Annals of Physics 457 (2023) 169389
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Contents lists available at ScienceDirect
Annals of Physics
journal homepage: www.elsevier.com/locate/aop
Path integral in position-deformedHeisenberg
algebrawithmaximal length uncertainty
Latévi M. Lawson a,b,∗, Prince K. Osei a,c, Komi Sodoga b,
red Soglohu a,c
a African Institute for Mathematical Sciences (AIMS) Ghana
Accra P.O. Box LG DTD 20046, Legon, Accra, Ghana
b Université de Lomé, Faculté des Sciences, Departement de Physique, Laboratoire de Physique des
Matériaux et des Composants à Semi-Conducteurs, 01 BP 1515 Lomé, Togo
c Department of Mathematics, University of Ghana, P.O. Box LG 62, Legon, Ghana
a r t i c l e i n f o a b s t r a c t
Article history: In this work, we study the path integral in a position-deformed
Received 12 January 2023 Heisenberg algebra with quadratic deformation which imple-
Accepted 6 June 2023 ments both minimal momentum and maximal length uncer-
Available online 12 June 2023 tainties. We construct propagators of path integrals within this
Keywords: deformed algebra using the position space representation on the
Generalized uncertainty principle one hand and the Fourier transform and its inverse representa-
Quantum gravity tions on the other. The result is remarkably similar to the one
Path integral obtained by Pramanik (2022) from the Perivolaropoulos’s de-
Propagator and action formed algebra (Perivolaropoulos, 2017). Then, the propagators
and the corresponding actions of a free particle and a simple
harmonic oscillator are discussed as examples. We also show
that the actions which describe the classical trajectories of both
systems are bounded by the ordinary ones of classical mechanics
due to the existence of this maximal length. Consequently, par-
ticles of these systems travel faster from one point to another
with low kinetic and mechanical energies.
© 2023 Elsevier Inc. All rights reserved.
∗ Corresponding author at: African Institute for Mathematical Sciences (AIMS) Ghana
ccra P.O. Box LG DTD 20046, Legon, Accra, Ghana.
E-mail addresses: latevi@aims.edu.gh (L.M. Lawson), posei@nexteinstein.org (P.K. Osei),
ntoinekomisodoga@gmail.com (K. Sodoga), fred@aims.edu.gh (F. Soglohu).ttps://doi.org/10.1016/j.aop.2023.169389
003-4916/© 2023 Elsevier Inc. All rights reserved.
L.M. Lawson, P.K. Osei, K. Sodoga et al. Annals of Physics 457 (2023) 169389
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1. Introduction
It has long been argued that quantum gravity should lead to a minimal observable length.
his minimal length should however be described quantum mechanically as a nonzero minimal
ncertainty in position measurements. The past two decades has seen the development and
xploration of the general framework for the implementation of the appearance of a nonzero
inimal uncertainty [1–19]. It is known that, the existence of this minimal length uncertainty
resents the issue of high energy requirements that are beyond the scope of any experimental
easibility. To circumvent this requirement, the first author has recently proposed a position-
eformed Heisenberg algebra [20] in two dimensions (2D) that introduces a simultaneous existence
f minimal and maximal length uncertainties. The emergence of this maximal length demonstrated
uantum deformation effects in this space and predicted the detection of low-energy gravity
articles [21,22]. These effects have been confirmed in [23] by the study of statistical properties of
deal gas in this deformed-Heisenberg algebra with maximal length uncertainty [21]. This maximal
ength induces logarithmic corrections to the thermodynamic quantities of this gas which are
onsequences of strong quantum deformation effects. Furthermore, the mathematical and statistical
roperties of Gazeau-Klauder coherent states for a free particle in a square well potential have also
een investigated within this position-deformed Heisenberg algebra [24].
In the present work, we investigate the effects of this maximal length uncertainty on the trajec-
ories of systems by studying the path integral in the position-deformed Heisenberg algebra [21].
o do so, we construct the position space representation describing this maximal length, as well as
he corresponding Fourier transform and its inverse representations. We derive the propagators
f path integrals and the classical action in these different representations. The results in the
osition representation are consistent with the recent one obtained by Pramanick in [25] from the
erivolaropoulos’s position-deformed Heisenberg algebra [26]. Then, the Hamiltonian’s principle of
east action is used to generate the classical equations of motion. We compute the propagators
nd the actions of a free particle and a simple harmonic oscillator as applications. We show that
hese deformed actions which describe the classical trajectories of both systems are bounded
y the standard ones of classical mechanics. This indicates that particles of these systems travel
uickly from one point to another with low kinetic and mechanical energies. This result perfectly
trengthens the claim that the recently proposed position-deformed algebra [21,22] induces strong
eformation of the quantum levels allowing particles to jump from one state to another with low
nergies.
This paper is outlined as follows: in Section 2, we establish the Hilbert space representations of
ave functions associated with this deformed algebra. In Section 3, we construct the path integrals
n these wave function representations and deduce the corresponding quantum propagators and
lassical actions. As examples, we compute the propagators and the actions for some simple models
uch as the free particle and the harmonic oscillator. In the last section, we present our conclusion.
. Position deformed Heisenberg algebra with maximal length
Let H L2= (R) be the Hilbert space of square integrable functions. The Hermitian operators x̂
nd p̂ that act on this space satisfy the condition
[x̂, p̂] = ih̄I. (1)
The corresponding Heisenberg uncertainty principle is given by
h̄
∆x∆p ≥ . (2)
2
Let {|x⟩} ∈ H be the complete position basis vectors. The action operators in Eq. (1) on this basis
vector reads as follows
d
x̂|x⟩ = x|x⟩ and p̂|x⟩ = −ih̄ |x⟩, x ∈ R. (3)dx
2
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The completeness and orthogon∫ality relations are given by [27]
+∞
x′⟨ |x⟩ = δ(x x′− ) and dx|x⟩⟨x| = I. (4)
−∞
nother useful choi∫ce of basis vectors is the momentum vector {|p⟩} ∈ H defined by taking Fouriertransforms
1 +∞ i px
|p⟩ = √ dxe h̄ |x⟩ with p ∈ R (5)
2π h̄ −∞
and its inver∫se is defined as follows
+∞ dp i
x e− h̄ px| ⟩ = √ |p⟩. (6)
−∞ 2π h̄
he inner product and complete∫ness relations are given by [27]
+∞
p′⟨ |p⟩ = δ(p ′− p ) and dp|p⟩⟨p| = I. (7)
−∞
he action of the ope∫rators in((1) on the v)ector |p⟩ is given by
1 +∞ d i
p̂|p⟩ = √
2π h̄ ∫ dx ih̄ e h̄ px− ( ) |x⟩ = p|p⟩, (8)−∞ dx1 +∞ d i d
x̂|p⟩ = √ dxih̄ e h̄ px |x⟩ = ih̄ |p⟩. (9)
2π h̄ −∞ dp dp
We introduce new operators X̂ and P̂ acting on H. They are defined by
X̂ x̂, P̂ (I τ x̂ τ 2 2= = − + x̂ )p̂. (10)
hey satisfy the following relation [25]
[X̂, P̂] = ih̄(I − τ X̂ + τ 2X̂2), (11)
here τ ∈ (0, 1) is the generalized uncertainty principle parameter related to quantum deformation
ffects in this space [4,20–22]. Obviously by taking τ → 0, we recover the algebra (1). This algebra
(11) is consistent with the one proposed by Perivolaropoulos [26].
The action of the operators (10) on the following unit basis vectors {|x⟩}, {|p⟩} reads as follows
X̂ |x⟩ = x|x⟩ and P̂|x⟩ = −ih̄(1 − τx τ 2+ x2)∂x|x⟩, x ∈ R. (12)
X̂ |p⟩ = ih̄∂p|p⟩ and P̂|p (1 iτ h̄∂ τ 2h̄2∂2⟩ = − p − p )p|p⟩, p ∈ R. (13)
Let us consider an arbitrary vector |φ⟩ ∈ H, the projection of this vector on the unit vectors {|x⟩}
and {|p⟩} generates the functions φ(x) = ⟨x|φ⟩ and φ(p) = ⟨p|φ⟩. As a result, we can write the above
equations as follows
X̂φ(x) = xφ(x) and P̂φ(x) ih̄(1 τx τ 2= − − + x2)∂xφ(x), (14)
X̂φ(p) = ih̄∂pφ(p) and P̂φ(p) 2 2 2= (1 − iτ h̄∂p − τ h̄ ∂p )pφ(p). (15)
An interesting feature can be observed from the commutation relation (11) through the following
uncertainty relatio(n:h̄ )
∆X∆P ≥ 1 τ 2 2− ⟨X̂⟩ + τ ⟨X̂ ⟩ , (16)
2
here ⟨X̂⟩ and ⟨X̂2⟩ are the expectation values of the operators X̂ and X̂2 respectively for any space
epresentations. Using the relation X̂2 (∆X)2 X̂ 2⟨ ⟩ = +⟨ ⟩ , Eq. (16) can be rewritten as a second order
quation for ∆X
2 1 1
∆X2 2− ∆P∆X + ⟨X̂⟩ − ⟨X̂⟩ + ≤ 0. (17)h̄τ 2 τ τ 2
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By setting Eq. (17) into
∆X2
2 1 1
− 2 ∆P∆X X̂
2
+ ⟨ ⟩ − ⟨X̂⟩ + = 0, (18)
h̄τ τ τ 2
the solutions ∆X are√g(iven b)y
P P 2∆ ∆ 1 1
∆X = 2 ±  2 + [⟨X̂⟩ − ⟨X̂⟩
2 −
h̄τ (h̄τ τ√ )2 ( )
τ 2 ]
∆P ∆P 1 2 3
= 2 ± 2 − ⟨X̂⟩ − + . (19)h̄τ h̄τ 2τ 4τ 2
The reality of solut√io(ns (19) give)s the following minimum value for ∆P
2 1
2 3
∆Pmin = h̄τ ⟨X̂⟩ − + . (20)2τ 4τ 2
In order to determine the absolute minimum measurable momentum of this deformed algebra, we
take only the physical states into account which satisfy the condition ⟨X̂ 1⟩ = 2 . Then, the aboveτ
Eq. (19) and (20) are reduced into the absolute minimum momentum ∆Pabsmin and maximal length
∆Xabsmax respectevely
√ √
abs h̄τ 3 3∆Pmin = and ∆X
abs
= = l . (21)
2 max 2 maxτ
hese provides the precise scale for the maximum length and minimum momentum which are
ignificantly different from the physical condition imposed in [20–24]
It is well known in [4] that, the existence of minimal uncertainty raises the question of the loss of
epresentation i.e., the space is inevitably bounded by minimal quantity beyond which any further
ocalization of particles is not possible. In the present situation, the minimal momentum ∆Pabsmin
eads to a loss of φ(p)-representation and a maximal φ(x)-representation. Thus, the corresponding
epresentation of operators are given by
X̂φ(x) = xφ(x) and P̂φ(x) = −ih̄Dxφ(x), (22)
where Dx = (1− τx+ τ 2x2)∂x is a deformed derivative. Using Eq. (22), one can recover the algebra
(11). As one can see from the representation of operators in Eq. (10) or in Eq. (22), the position
operator X̂ is Hermitian while the momentum operator P̂ is not
X̂† = X̂ and P̂† = P̂ + ih̄τ (I − 2τ X̂) P̂†H⇒ ̸= P̂ . (23)
In order to guarantee the Hermicity of this operator, we arbitrarily restrict the study from the
infinite dimensional Hilbert space H into its bounded dense domain D 2τ = L (−lmax, +lmax) in such
away that, for τ → 0, one recovers the entire space H. As will be shown in the forthcoming
evelopment, the restriction of the domain guarantees the physical meaning of the eigenstates
f the momentum operator (see more detail information in the Appendix). Furthermore, this
estriction perfectly fits with the work of Nozari and Etemadi did in momentum space [12]. In
ddition to this condition, we propose the following deformed completeness relation to get the
ermic∫ity of the operator P̂
+lmax dx
2 2 |x⟩⟨x| = I. (24)
−lmax 1 − τx + τ x
Consequently, the scalar product between two states |Ψ ⟩, |Φ⟩ ∈ Dτ and the orthogonality of
eigenstates beco∫me [12]
+lmax dx
∗
⟨Ψ |Φ⟩ = 2 2 Ψ (x)Φ(x), (25)
−lmax 1 − τx + τ x
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⟨x|x′⟩ = (1 − τx + τ 2x2)δ(x x′− ). (26)
Now let us consider the operator P̂ in its closed interval
Dτ (P̂) = {ϕ, −ih̄Dxϕ 2∈ L (−lmax, +lmax), ϕ(−lmax) = 0 = ϕ(+lmax)}, (27)
nd its adjoint domain defined by
D †τ (P̂ ) = {ξ, −ih̄Dxξ ∈ L2(−lmax, +lmax)}. (28)
hus, we may write Dτ (P̂) ⊂ D †τ (P̂ ), which means that the domain of P̂ is a proper subset of the
omain of its adjoint P̂†. To show the Hermicity of the operator P̂ , we consider a functional F (φ, ψ)
efined by
F (ϕ, ξ ) := ⟨ξ |P̂ϕ⟩ − ⟨P̂†ξ |ϕ⟩. (29)
sing the relation∫ (24) and by a straigh[tforward computation of this function]al, we have+lmax dx
F (ϕ, ξ ) ∗= ∫ 2 2 ξ (x) (−ih̄Dxϕ(x)) − (−ih̄Dxξ (x))
∗ ϕ(x)
−lmax 1 − τ(x + τ x+lmax ) [ ]
∗ ∗
= −ih̄ d ξ (x)ϕ(x) = −ih̄ ξ (x)ϕ(x) +lmaxl . (30)−
l max− max
Since ϕ(±lmax) = 0, and ξ (x) can reach any arbitrary value at the boundaries. This lead to the
vanishing of F (φ, ψ) i.e., F (φ, ψ) = 0. Consequently, the operator P̂ is symmetric in D(P̂) such that
⟨ξ |P̂ϕ † †⟩ = ⟨P̂ ξ |ϕ⟩ H⇒ P̂ = P̂ . (31)
Despite the fact that the momentum is Hermitian, it is not always a self-adjoint operator because
its domain includes the domain of P̂†. It could have none or have an infinite number of self-adjoint
extensions. Note that, as a rule in quantum mechanics, the operators that act on square integrable
functions are essentially self-adjoint. However, there are exceptions to this rule. This is because
the basic quantization requirement that operators whose expectation values are real do not strictly
require these operators be self-adjoint. Indeed, the Hermicity result (31) is a sufficient condition to
ensure that all expectation values of the momentum operator are real.
To construct a Hilbert space representation that describes the maximal length and the minimal
momentum uncertainties, one has to solve the eigenvalue problem
−ih̄Dxφρ(x) = ρφρ(x) with ρ ∈ R. (32)
The solution of this e(quation [is given(by ) ])
2ρ 2τx − 1 π
φρ(x) = A exp i √ arctan √ + , (33)
τ h̄ 3 3 6
here A is(a∫n abritrary constant.)Then by√normalization, ⟨φρ |φρ⟩ = 1, we have1 √lmax −dx 2 τ 3
A = = (34)
−lmax 1 − τx + τ 2x2 2 arctan(6)
Substituting Eq√. (34) into Eq. (33() gives√ [ ( ) ])
τ 3 2ρ 2τx − 1 π
φρ(x) = exp i √ arctan √ + . (35)2 arctan(6) τ h̄ 3 3 6
This wave function describes simultaneously the maximal length and the minimal momentum un-
certainties. As one can see, the eigenvectors |φρ⟩ are physical states. This is because the expectation
values of position energy operat∫or X̂n (n ∈ N) is finite:√
τ 3 +lmax xnn dx
⟨φρ |X̂ |φρ⟩ = 2 2 < ∞. (36)2 arctan(6) −lmax 1 − τx + τ x
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In comparison with Kempf et al. formalism [4], the expectation value of the operator X̂2 in this
ramework does not diverge as the one obtained in the momentum space [28]. According to this
ormalism, any state that has a well-defined minimal uncertainty measurement which is inside of
forbidden gap cannot have a finite energy, so cannot be accepted as physical states. Conversely,
ere the energy operator X̂2 is well defined therefore the states |φρ⟩ are physically relevant.
As a consequence of this fact, we can define a new identity operator from this position wave
unction (35) which will play the role of the completeness relation of the momentum eigenstates
n the ∫derivation of the path-integral. It reads as follows
+∞ arctan(6)
√ dρ|ρ⟩⟨ρ| = I. (37)
−∞ π h̄τ 3
o prove this, we refer to the work of Bernardo and Esguerra [29,30] and computing the orthogo-
ality of the states∫ x x′⟨ | ⟩, we have
+∞ arctan(6)
⟨x|x′⟩ = ∫⟨x| √ dρ|ρ⟩⟨ρ ′|x ⟩−∞ π h̄τ 3 ∫
+∞ arctan(6) +∞ arctan(6)
dρ x ρ ρ x′ ∗ ′= √ ⟨ | ⟩⟨ | ⟩ = √ dρφρ(x)φ (x ). (38)
−∞ π h̄τ 3
ρ
−∞ π h̄τ 3
By using Eq. (35), we∫ recover Eq.((26) as fo[llows ( ) ( )])
1 +∞ 2ρ 2τx 1 2τx′− − 1
⟨x ′|x ⟩ = ( dρ e(xp i √) arctan ( √ )−)arctan √2π h̄ −∞ τ h̄ 3 3 3√
τ 3 2τx − 1 2τx′ − 1
= δ arctan √ − arctan √
2 3 3
= (1 − τx 2+ τ x2)δ(x ′− x ). (39)
This confirm the claim that Eq. (37) is a correct expression for the identity.
Since the states φρ(x) are physically meaningful and are well localized, one can obtain the quasi-
momentum representation by projecting an arbitrary state |ψ⟩ ∈ H onto these localized states |φρ⟩
and one can obtain the q√uasi-moment∫um representation, that is√ l [ ( ) ]τ 3 + max dxψ(x) i 2ρ− √ arctan 2τ√x−1 π+
ψ(ρ) = ⟨φ |ψ⟩ = e τ h̄ 3 3 6ρ . (40)2 arctan(6) −lmax 1 − τx + τ 2x2
his mapping defined the generalized Fourier transform of the representation in Eq. (35). Its inverse
epresentation is given by
√
a√ ∫ [ ( ) ]rctan(6) +∞ i 2ρ√ arctan 2τ√x−1 π+ψ(x) = √ dρψ(ρ)e τ h̄ 3 3 6 . (41)
π h̄ 2τ 3 −∞
Similar to the action of X̂ on ψ(p) in Eq. (15), here the action of the operator X̂ on the quasi-
momentum wavefunction (40) r√eads as follow∫s√
d τ 3 +lmax dxψ(x)
X̂ψ(ρ) = ih̄ ψ(ρ) = ih̄
dρ 2 arctan([6) ( 2 2−lmax )1 −]τx + τ x
d i 2ρ√ arctan 2τ− √x−1 π+
× e τ h̄[ 3 ( 3 6 ) ] (42)dρ
d 2 2τx − 1 π
ψ(ρ) = −i √ arctan √ + ψ(ρ). (43)
dρ τ h̄ 3 3 6
his equation is equ[ivalent√ (to ) ] [ ( ) ( )]
τ h̄ 3 d 2τx − 1 π 2τx − 1 1
i = arctan √ + = arctan √ + arctan √ . (44)2 dρ 3 6 3 3
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From the following relation [31] ( )
α + β
arctanα + arctanβ = arctan , with αβ < 1, (45)
1 − αβ
e deduce[ that ( ) ( )] √
2τx − 1 1 τx 3
tan arctan √ + arctan √ = . (46)
3 3 2 − τx
In Eq. (44), we can see th(at the po)sition operator is represented as√
2 tan i
τ(h̄ 3 d2 dρX̂ = √ √ ) I, (47)
τ 3 t(an i τ h̄ 3)d+ 2 dρ√
tan i τ(h̄ 3 d2 2 dρX̂ψ(ρ) = √ √ )ψ(ρ). (48)
τ 3 tan i τ h̄ 3 d+ 2 dρ
From the action of P̂ on the quasi-representation (41) and using Eq. (14), we have
P̂ψ(ρ) = ρψ(ρ). (49)
Note that in the limit τ → 0, we recover the corresponding ordinary quantum mechanics results
in momentum space (8)
d
lim X̂ψ(ρ) = ih̄ ψ(ρ) and lim P̂ψ(ρ) = ρψ(ρ). (50)
τ→0 dρ τ→0
. Path integral and propagator in position-deformed algebra
From the path integrals within this position-deformed Heisenberg algebra, we construct the
ropagator depending on the position-representation and on the Fourier transform and its inverse
epresentations. We compute propagators and deduce the actions of a free particle and a harmonic
scillator as applications.
.1. Path integral and propagator in position-space representation
The Hamiltonian operator for a particle with mass m living in one spatial dimension is given by
P̂2
Ĥ = + V (X̂), (51)
2m
where V is the potential energy of the system. The time-dependent deformed Schrödinger equation
in the position representation is given by
h̄2
Ĥ 2|φρ(t)⟩ = − Dx |φρ(t)⟩ + V (x)|φρ(t)⟩ = ih̄∂t |φρ(t)⟩. (52)2m
The time-evolution process is described by
i
− Ĥ(t t ′− )
|φρ(t)⟩ = e h̄ |φρ(t ′)⟩, (53)
Multiplication of∫⟨x| from the left of Eq. (53) gives
+lmax dx′
φρ(x, t) = 2 2 K (x, t, x
′, t ′)φ (x′, t ′ρ ) (54)
′ ′
−lmax 1 − τx + τ x
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where K is the kernel in the Hamiltonian or the amplitude for a particle to propagate from the state
with position x′ to the state with position x (x > x′) in a time interval ∆t t t ′= − (t > t ′) [32,33]
nd it is defined as
i Ĥ(t t ′K (x − − ), t, x′, t ′) ′= ⟨x|e h̄ |x ⟩. (55)
Splitting the interval t−t ′ into N intervals of length ϵ = (tk−tk−1)/N and inserting the completeness
elations in (25) and (∫37), the(p∏ropagator (90) bec)omes ( )+l ∫max N−1 dx +∞ ∏N arctan(6)
K (x, t, x′ t ′) k, = √ dρ
−lmax 1 x
2 k
− τ + τ 2x
k 1 k k −∞ k 1 π h̄τ 3= =
i
x − ϵĤ×⟨ k|ρk⟩⟨ρk|e h̄ |xk−1⟩. (56)
Recall that √
√ ( [ ( ) ])
τ 3 i 2ρ√k arctan 2τxk−1√ π+
⟨xk|ρk⟩ = φ τ h̄ 3 3 6ρk (xk) = e , (57)2 arctan(6)
i i
⟨ρk|e− h̄ ϵĤ x e− h̄ ϵH(ρk,x| ⟩ ≃ k−1) 2k−1 ⟨ρk|xk−1⟩ + O(ϵ )
i
e− h̄ ϵH(ρk,x≃ k−1)φ∗ρ (xk−1) + O(ϵ
2). (58)
k
Substituting these expre[ss∫ions in(to Eq. (56) gives )][ ( )]
+lmax N∏−1 ∫dx +∞ ∏N dρ i
Kdisc(x t x′
k k ϵS
, , , t ′) = e h̄ disc , (59)
−lmax 1 − τx + τ 2x
2 2π h̄
k=1 k k −∞ k=1
where the discrete acti⎡on Sdisc is(given by∑ ) ( )⎤N−1 2ρ ⎣arctan 2τxk−1 arctan 2τxk−1−1√ − √ ∑N−1S k 3 3disc = √ ⎦ − H(ρk, xk−1) (60)
k 1 τ 3 ϵ= k=1
inally, we take the limit N → ∞, so that ϵ → 0. We obtain the final expression for the propagator
as follows ∫
i
K (x, t, x′, t ′) = DxD e h̄ Sρ , (61)
where the integra∏tion measures Dx and Dρ are defined asN−1 ( )dx Nk ∏ dρDx = lim 2 and D kρ = lim . (62)N→∞ 1 − τx + τ 2x N→∞ 2π h̄
k=1 k k k=1
and the[continuou]s ac∫tion St [is given by ]ẋ(ν)
S x(t), x(t ′) = dν 2 2 ρ(ν) − H(ρ(ν), x(ν)) , (63)
t ′ 1 − τx(ν) + τ x (ν)
here ẋ(ν) = dx/dν. These results (61), (62) and (63) are remarkably similar to the one obtained
y Pramanick [25] from the Perivolaropoulos space [26].
Now, taking the lim∫it τ → ∏0, the d∏efo(rmed p)ropagator (61) is reduced toN−1 N
0 dρk i 0K (x, t, x′, t ′) = lim dxk e h̄ S , (64)
N→∞ 2π h̄
k=1 k=1
where th[e undefor]med∫action S0 is given byt
S0 x(t), x(t ′) = dν [ẋ(ν)ρ(ν) − H(ρ(ν), x(ν))] . (65)t ′
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It is straightforward to show that the following relations
K (x, x′, t, t ′) K 0≤ (x, x′, t, t ′) 0H⇒ S ≤ S . (66)
The stationary∫ path (63) is obtaine∫d by u(sing the variational princip)let t ∂L ∂L
δS = δ dνL [ẋ(ν), x(ν)] = dν δx(ν) + δẋ(ν) = 0, (67)
t ′ t ′ ∂x(ν) ∂ ẋ(ν)
here the Lagrangian L of the system is given by
ẋ(ν)
L [ẋ(ν), x(ν)] = 2 2 ρ(ν) − H(ρ(ν), x(ν)). (68)1 − τx(ν) + τ x (ν)
he solutions of Eq. (67) generates the following differential equations
∂H ∂H
ẋ = (1 2 2− τx + τ x ) = {x, ρ}τ , (69)
∂ρ ∂ρ
2 2 ∂H ∂Hρ̇ = −(1 − τx + τ x ) = −{x, ρ}τ , (70)
∂x ∂x
where {x, ρ 2 2}τ = (1−τx+τ x ) is the position-deformed Poisson bracket. By taking the limit τ → 0,
we recover the ordinary Hamilton’s equations of motion.
3.2. Path integral and propagator in Fourier transform and its inverse representions
Using the generalized Fourier transform and its inverse representations (40), (41) and taking into
account Eq. (54),√we have
√ ∫
τ 3 +l
[ ( ) ]
max dx i 2ρ√ arctan 2τ√x−1 π− +
ψ(ρ, t) = e τ h̄ 3 3 6∫2 arctan(6) 2−lmax 1 − τx + τ x2
+l √max K (x, t, x′, t ′) a√rctan(6)× ∫ dx
′
√
−lmax 1 − τx′[+ τ 2(x′2 π) h̄ ]2τ 3
+∞
i 2ρ
′ ′
√ arctan 2τ√x −1 π′ +
× dρ e τ h̄ 3 3 6 ψ(ρ ′, t ′). (71)
−∞
This path integra∫l can be rewritten as follows
+∞
ψ(ρ, t) ′= dρ K(ρ, t, ρ ′, t ′)ψ(ρ ′, t ′), (72)
−∞
where K is the propagator in Fourier transform and its inverse representions for a particle to go
from a state ψ(ρ ′) to a state∫ψ(ρ) in a time interval t t ′− is
1 +lmax dx dx′
K(ρ, t, ρ ′, t ′) =
2π h̄ 2 2 ′ 2 ′2−lm[ax 1 −( τx +) τ x 1 −( τx +)]τ x
i ∫2 ′− √ ρ arctan 2τ√x−1 ′−ρ arctan 2τ√x −1×e τ h̄ 3 3 3 K (x, t, x′, t ′),
1 dx dx′ i
DxD e h̄S= ρ , (73)
2π h̄ 1 − τx + τ 2x2 1 − τx′ + τ 2x′2
with S the action given by [ ( ) ( )]
2 2τx − 1 2τx′ − 1
S(ρ, t, ρ ′, t ′) S ρ arctan ρ ′= − √ √ − arctan √ . (74)
τ 3 3 3
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3.3. Propagators for a free particle and for a harmonic oscillator
In this section, we compute the propagator in position-space (55) and the one in Fourier
ransform and its inverse representations (73) for the Hamiltonians of a free particle and a simple
armonic oscillator. From these propagators, we deduce the actions of both systems.
The generalized form of the Hamiltonian we chose is
P̂2
Ĥ = + V (X̂). (75)
2m
The action of this(Hamiltonian of)the functions φ(x) and ψ(ρ) reads as follows
h̄2
Ĥφ(x) = ⎛− D
2
x +⎛V (x) φ(x),2m ( )⎝ √ ⎞⎞
(76)
τ h̄ 3 d
ρ2 ⎝2 tan i (2 dρĤψ(ρ) = + V )⎠⎠√ √ ψ(ρ). (77)2m τ 3 + tan i τ h̄ 3 d2 dρ
.3.1. Propagator of a free particle
The free particle problem defined by the Hamiltonian is given by
P̂2
Ĥfp = . (78)2m
The propagator in position-representation in the time interval ∆t = t t ′− is given by
i P̂2
K (x, x′fp , ∆t) −= ⟨x|e h̄ 2m ∆t |x′∫⟩
arctan(6) +∞ i P̂2−
= √ ⟨∫x| dρe h̄ 2m ∆t ρ ′| ⟩⟨ρ|x ⟩π h̄τ 3 −∞
arctan(6) +∞ i ρ2−
= h̄ 2m
∆t ′
∫ √ ( dρe ⟨x|ρ⟩⟨ρ|x ⟩π h̄τ 3 −∞ [ ( ) ( )] )
+∞ dρ i 2ρ√ arctan 2τ√x−1 arctan 2τ√x′−1 i ρ
2
− −
e τ h̄ 3 3 3 h̄ 2m
∆t
= . (79)
−∞ 2π h̄
ompleting this Gauss√ian integral (79),[we h(ave ) ( )]
m 2m ′ 2
′ i 2 arctan
2τ√x−1 −arctan 2τ√x −1
Kfp(x, x , ∆t) = e h̄3τ ∆t 3 3 . (80)2π h̄i∆t
hus the deformed-[classica(l action is)given by ( )]
2m 2τx − 1 2τx′ − 1 2
Sfp = 2 arctan √ − arctan √ . (81)3τ ∆t 3 3
The limit τ → 0, the latter propagator properly reduces to the well-known result in ordinary
quantum mechanics for a free particle [32√,33] that is
m i m(x−x′)2
lim K ′fp(x, x , ∆t) K 0= fp(x, x
′, ∆t) = e h̄ 2∆t , (82)
τ→0 2π h̄i∆t
and the corresponding classical action is given by
′ 2
lim S S0
m (x − x )
fp = fp = , (83)
τ→0 2 ∆t
S0fp m (x x′− )2
= = T 0, (84)∆t 2 (∆t)2
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a
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c
(
a
where T 0 is the standard kinetic energy of the particle. Also, it is straightforward to show the
following relations
Kfp(x, x′, ∆t) ≤ K 0fp(x, x
′, ∆t) H⇒ S 0 0fp ≤ Sfp H⇒ T ≤ T , (85)
here T is the deform[ ed kin(etic energy)of the par(ticle )]
2m 2 x 1 2 x′ 1 2τ − τ −
T = 2 2 arctan √ − arctan √ . (86)3τ (∆t) 3 3
his indicates that the deformed propagator and actions of the free particle are dominated by the
tandard ones without quantum deformation. These results indicate that the quantum deformation
ffects in this space shortens the paths of particles, allowing them to move from one point to
nother in a short time. In one way or another, as one can see from Eq. (85), these results can
e understood as free particles use low kinetic energies to travel faster in this deformed space. This
onfirms our recent results [21,22] and strengthens the claim that the position deformed-algebra
11) induces strong deformation of the quantum levels allowing particles to jump from state to
nother with low energy transitions [21,22].
The propagator for the √Fourier tra∫nsform and its inverse∫representions is given by
1 m +lmax dx +lmax dx′
K(ρ, ρ ′, ∆t) =
2π h̄ [2h̄π i∆(t ) 1 −(τx + τ)]2x2 1 − τx′ + τ 2x′2−lmax −lmax
i 2√ ρ arctan 2τ√x−1 ′ 2τx
′
−1
−
×e τ h̄ 3 [ ( 3) −ρ ar(ctan √)3]
2m ′ 2i 2 arctan
2τ√x−1 −arctan 2τ√x −1
×e h̄3τ ∆t 3 3 . (87)
The corresponding actio[n is given(by ) ( )]
2 2τx − 1 2τx′ − 1
Sfp = Sfp − √ ρ arctan ′√ − ρ arctan √ . (88)
τ 3 3 3
3.3.2. Reduced propagator of a simple harmonic oscillator
The simple harmonic oscillator problem is defined by the Hamiltonian
P̂2 1
Ĥ 2 2= + mω X̂ . (89)
2m 2
The propagator in position re(presentatio)n is given by
i P̂2 1 2 2
− +
K (x x′ t) x e h̄ 2m 2
mω X̂ ∆t
, , ∆ x′ho = ⟨ | | ⟩ (90)
For a sufficiently small time ∆t = ϵ, the time evolution operator is factorizable as a consequence
of the Baker-Campbell-Hausdorff formula [32]. The propagator (90) is rewritten as follows
P̂2 i 2 2
K (x x′ ) x e−iho , , ϵ = ⟨ | 2mh̄ ϵe− 2h̄mω X̂ ϵ x′| ⟩∫+ O(ϵ2)
arctan(6) +∞i 2 2′2
∫ x e− 2h̄mω x ϵ
i P̂
−
= √ ⟨ | dρe h̄ 2m ϵ ′|ρ⟩⟨ρ|x ⟩ + O(ϵ2)
π h̄τ 3 −∞
+∞ dρ
=
−∞( 2π[h̄ ( ) ( )] ( ) )
2ρ ′ 2i √ arctan 2τ√x−1 arctan 2τ√x −1 i ρ 1 2 ′2− − + mω x ϵ
e τ h̄ 3 3 3 h̄ 2m 2× . (91)
Computing the Gaus√sian integral (9[1), w(e hav)e ( )]
m i 2m arctan 2τ√x ′ 2−1′ 2 −arctan 2τ√x −1 i− mω2x′2ϵKho(x, x , ϵ) = e h̄3τ ϵ 3 3 2h̄ , (92)2π h̄iϵ
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a
d
r
a
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t
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p
a
and the correspon[ding def(ormed cla)ssical action(is given b)y]
2m 2τx − 1 2 2τx′ − 1 1
S 2 ′2ho = 2 arctan √ − arctan √ − mω x ϵ. (93)3τ ϵ 3 3 2
t the limit τ → 0, we recover the ordinary propagator and the classical action of the simple
armonic oscillator [32,33] √ ( )
m i m(x
′ 2
−x ) 1 2 ′2
lim K (x x′ ) K 0 (x x′ ) e h̄ 2
−
ϵ 2mω x ϵ
ho , , ϵ =
0 ho
, , ϵ = ,
τ→ 2π h̄iϵ
0 m(x − x
′)2 1
lim S 2 ′2ho = Sho = − mω x ϵ, (94)
τ→0 2ϵ 2
S0 m(x ′ 2ho − x ) 1
= 2 − mω
2x′2 0= E
ϵ 2(ϵ) 2 m
, (95)
where E0m is the standard mechanical energy of a simple harmonic mechanics. Like in the prior
instance (85) it is simple to demonstrate that
K (x, x′ho , ϵ) 0≤ Kho(x, x
′, ϵ) 0 0H⇒ Sho ≤ Sho H⇒ Em ≤ Em, (96)
where Em is the de[formed (mechanica)l energy of(harmonic)os]cillator
2m 2τx − 1 2τx′ 2− 1 1
E 2 ′2m = 2 2 arctan √ − arctan √ − mω x . (97)3τ ϵ 3 3 2
his also strengthens our obtained result in (85). In more general case, we can see that the harmonic
scillator potential does not affect the motion of the deformed free particle such that
K 0 0 0 0ho ≈ Kfp ≤ Kfp ≈ Kho H⇒ Sho ≈ Sfp ≤ Sfp ≈ Sho. (98)
The propagator in Fou√rier tran∫sform and its inverse r∫epresentions is given by
1 m +lmax dx +lmax dx′
K(ρ, ρ ′, ϵ) =
[ 2π( h̄ 2)π h̄iϵ ( 1)]− τx + τ 2x2 1 − τx′ + τ 2x′2−lmax −lmax
i 2√ ρ arctan 2τ
′
− √
x−1 ′
−ρ arctan 2τ√x −1
×e τ h̄ 3[ ( )3 ( )]3
i 2m arctan 2 x
′ 2τ√−1 −arctan 2τ√x −1 i 2 ′2−
×e h̄3τ2ϵ 3 3 2h̄
mω x ϵ
, (99)
and its action is given b[y ( ) ( )]
2 2τx ′− 1 2τx − 1
Sho = Sho − √ ρ arctan √ − ρ ′ arctan √ . (100)
τ 3 3 3
. Conclusion
We have constructed path integrals in Euclidean position representation and in Fourier transform
nd its inverse representations within a position-deformed Heisenberg algebra (11). We have
erived from these path integrals the propagators and the corresponding classical actions. These
esults are remarkably similar to the one obtained by Pramanick [25] from the Perivolaropoulos’s
lgebra [26]. Then, the classical equations of motion are obtained by the principle of least action. The
amiltonians of a free particle and a simple harmonic oscillator are used as examples to compute
he propagators and the actions in position representation and in Fourier transform and its inverse
epresentations. We have shown through these results that, the propagators and the actions of these
ystems in position space representation are properly bounded by the well-known results in the
→ 0 limit. This has indicated that the deformation induced by the maximal length shortens
article pathways, allowing them to travel faster from one point to the next with low kinetic
nd mechanical energies. This confirms our recent results [21,22] and strengthens the claim that
12
L.M. Lawson, P.K. Osei, K. Sodoga et al. Annals of Physics 457 (2023) 169389
,
D
a
S
I
R
D
s
D
A
p
f
f
A
(
m
T
A
H
s
H
the position deformed-algebra (11) induces deformation of the quantum levels allowing particles
to jump from state to another with low transition energies. Finally, the propagators for Fourier
transform and its inverse representations for both systems are given as integral expressions and
we have deduced the corresponding actions.
CRediT authorship contribution statement
Latévi M. Lawson: Initiated the project, Performed the computations, Wrote the main manuscript
iscussion, Results analysis and Interpretation. Prince K. Osei: Initiated the project, Supervised
nd finalized the writing of the manuscript, Discussion, Results analysis and Interpretation. Komi
odoga: Supervised and finalized the writing of the manuscript, Discussion, Results analysis and
nterpretation. Fred Soglohu: Performed the computations, Wrote the main manuscript, Discussion,
esults analysis and Interpretation.
eclaration of competing interest
The authors declare that they have no known competing financial interests or personal relation-
hips that could have appeared to influence the work reported in this paper.
ata availability
No data was used for the research described in the article.
cknowledgments
LML acknowledges support from DAAD (German Academic Exchange Service) under the DAAD
ostdoctoral in region grant. He would also like to thank Tevian Dray who supported his application
or this DAAD financial grant. Thanks for Andreas Fring, Sebastián Franchino-Viñas and Vishnu Jejjala
or providing us some materials which considerably improved the quality of this paper.
ppendix
In this appendix we provide a detail information on the physically relevance of the states |φρ⟩
35) by restricting the Hilbert H = L2(R) into its dense bounded domain D 2τ = L (−lmax, +lmax).
To construct a Hilbert space representation that describes the maximal length and the minimal
omentum uncertainties, one has to solve the eigenvalue problem
− ih̄Dxφρ(x) = ρφρ(x), ρ ∈ R, φρ(x) ∈ H. (101)
he solution of√this equati(on is give√ [n by ( ) ])
τ 3 2ρ 2τx − 1 π
φρ(x) = exp i √ arctan √ + .2π τ h̄ 3 3 6
s we can see, the expectation value of operator (energy) X̂n (n ≥ 2) in this infinite dimensional
ilbert space diverges
√ ∫
+∞ n
n τ 3 x dx
⟨φρ |X̂ |φρ⟩ =
π 1 − τx + τ 2x2
> ∞.
−∞
In comparaison to Kempf et al. formalism [4], the momentum eigenvectors |φρ⟩ are not physical
tates. To circumvent this problem, we restrict the study from the infinite dimensional Hilbert space
into its bounded dense domaine D 2= L (−l , +l ) in such away that, for τ → 0, one recoversτ max max
13
L.M. Lawson, P.K. Osei, K. Sodoga et al. Annals of Physics 457 (2023) 169389
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the entire space H L2= (R). In this domaine, repeating the resolution of Eq. (101), we obtain the
solution (35) o√f the paper given
√ (by [ ( ) ])
τ 3 2ρ 2τx − 1 π
φρ(x) = exp i √ arctan √ + .2 arctan(6) τ h̄ 3 3 6
ith this solution in hand, we show that expectation values of position energy operator X̂n (n ∈ N)
s finite:
√ ∫
τ 3 +lmax xndx
⟨φ nρ |X̂ |φρ⟩ = < ∞. (102)2 arctan(6) 2 2−lmax 1 − τx + τ x
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