Abstract

Quantum speed limits of relativistic charged spin-0 and spin-1 bosons in the background of a homogeneous magnetic field are studied on both commutative and noncommutative planes. We show that, on the commutative plane, the average speeds of wave packets along the radial direction during the interval in which a quantum state is evolving from an initial state to the orthogonal final one can not exceed the speed of light, regardless of the intensities of the magnetic field. However, due to the noncommutativity, the average speeds of the wave packets on noncommutative plane will exceed the speed of light in vacuum provided the intensity of the magnetic field is strong enough. It is a clear signature of violating Lorentz invariance in the relativistic quantum mechanics region.

1. Introduction

Duffin-Kemmer-Petiau (DKP) equation is a first-order relativistic wave equation [1–3]. Different from Dirac equation which describes spin- fermions, DKP equation describes spin-0 and spin-1 bosons. DKP equation takes the formwhere is the rest mass and the matrices satisfy the algebraic relationHere, is the metric tensor. The algebra (2) has three different representations: a (one-dimensional) trivial representation, a five-dimensional representation describing spin-0 bosons, and a ten-dimensional representation describing spin-1 bosons. As a Dirac-type relativistic quantum mechanical model, DKP equation with various potentials has been studied in the past years [4–8]. The magnetic coupling in DKP equation is also considered [9]. When magnetic field is taken into consideration, one should introduce the magnetic potentials by the minimal coupling (we set ), , where and are the charge and the magnetic potentials, respectively. Since DKP equation is analogous with Dirac equation, the author of [9] compares it with Dirac equation in detail in that paper.

On the other hand, the minimum time of a quantum state evolving from an initial state to the orthogonal final one in Hilbert space is of great importance in the field of quantum computation, quantum control, and quantum metrology. In fact, it has attracted attention for a long time [10]. At present, there are two different descriptions of the minimum time for a quantum system evolving from an initial state to the orthogonal final state. One is given by the expression , in which is the energy variance, defined by , with being the Hamiltonian of the system and being a specific superposition of eigenstates of [11]. The other is given in [12], which states that the minimum time is given by , where and are the mean energy and the lowest energy of the state which participates in the superposition. Obviously, the results of [11, 12] will be equivalent if the condition is satisfied. This condition can be simply satisfied by superposing two steady states homogeneously. According to the results in [11, 12], it is natural to assume that the minimum time should be given by [13]. A unified bound which contains both and is considered [14].

An interesting connection between the minimum time of the quantum state evolving in Hilbert space and the average speed of an electron wave packet travelling in spatial space is constructed in a recent paper [15]. In this paper, the authors study a relativistic electron coupling to a homogeneous magnetic field. They find that the average speed of this electron wave packet moving in radial direction during the interval in which a quantum state is evolving from an initial state to the orthogonal final one in Hilbert space is less than the speed of light in vacuum, regardless of intensities of the magnetic field one applies. It seems that, as expected, Lorentz invariance is not violated in this relativistic quantum mechanical model. However, Lorentz invariance would be violated in the nonrelativistic limit of this model since the average speed of the electron wave packet in radial direction exceeds the speed of light in vacuum provided the intensity of the magnetic field is strong enough.

The work of [15] is generalized to noncommutative case recently. Noncommutativity becomes one of the foci of theoretical research due to superstring theories in recent years [16–19]. There are a large number of papers studying quantum field theories in noncommutative space [20–25]. Nonrelativistic noncommutative quantum mechanical models, such as noncommutative harmonic oscillator and noncommutative Landau problem, have been studied extensively [26–32]. The relativistic quantum mechanical models on noncommutative space are investigated since the work of [33]. Some geometrical phases in noncommutative relativistic quantum theory are studied in [34–36] recently. DKP equation in noncommutative space, especially the DKP oscillator in noncommutative space, has also been investigated by some authors [37–42]. Interestingly, it is found that the noncommutativity even has some relationships with Jaynes-Cumming model in quantum optics context [43].

In [44], the authors study the noncommutative (both the coordinates and momenta are noncommutative simultaneously) Dirac equation. They find that Lorentz invariance will be violated in noncommutative Dirac equation since the average speed of an electron wave packet exceeds the speed of light in vacuum if the magnetic field is strong enough. In fact, the problem of violating Lorentz invariance has been considered [45–47]. In [46], the authors find that because of noncommutativity, Lorentz invariance will be violated in the noncommutative quantum electrodynamics (QED) since the electromagnetic wave travels in different speeds along different directions at the presence of a background magnetic field. A similar result is also obtained in [47]. This problem is also considered semiclassically from both noncommutative and gravitational points of view [48–50].

In this paper, we investigate the problems of whether Lorentz invariance is violated for spin-0 or spin-1 relativistic bosons in two-dimensional spatial space. We study the commutative case firstly and then generalize to the noncommutative case. The organization of this paper is as follows. In the next section, we study spin-0 and spin-1 charged bosons coupling to homogeneous magnetic fields on commutative plane. Then, in Section 3, we study the noncommutative case. Some remarks and further discussions will be presented in the last section.

2. Spin-0 and Spin-1 Charged Bosons Coupling to a Homogenous Magnetic Field on Commutative Plane

In this section, we study spin-0 and spin-1 bosons coupling to a homogeneous magnetic field on a commutative plane. We start our studies from the spin-0 bosons.

As stated before, the five-dimensional representation of the algebra (2) describes spin-0 boson. The explicit expressions of five-dimensional matrices arein whichHere , , and are , and zero matrices, respectively. Choosing symmetric gauge and introducing Larmor frequency , we write (1) in the formwhere is a five-component wave function .

Substituting the explicit expressions of matrices (3) into the above equation, we get a set of equationsObviously, the five components are not independent to each other.

Combining the above equations, we get the dynamical equation of the component . It iswhere is the Landau Hamiltonian:with being the angular momentum along direction. Obviously, the Hamiltonian (8) describes a planar nonrelativistic charged particle interacting with a homogeneous perpendicular magnetic field.

Equation (7) is easily solved. The eigenvalues and eigenfunctions, respectively, are [51]where , , and is Laguerre’s polynomials. Thus, the multicomponent wave function can be obtained if one chooses a specific solution of the component .

In order to simplify our calculation further and make a comparison with the spin- case [15], we choose two steady states whose first components, respectively, are (we only consider the positive-energy sector)

Then, after some direct calculations, we get these two steady states. They arewhereare two normalization constants.

As we have mentioned, there are some controversies on the minimum time of a quantum state evolving from an initial state to the orthogonal one. In order to avoid these problems, we superpose these two steady states (13) homogeneously; that is,

According to [11, 12], the minimum time for a state evolving from the initial state to the final orthogonal state is given by , or, equivalently, , where and are the mean energy and energy variance on the state , respectively. After direct calculation, we get

The average radial displacement of the spinless boson in the interval is given bySubstituting (13) and (15) into the above equation and after some direct calculations, we arrive atThus, the average speed of the wave packet of this charged boson along the radial direction during this interval is given by . The result isThe average speed reaches its maximum when . It isIt shows that the average radial speed of this spinless boson wave packet is less than the speed of light in vacuum () no matter the intensities of the magnetic field. Therefore, Lorentz invariance is not violated in this relativistic quantum mechanics model.

Now, we study the DKP equation with -dimensional representation of the algebra (2). As is known, it describes spin-1 bosons. The explicit expressions of matrices arein which

Using the explicit expressions of matrices, we write the DKP equation in the component formAccording to (23d), (23g), (23h), and (23i), we get the dynamical equation of the component where is given in (8).

Therefore, the solutions of in spin-1 case take the same form as (9) and (10); that is,

We choose a special solution for components and , that is, and As a result, the components . Thus, the ten-component eigenfunction can be obtained if we select a specific solution for from (26). In order to simplify our calculations and make a comparison with [15], we choose two specific solutions for . They areThus, the explicit expressions for two steady states we want to superpose arewhere , are identical to (14).

We superpose these two steady states (28) homogeneously in the same manner as (15). The eigenvalues are equal to the ones in spin-0 case; thus the minimum time for the superposition state evolving from the initial state to the final orthogonal state is the same as (16). The radial displacement of the wave packet during the interval can be calculated straightforwardly. The result is identical to (17). Therefore, the average velocity along radial direction of this spin-1 boson is the same as in (19). The maximum average speed is achieved when the intensity of the magnetic field tends to infinity. The result is identical to (20). It is less than the speed of light in vacuum. Thus, it shows that Lorentz invariance is not violated.

3. Spin-0 and Spin-1 Bosons on Noncommutative Plane

As we have shown, the average speeds of the wave packets of spin-0 and spin-1 bosons along the radial direction will not exceed the speed of light in vacuum, regardless of the intensity of the magnetic field. A natural question is does the Lorentz invariance get violated in the noncommutative DKP equation? In the following, we will investigate DKP equation in noncommutative -dimensional phase space.

There are two ways to study noncommutative theories in noncommutative space. One is to replace the ordinary product by the Moyal () product [52]:where , are two -dimensional functions and , is a rank-two antisymmetrical tensor. When the noncommutativity between momenta is taken into account, the above -product should be generalized to the formin which , is also a rank-two antisymmetrical tensor.

The other equivalent way is to introduce the noncommutative algebra among variables . For the noncommutative -dimensional phase space, the noncommutative algebra is given bywhere and are two real parameters and is the 2-dimensional antisymmetric tensor. It can be easily verified that the -product (30) can reproduce the commutator (31) directly for 2-dimensional phase space.

For the sake of consistency, the commutator among coordinates and momenta should be modified as [53] (in order to avoid the problem of unitarity, we only consider noncommutativity among coordinates and momenta)

By applying the latter way to study the noncommutative theories, one assumes that the dynamical equations in noncommutative quantum mechanics take the same form as their commutative counterparts. However, variables in dynamical equation are replaced by the corresponding noncommutative ones. Therefore, the noncommutative version of spin-0 DKP equations is nothing but to replace the variables by which satisfy the algebraic relations (31) and (32) in (6a), (6b), (6c), (6d), and (6e). Choosing the symmetric gauge [54] and combining them, we getwherewith being the noncommutative angular momentum along direction. Hamiltonian (34) describes a planar nonrelativistic charged particle interacting with a homogeneous perpendicular magnetic field on the noncommutative planes (31) and (32).

We map noncommutative variables to commutative ones which satisfy the standard Heisenberg algebraIt is straightforward to check that the map from noncommutative variables to commutative ones can be realized by [53]

In terms of commutative variables , we find that the dynamical equation for component takes the same form as (33). The corresponding Hamiltonian (34) in terms of commutative variables becomesin whichwhere , are two parameters.

The eigenvalues and eigenfunctions of Hamiltonian (37) arewhere and Other components can be calculated directly from the commutative version of (6a), (6b), (6c), (6d), and (6e).

In order to make a comparison with commutative version, we choose two solutions of asThen, the corresponding two steady states arewhere and are two normalization constants. They arerespectively.