isofts.kiev.ua/handle/123456789/149130 We demonstrate that the recent paper by Jana and Roy entitled ''Non-Hermitian quantum mechanics with minimal length uncertainty'' [SIGMA 5 (2009), 083, 7 pages, arXiv:0908.1755] contains various misconceptions. We compare with an analysis on the same topic carried out previously in our manuscript [arXiv:0907.5354]. In particular, we show that the metric operators computed for the deformed non-Hermitian Swanson models differs in both cases and is inconsistent in the former. This paper is a contribution to the Proceedings of the 5-th Microconference “Analytic and Algebraic Methods V”. en Інститут математики НАН України Symmetry, Integrability and Geometry: Methods and Applications Comment on ''Non-Hermitian Quantum Mechanics with Minimal Length Uncertainty'' Article published earlier |
| spellingShingle | Comment on ''Non-Hermitian Quantum Mechanics with Minimal Length Uncertainty'' Bagchi, B. Fring, A. |
| title | Comment on ''Non-Hermitian Quantum Mechanics with Minimal Length Uncertainty'' |
| title_full | Comment on ''Non-Hermitian Quantum Mechanics with Minimal Length Uncertainty'' |
| title_fullStr | Comment on ''Non-Hermitian Quantum Mechanics with Minimal Length Uncertainty'' |
| title_full_unstemmed | Comment on ''Non-Hermitian Quantum Mechanics with Minimal Length Uncertainty'' |
| title_short | Comment on ''Non-Hermitian Quantum Mechanics with Minimal Length Uncertainty'' |
| title_sort | comment on ''non-hermitian quantum mechanics with minimal length uncertainty'' |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/149130 |
| work_keys_str_mv | AT bagchib commentonnonhermitianquantummechanicswithminimallengthuncertainty AT fringa commentonnonhermitianquantummechanicswithminimallengthuncertainty |