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CHINESE JOURNAL OF PHYSICS
VOL. 50, NO. 4
August 2012
Natural Cutoffs and Dynamics of Harmonic Oscillations
Kourosh Nozari,∗ S. Namdari,† and J. Vahedi‡
Department of Physics, Islamic Azad University, Sari Branch, Sari, Iran
(Received June 18, 2011; Revised October 25, 2011)
Different proposed candidates of quantum gravity, such as string theory, noncommutative
geometry, loop quantum gravity, and doubly special relativity, all predict the existence of a
minimum observable length and/or a maximum observable momentum. These natural cutoffs
modify the standard Heisenberg uncertainty principle leading to the so-called Generalized
Uncertainty Principle (GUP). In this paper, we study the effects of minimal length and
maximal momentum on the dynamics of harmonic oscillations. Within this framework we
also study the status of Ehrenfest’s theorem with the GUP.
PACS numbers: 04.60.-m
I. INTRODUCTION
Various approaches to quantum gravity, such as string theory and loop quantum
gravity as well as black holeqphysics, predict a minimum measurable length of the order
Gh̄
−35 m. In the presence of this minimal observable
of the Planck length, ℓp =
c3 ∼ 10
length, the standard Heisenberg Uncertainty Principle attains an important modification
leading to the Generalized Uncertainty Principle (GUP) [1]. As a result, the corresponding
commutation relations between position and momenta are generalized too. In recent years
a lot of attention has been directed to extending the fundamental problems of physics in
the GUP framework (see for instance [2–10]). Since within the GUP framework one cannot
probe distances smaller than the minimum measurable length in finite time, we expect that
it modifies the Hamiltonian of physical systems too (see [5] for instance). Recently, a GUP
was proposed by Ali et al. which is consistent with the existence of a minimal measurable
length and a maximal measurable momentum [11, 12]. This kind of GUP has its origin
in Doubly Special Relativity [13]. In the streamline of these researches, in this paper we
study the effects of minimal length and maximal momentum on the dynamics of harmonic
oscillations. We also study the status of Ehrenfest’s theorem within this framework.
∗
Electronic address: [email protected]
Electronic address: [email protected]
‡
Electronic address: [email protected]
†
http://PSROC.phys.ntu.edu.tw/cjp
554
c 2012 THE PHYSICAL SOCIETY
OF THE REPUBLIC OF CHINA
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KOUROSH NOZARI, S. NAMDARI, AND J. VAHEDI
555
II. A GENERALIZED UNCERTAINTY PRINCIPLE
One of the implications of the existence of a minimal observable length is the modification of the commutation relation between position and momentum in usual quantum
mechanics leading to the so-called generalized uncertainty principle. Recently, a GUP was
proposed by Ali et al. which is consistent with the existence of both a minimal measurable
length and a maximal measurable momentum [11, 12]. The motivation behind this kind
of GUP is the fact that within the framework of Doubly Special Relativity, the existence
of both a minimal length and a maximal momentum is guaranteed [13]. In this new GUP
framework, the spaces of position and momentum are assumed to be commutative separately, i.e., [Xi , Xj ] = [Pi , Pj ] = 0. Also the following deformed Heisenberg algebra is
satisfied:
Pi Pj
2
2
(1)
[Xi , Pj ] = ih̄ δij − α P δij +
+ α P δij + 3Pi Pj ,
P
where α = α0 /MP c = α0 ℓP /h̄, P 2 = Σ3j=1 Pj Pj , MP is the Planck mass, ℓP is the Planck
length, and MP c2 is the Planck energy. Using the above commutation relations, we can
obtain the generalized uncertainty relation in one dimension and up to the second order of
the GUP parameter as follows [11, 12]:
h̄ 1 − 2αhP i + 4α2 hP 2 i
∆X∆P ≥
2"
!
#
p
α
h̄
+ 4α2 (∆P )2 + 4α2 hP i2 − 2α hP 2 i .
1+ p
≥
(2)
2
hP 2 i
The above inequality implies both a minimum length and a maximum momentum at
the same time, namely [11, 12]
∆X ≥ (∆X)min ≈ α0 ℓP ,
MP c
∆P ≤ (∆P )max ≈
.
α0
(3)
We can also rewrite the position and momentum operators in terms of new variables [11]
Xi = xi ,
(4)
Pi = pi 1 − αp + 2α2 p2 ,
where xi and pi obey the usual commutation relations [xi , pj ] = ih̄δij . It is straightforward
to check that with this definition, Equation (1) is satisfied up to O(α2 ). Therefore, we can
interpret pi and Pi as follows: pi is the momentum operator at low energies (pi = −ih̄∂/∂xi )
and Pi is the P
momentum operator at high energies. Moreover, p is the magnitude of the pi
vector (p2 = 3i pi pi ).
To study the effects of this kind of GUP on quantum mechanical systems, let us
consider the following general Hamiltonian:
H=
P2
+ V (x),
2m
(5)
NATURAL CUTOFFS AND DYNAMICS . . .
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VOL. 50
which by using Equation (4) can be written as
H = H0 + αH1 + α2 H2 + O(α3 ),
(6)
p2
where H0 = 2m + V (x) and
H1 =
−p3
,
m
H2 =
5p4
.
2m
(7)
From this generalization of the Hamiltonian, one can obtain the eigenfunctions and eigenvalues of a harmonic oscillator in the quasi-space representation within the Schrödinger
picture of quantum mechanics. Here we adopt another strategy: we study the dynamics of
harmonic oscillation in the Heisenberg picture through the deformed commutators of the
model. Before starting to perform our calculations, we note that due to the presence of the
cubic term of the particles’ momentum, time reversal invariance is violated as a result of
the GUP with maximal momentum. In this situation, the Hamiltonian is not certainly the
physical energy of the system under consideration. Nevertheless, the Heisenberg equation
of motion is still valid. So, our strategy is sensible.
III. EQUATION OF MOTION FOR HARMONIC OSCILLATIONS
The equation of motion for an observable A in the Heisenberg picture of quantum
mechanics is as follows [14]:
dA
i
= [H, A] .
dt
h̄
(8)
The Hamiltonian of a linear harmonic oscillator is given by
H=
1
P2
+ mω 2 X 2 .
2m 2
(9)
We start with the following GUP
[X, P ] = ih̄ 1 − 2αP + 4α2 P 2 .
(10)
Therefore, the equations of motion For X and P in the Heisenberg picture are
and
P
dX
=
1 − 2αP + 4α2 P 2 ,
dt
m
dP
= −mω 2 X − α (XP + P X) + 2α2 XP 2 + P 2 X ,
dt
(11)
(12)
respectively. Before solving these equations of motion, we note an important point: Equation (12) implicitly contains a violation of the Weak Equivalence Principle in the quantum
VOL. 50
KOUROSH NOZARI, S. NAMDARI, AND J. VAHEDI
557
gravity regime. In fact, as we have shown by studying the Heisenberg equations of motions in the presence of the GUP, the acceleration is no longer mass-independent because
of the mass-dependence through the momentum P . Therefore, the equivalence principle
is dynamically violated in the GUP framework. This dynamical violation of the equivalence principle probably has its origin on the very nature of the spacetime manifold at the
quantum gravity level.
Now we can obtain the time evolution of X and P by using the Baker-Hausdorff
lemma as
P 2 (0)
P (0)
sin ωt + α −2
(ωt)
X(t) = X(0) cos ωt +
mω
mω
3
4 P 2 (0)
2
2
+ [X(0)P (0) + P (0)X(0)] (ωt) +
− mωX (0) (ωt)3
2
3 mω
3
5
P (0)
4
2
− [X(0)P (0) + P (0)X(0)] (ωt) + α 4
(ωt)
8
mω
− 5 X(0)P 2 (0) + P 2 (0)X(0) + 2P (0)X(0)P (0) (ωt)2
20 P 3 (0) 17
+ mω[X 2 (0)P (0) + P (0)X 2 (0)]
+ −
3 mω
6
13
107
3
+ mωX(0)P (0)X(0) (ωt) +
[X(0)P 2 (0) + P 2 (0)X(0)]
3
24
37
5
+ P (0)X(0)P (0) − m2 ω 2 X 3 (0) (ωt)4 ,
(13)
12
2
and
P (t) = P (0) cos ωt − mωX(0) sin ωt + α {mω [X(0)P (0) + P (0)X(0)] (ωt)
7
+ 2P 2 (0) − m2 ω 2 X 2 (0) (ωt)2 − mω [X(0)P (0) + P (0)X(0)] (ωt)3
6
2 2
7 2 2 2
4
+ − P (0) + m ω X (0) (ωt)
3
12
2
+α −2mω X(0)P 2 (0) + P 2 (0)X(0) (ωt)
3
11
+ − P 3 (0) + m2 ω 2 X 2 (0)P (0) + P (0)X 2 (0)
2
2
17
2 2
2
+3m ω X(0)P (0)X(0) (ωt) + mω
X(0)P 2 (0) + P 2 (0)X(0)
3
17
117 3
35
2 2 3
3
+ P (0)X(0)P (0) − 2m ω X (0) (ωt) +
P (0) − m2 ω 2
6
24
12
47 2 2
4
2
2
X (0)P (0) + P (0)X (0) − m ω X(0)P (0)X(0) (ωt) ,
12
(14)
respectively, where only terms up to the second order of α are considered. X(0) and P (0) are
the initial values of the position and momentum operators, respectively. It is evident that in
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NATURAL CUTOFFS AND DYNAMICS . . .
VOL. 50
the limit of α → 0 one recovers the usual results of ordinary quantum mechanics. The terms
proportional to α and α2 indicate that in the GUP framework the usual harmonic oscillator
is no longer harmonic, in the sense that its time evolution is not completely oscillatory.
Practically, one needs to know the expectation values of physical quantities. To
compute expectation values, we need a well-defined physical state. Note that the eigenstates
of the position operator are no longer physical states because of the existence of a minimal
length which completely destroys the notion of locality. So, we should consider a physical
state such as |αi where |αi is, for example, a momentum space eigenstate (note also that
one can use the quasi-space representation introduced by Kempf et al. in Ref. [1]). Suppose
that Pα (0) = hα|P (0)|αi, Xα (0) = hα|X(0)|αi. Now, the time evolution of the expectation
value of the momentum operator is
Pα (0)
hα|P (t)|αi
=
cos ωt − ωXα (0) sin ωt
m
m
+α {ω (Xα (0)Pα (0) + Pα (0)Xα (0)) (ωt)
2
Pα (0)
2 2
+ 2
− mω Xα (0) (ωt)2
m
7
− ω (Xα (0)Pα (0) + Pα (0)Xα (0)) (ωt)3
6
7
2 Pα2 (0)
2 2
4
+ mω Xα (0) (ωt)
+ −
3 m
12
2
+α −2ω Xα (0)Pα2 (0) + Pα2 (0)Xα (0) (ωt)
11 Pα3 (0) 3
+ −
+ mω 2 Xα2 (0)Pα (0) + Pα (0)Xα2 (0)
2 m
2
2
+3mω Xα (0)Pα (0)Xα (0) (ωt)2
17
+ω
Xα (0)Pα2 (0) + Pα2 (0)Xα (0)
3
17
2 3
+ Pα (0)Xα (0)Pα (0) − 2mω Xα (0) (ωt)3
6
117 Pα3 (0) 35
+
− mω 2 Xα2 (0)Pα (0) + Pα (0)Xα2 (0)
24 m
12
47
2
4
− mω Xα (0)Pα (0)Xα (0) (ωt) .
12
(15)
This relation shows that there is a complicated mass-dependence of the expectation
value of the momentum operator. In ordinary quantum mechanics, hα|Pm(t)|αi and Pαm(0)
are mass independent. Here, although Pαm(0) is still mass independent, but now hα|Pm(t)|αi
has a complicated mass dependence. This is a novel phenomenon which has its origin on
the nature of spacetime at the Planck scale. Physically, it is reasonable to say that the
expectation value of the momentum operator should to be a function of the particle’s mass,
but the mass dependence here is more complicated than the ordinary quantum mechanics
case.
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KOUROSH NOZARI, S. NAMDARI, AND J. VAHEDI
559
In ordinary quantum mechanics (that is, the α → 0 limit), the expectation value
of position operator defined as x(t) = x(0) cos ωt + p(0)
mω sin ωt , taken with respect to a
stationary state is zero:
hp(0)i
sin ωt = 0.
(16)
mω
q
q
h̄
mωh̄ †
† + a) and p(0) = i
†
Now, we define x(0) =
(a
2mω
2 (a − a), where a and a
are lowering and raising operators, respectively. To compute the expectation values of
operators in the presence of the quantum gravity effects encoded in the GUP, we simply
introduce an energy eigenstate as
hx(t)i = hx(0)i cos ωt +
(a† )n
|ni = √ |0i.
n!
(17)
With this eigenstate, the expectation value of X(t) and P (t) in the GUP framework now
are non-vanishing, since
1
3
(18)
hn|X(t)|ni = α(2n + 1)h̄ −ωt + (ωt)
6
and
1
1
2
4
hn|P (t)|ni = α(2n + 1)mωh̄ ( ωt) − (ωt) .
2
24
(19)
Amazingly, we see that the expectation value of an observable taken with respect to
a stationary state varies with time. This is a consequence of the GUP.
IV. EHRENFEST THEOREM
In this section we show that the Ehrenfest theorem is not valid in the GUP framework.
In ordinary quantum mechanics, Ehrenfest’s theorem in one space dimension is described
by the following equations
p
dhxi
=h i
dt
m
(20)
d
dhpi
= h− V (x)i.
dt
dx
(21)
and
In the presence of a minimal length and maximal momentum, a simple calculation
NATURAL CUTOFFS AND DYNAMICS . . .
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VOL. 50
shows that
d
hP (0)i
hP 2 (0)i
hX(t)i = −ωhX(0)i sin ωt +
cos ωt + α −2
dt
m
m
hP 2 (0)i
+3 (hX(0)ihP (0)i + hP (0)ihX(0)i) ω 2 t + 4
mω
5
2
3 2
4 3
−3mωhX (0)i ω t − (hX(0)ihP (0)i + hP (0)ihX(0)i) ω t
2
3
P (0) − 10 hX(0)ihP 2 (0)i + hP 2 (0)ihX(0)i
+α2 4
m
+4hP (0)ihX(0)ihP (0)i] ω 2 t
hP 3 (0)i 17
+ −20
+ mω hX 2 (0)ihP (0)i + hP (0)ihX 2 (0)i
mω
2
+13mωhX(0)ihP (0)ihX(0)i] ω 3 t2
107
hX(0)ihP 2 (0)i + hP 2 (0)ihX(0)i
+
6
37
2 2
3
4 3
+ hP (0)ihX(0)ihP (0)i − 10m ω hX (0)i ω t
3
(22)
and
h
hP (0)i
P (t)
i =
cos ωt − ωhX(0)i sin ωt + α {(hX(0)ihP (0)i
m
m
hP 2 (0)i
2
2
2
+hP (0)ihX(0)i) ω t + 2
− mω hX i (ωt)2
m
7
− (hX(0)ihP (0)i + hP (0)ihX(0)i) ω 4 t3
6
2 hP 2 (0)i
7
2
2
4
+ −
+ mω hX (0)i (ωt)
3 m
12
2
+α −2 hX(0)ihP 2 (0)i + hP 2 (0)ihX(0)i ω 2 t
11 hP 3 (0)i 3
+ mω 2 hX 2 (0)ihP (0)i + hP (0)ihX 2 (0)i
+ −
2
m
2
+2hX(0)ihP (0)ihX(0)i)] (ωt)2
17
+
hX(0)ihP 2 (0)i + hP 2 (0)ihX(0)i
3
17
3
+ hP (0)ihX(0)ihP (0)i − 2mωhX (0)i ω 4 t3
6
117 hP 3 (0)i 35
+
− mω 2 hX 2 (0)ihP (0)i + hP (0)ihX 2 (0)i
24
m
12
47
2
4
− mω hX(0)ihP (0)ihX(0)i (ωt) .
12
(23)
VOL. 50
KOUROSH NOZARI, S. NAMDARI, AND J. VAHEDI
561
It is evident that the last two equations are equal (in order to satisfy Equation (20))
only in the limit of α → 0. Therefore, in the framework of the GUP, Equation (20) is
not valid. In the same manner, one can show also that Equation (21) is not valid in
the framework of the GUP. So, we conclude that Ehrenfest’s theorem is not valid in the
quantum gravity era. That Ehrenfest’s theorem breaks down probably has its origin on the
very nature of spacetime at the Planck scale.
V. SUMMARY
The very notion of locality breaks down at the Planck scale. There is a fundamental
length scale of the order of the Planck length (ℓp ∼ 10−35 m) that cannot be probed in
a finite time. This reflects the fact that the spacetime manifold has a noncommutative,
foamy structure at the very high energy limit. This feature gives also a fuzzy nature to
spacetime points at this scale, due to the finite resolution of adjacent points. Recently, in
the framework of Doubly Special relativity as a route to quantum gravity, it has been shown
that there is a maximal observable momentum too. The existence of a minimal observable
length and/or a maximal observable momentum brings a lot of new phenomenology into the
rest of the physics. In this paper, following our previous studies in this field, we studied the
mutual effects of a minimal length and maximal momentum on the dynamics of harmonic
oscillations. We have shown that in the quantum gravity era, the notion of harmonic
oscillation is no longer valid. Only in the limit of α → 0 one recovers the usual results of
ordinary quantum mechanics. In this respect, in the GUP framework, the usual harmonic
oscillator is no longer harmonic since its time evolution is not completely oscillatory. By
treating the expectation value of the momentum operator, we have shown that there is a
complicated mass-dependence of this expectation value. This complicated mass-dependence
of the expectation value of the momentum operator is a novel phenomenon as a direct
implication of the adopted GUP. As another important aspect of the problem, we have
shown also that Ehrenfest’s theorem breaks down at the Planck scale, and this break down
probably has its origin on the very nature of spacetime at the Planck scale. Other important
outcomes of our study are a violation of the time reversal symmetry and also a break down
of the weak equivalence principle at the quantum gravity level. In fact, in the presence of
the GUP, the acceleration is no longer mass-independent because of the mass-dependence
through the momentum. Therefore, the weak equivalence principle is dynamically violated
in the GUP framework. On the other hand, due to the presence of the cubic term of the
particles’ momentum in the Hamiltonian with minimal length and maximal momentum,
time reversal invariance is violated. In this situation, the Hamiltonian is certainly not the
physical energy of the system under consideration. However, the Heisenberg equation of
motion is still valid in this case. Finally, we note that the minimal length and maximal
momentum provide natural cutoffs for regularization of field theories at the quantum gravity
level.
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NATURAL CUTOFFS AND DYNAMICS . . .
VOL. 50
Acknowledgements
This work is supported financially by the Research Deputy of the Islamic Azad University, Sari Branch, Sari, Iran.
References
[1] G. Veneziano, Europhys. Lett. 2, 199 (1986).
D. Amati, M. Ciafaloni, and G. Veneziano, Phys. Lett. B 197, 81 (1987).
D. Amati, M. Ciafaloni, and G. Veneziano, Int. J. Mod. Phys. A 3, 1615 (1988).
D. Amati, M. Ciafaloni, and G. Veneziano, Nucl. Phys. B 347, 530 (1990).
D. J. Gross and P. F. Mende, Nucl. Phys. B 303, 407 (1988).
D. Amati, M. Ciafaloni, and G. Veneziano, Phys. Lett. B 216, 41 (1989).
T. Yoneya, Mod. Phys. Lett. A 4, 1587 (1989).
K. Konishi, G. Paffuti, and P. Provero, Phys. Lett. B 234, 276 (1990).
M. Maggiore, Phys. Lett. B 304, 65 (1993).
M. Maggiore, Phys. Lett. B 319, 83 (1993).
M. Maggiore, Phys. Rev. D 49, 5182 (1994).
L. J. Garay, Int. J. Mod. Phys. A 10, 145 (1995).
A. Kempf, G. Mangano, and R. B. Mann, Phys. Rev. D 52, 1108 (1995).
F. Scardigli, Phys. Lett. B 452, 39 (1999).
R. J. Adler and D. I. Santiago, Mod. Phys. Lett. A 14, 1371 (1999).
R. J. Adler, Am. J. Phys. 78, 925 (2010).
[2] K. Nozari, Phys. Lett. B 629, 41 (2005).
K. Nozari and B. Fazlpour, Gen. Relat. Gravit. 38, 1661 (2006).
K. Nozari and S. H. Mehdipour, Europhys. Lett. 84, 20008 (2008).
[3] K. Nozari and T. Azizi, Gen. Relat. Gravit. 38, 735 (2006).
[4] K. Nozari and S. H. Mehdipour, Chaos, Solitons and Fractals 32, 1637 (2007).
[5] P. Pedram, Int. J. Mod. Phys. D 19, 2003 (2010).
[6] P. Wang, H. Yang, and X. Zhang, JHEP 08, 043 (2010).
[7] K. Nozari and P. Pedram, Europhys. Lett. 92, 50013 (2010).
[8] S. Das, E. C. Vagenas, and A. F. Ali, Phys. Lett. B 690, 407 (2010).
[9] S. Das and E. C. Vagenas, Can. J. Phys. 87, 233 (2009).
[10] S. Das and E. C. Vagenas, Phys. Rev. Lett. 101, 221301 (2008).
S. Das and E. C. Vagenas, Phys. Rev. Lett. 104, 119002 (2010).
[11] A. F. Ali, S. Das, and E. C. Vagenas, Phys. Lett. B 678, 497 (2009).
[12] P. Pedram, K. Nozari, and S. H. Taheri, JHEP 1103, 093 (2011)
K. Nozari, P. Pedram, and M. Molkara, [arXiv:1111.2204] (to appear in Int. J. Theor. Phys.).
[13] J. Magueijo and L. Smolin, Phys. Rev. Lett. 88, 190403 (2002).
J. Magueijo and L. Smolin, Phys. Rev. D 71, 026010 (2005).
J. L. Cortes and J. Gamboa, Phys. Rev. D 71, 065015 (2005).
[14] L. D. Landau and E. M. Lifshitz, Quantum Mechanics (Non-relativistic theory) (Elsevier Science Ltd., 2003).