- Published: December 31, 2021
- Updated: December 31, 2021
- University / College: The Rockefeller University
- Language: English
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Noncommutative space-time is introduced to the standard model of particle physics through noncommutative quantum field theory based on star products and Seiberg-Witten maps [ 1 ]. Since the standard model of particle physics has many unsolved problems and quantum field theory on a noncommutative space-time may provide some answers to these puzzles, certain amount of work have been done concerning noncommutative quantum field theory [ 2 – 6 ]. In addition, because finding the measurable effects of space noncommutativity is the central topic in noncommutative quantum mechanics (NCQM), various aspects of quantum mechanics (QM) have been studied to a very great extent on noncommutative space (NCS) and noncommutative phase space (NCPS). For example, NCQM is applied to the hydrogen-like atom and the corrections to the Lamb shift are calculated accordingly and the constraint on θ is obtained to be [ 7 – 9 ]. Transitions in the helium atom is studied and the constraint: is provided in Haghighat and Loran [ 10 ]. The authors of Zhang [ 11 ] have suggested the possibility of testing spatial noncommutativity via cold Rydberg atoms. In Chaichian et al. [ 12 ], Chaichian et al. [ 13 ], Falomir et al. [ 14 ], Li and Dulat [ 15 ], Mirza and Zarei [ 16 ], Li and Wang [ 17 ], Mirza et al. [ 18 ], Singlton and Vagenas [ 19 ], Dulat and Ma [ 20 ] and Singleton and Ulbricht [ 21 ], the Aharonov-Bohm type phase is studied on a NCS and a NCPS. A lower bound is shown to be for the space noncommutativity parameter in Chaichian et al. [ 12 ]. In Mirza and Zarei [ 16 ], Li and Wang [ 17 ] and Mirza et al. [ 18 , 22 ], the Aharonov-Casher phase on a NCS and a NCPS has been studied for a spin-1/2 and a spin-1 particle. And in Mirza and Zarei [ 16 ] the limit for the space noncommutativity parameter is . The authors of Harms and Micu [ 23 ], Dayi and Jellal [ 24 ] and Chakraborty et al. [ 25 – 30 ] have studied the noncommutative quantum Hall effect, and in Harms and Micu [ 23 ] a limit of is given. The noncommutative spin Hall effect (SHE) is discussed through a semiclassical constrained Hamiltonian and interesting results are obtained in Dayi and Elbistan [ 31 ]. By studying the SHE in the framework of NCQM a lower limit for the noncommutative parameter is shown to be in Ma and Dulat [ 32 ]. Also Yakup and Dulat [ 33 ] studied the Harmonic oscillator influenced by gravitational wave in noncommutative quantum phase space. Ma et al. [ 34 ] investigated the time-dependent Aharonov-Bohm effect on the NCS. Studies about the NCQM in three dimensions and rotational symmetry can be found in Sinha et al. [ 35 ].
In this paper, we focus on the quantum effect of an electric dipole on a NCS. First, the definition of noncommutative space-time is given in the following. NCS is a deformation of ordinary space in which the space coordinate operators satisfy the following relation:
θ ij is the totally antisymmetric real tensor, which represent the noncommutativity of the space. In addition, the product between the external fields on a NCS is
where and are two arbitrary, infinitely differentiable functions on a commutative space, and and are the corresponding functions on a NCS. In NCQM, one replaces the position and momentum operators with the θ-deformed one,
Here x i and p i are coordinate and momentum operators in usual quantum mechanics. In the presence of the electro-magnetic field, we also need to replace the electromagnetic potentials with the shifts given as follows
and then applies the usual rules of QM.
This paper is organized as follows: in Section 2, on a NCS the phase of an electric dipole is obtained as an application of the AB effect to a system composed of two charges. In Section 3, the behavior of an electric dipole is analyzed on a NCS by using a Lagrangian approach. In Section 4, we derive the phase for electric dipole once more by using the Lagrangian approach, furthermore we prove the gauge invariance of the phase on a NCS. Conclusions are given in the last section.
2. Aharonov-Bohm Phase for an Electric Dipole on a Noncommutative Space
A charged particle moving around a magnetic flux in force-free regions acquires the Aharonov and Bohm [ 36 ] topological phase,
For an electric dipole with total mass m = m 1 + m 2 may be considered as being composed of two charges ± q of mass m 1 , m 2 separated by small distancer= x 1 −x 2 . The phase for an electric dipole obtains through summing the AB phase of two charges ± q [ 37 ],
whered= qris electric dipole moment, x= ( m 1 x 1 + m 2 x 2 )/ m is the position of the center of mass.
We can obtain the topological AB phase for an electric dipole on a NCS by replacingA( x i ) in Equation (9) with the ,
is the electric dipole moment on a NCS; here the second term is the correction for the electric dipole moment on a NCS. Equation (12) can also be written as
where the additional phase δϕ NCS , related to the non-commutativity of space, is given by
The result (Equation 12) represents the noncommutative extension of the electric dipole phase (Equation 9). In the next section we will derive it by using a different approach.
3. Lagrangian on a Noncommutative Space
In this part we construct the Lagrangian for an electric dipole on NCS. The dipole Lagrangian on a commutative space is [ 37 ]
Using the expansion of Φ(x 1 ) = Φ(x 2 ) +r∇Φ(x 2 ) andA(x 1 ) = A(x 2 ) +r∇·A(x 2 ), with dipole approximation , Thus Equation (16) becomes
where m r = m 1 m 2 /( m 1 + m 2 ) is the reduced mass and Φ andAare evaluated atx 2 . This result is different from Gianfranco Spavieri’s work [ 37 ].
The Lagrangian for electric dipole in NCS can be obtained by using the Bopp’s to scalar potential Φ(x 2 ) and vector potentialA(x 2 ).
By inserting Equations (18) and Equation (19) into Equation (17), the Equation (17) becomes
comparing with Gianfranco Spavieri’s work [ 37 ], this is the correction term for Lagrangian in NCS. The difference depend on the non-commutativity θ of the space.
The canonical momentum for the center of mass in NCS is
is the deformed momentum of electric dipole in NCS. The second term δP NCS of Equation (22) is the correction momentum for electric dipole in NCS, and this also depend non-commutativity of space.
Expression (Equation 20) leads to the equation of motion
with d / dt = ∂ t +v·∇, and ∇ × ∇Φ = 0, and in terms of fields, Equation (25) reads
The second, third term is also the correction term for equation of motion in NCS.
Dipole moves in force-free region where the fields are uniform, so the force free condition , and this gives ; v· θ · ∇B= 0. This result is very important, we will make use of this in Section 4 to prove the gauge invariance of the phase.
The canonical momentum for relative coordinates is
The equation of motion for relative coordinates is
for the angular momentum
If the dipole moves in a region of space where fields are uniform, (d· ∇)E+ c −1 v× [∇ × (B×d)] = 0 in Equation (26) and ( d / dt )( m v+ c −1 B×d). In this case, v= −(1/ mc )B×d) + const may be inserted in Equation (28), which becomes an equation in the variabler′ only (here we assumev= v 1 = v 2 ). In most cases the solution of this equation represents a bound oscillatory motion and will contain terms of the type sin(ω t + φ) or cos(ω t + φ), where ω is the frequency and φ is a constant phase. The average of a dynamical variable is obtained by performing the average over the phase constants and one may expect that the variabled= q roscillated about the constant average equilibrium position
where is the correction term in NCS; α is the polarizability and is the equilibrium position in the absence of external fields.
Because of Equation (30), is constant and independent ofxonly when the fields are uniform. In this case is no longer a dynamical variable and, withE 0 = −∇Φ, the Lagrangian assumes the simple form
4. The Hamiltonian and Gauge Invariance of the Phase on Noncommutative Space
Gauge invariance of the AB phase in the time dependent AB efect is discussed in MacDougall and Singleton [ 38 ]. In this paper, we discuss the quantum phase for an electric dipole on a NCS by solving the Schroedinger equation. Let H ( x, p ) be the Hamiltonian operator of the usual quantum system, then the static Schrödinger equation on NC space is usually written as
where the Moyal-Weyl (or star) product between two functions is defined in Equation (4). On a NCS the star product can be changed into the ordinary product by replacing H ( x, p ) with . Thus the Schrödinger equation can be written as,
Thus the Equation (31) is actually defined on a commutative space, and the noncommutative effects can be evaluated through the Θ related terms. Note that the Θ term always can be treated as a perturbation in QM, since Θ ij < < 1. When magnetic field is involved, the Schrödinger equation (Equation 30) becomes
To replace the star product in Equation (32) with a usual product, we need to replace x i , p i , and A i with the shifts given in Equations (5) and (7). Thus the Schrödinger Equation (32) in the presence of magnetic field becomes
Now let us consider a particle of mass m and charge q moving in a magnetic field with magnetic potential A i , then the Schrödinger equation is (we choose unit of ℏ = c = 1),
Using the expressions of the canonical momenta Equations (22) and (27), the Hamiltonian Ĥ may be derived from the Lagrangian Equation (20). the corresponding Schrödinger equation in NCS reads
The main difference between classical and quantum behavior is due to the existence of the quantum phase ϕ of the wave function which, through the process of interference, may lead to an observable phase shift △ϕ. We obtain the solution of the Schrödinger equation (Equation 32) by direct substitution
where is the correction term in NCS,, and Ψ 0 solves the equation withA= 0, Φ = 0. Thus, the quantum phase ϕ coincides with Equation (14) in the Previous section.
In the interference experiments with particles possessing an electric dipole moment, the observable quantity is the phase shift Since
where . its expectation value reads
the relevant termB×d 0 known as Röntgen interaction. the second term is our correction term for Gianfranco Spavieri’s work [ 37 ] in NCS.
Our result (Equation 35) for the phase shift of an electric dipole in NCS differs from that proposed by other authors (Gianfranco Spavieri) for the presence of the extra term .
It can be shown that the phase shift is also gauge independent in NCS. Gauge theory in NCS is different from the case in usual commutative space. Gauge transformation in NCS is . Here we consider the three dimensional case: δ λ A= ∇λ( x ) + i [λ( x ), A]*.
Using the expression (Equation 35) in order to point out some properties of the AB phase shift, we write the contribution to the phase shift due to the gauge transformation as
This gauge independence based on the fact that the scalar function λ is a monovalued function for which ∮(∇λ)· d x= 0, and θ·(∇B) = 0 which we obtained under the force free condition.
In this paper we studied the noncommutative non-relativistic behavior of a neutral particle, which possessing electric dipole moments, in the presence of external electric and magnetic fields. For a special configuration of the field, we derived the phase of an electric dipole as an application of the AB effect to a system composed of two charges in NCS. We have shown that a topological phase of the AB type is a generic effect in dipole moment of neutral particles.
The result of this paper indicates that there is some difference between usual and NCQM studying same problem. For example, our Lagrangian, canonical momentum, phase, equation of motions are different from Gianfranco Spavieri’s work [ 37 ] who studied the same problem in usual commutative space. The difference depend on the noncommutativity θ and of the space. We may say that, our extra terms for theses quantities are the correction term for Gianfranco Spavieri’s work [ 37 ] in NCS. Obviously, if noncommutative parameter θ = 0, the phase in NCS is to be that in commutative space. If the present experimental situation is attainable, this effect in NCS might be tested at very high energy level, and the experimental observation of the effect remains to be further studied.
MA and RR designed the study; AA and MH carried out the theoretical calculations independently; MA performed the theoretical analysis; RR wrote the manuscript; MA edited the final manuscript.
Conflict of Interest Statement
The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.
This research was supported by the National Natural Science Foundation of China Grant Nos. 11465018, 61501026, 11347031, and XJEDU2014I058. We thank Prof. Sayipjamal Dulat for many helpful discussions.
13. Chaichian M, Demichev A, Presnajder P, Sheikh-Jabbari MM, Tureanu A. Quantum theories on noncommutative spaces with nontrivial topology: Aharonov-Bohm and Casimir effects. Nucl Phys.(2001)B611: 383. doi: 10. 1016/S0550-3213(01)00348-0