^{1}

^{1}

^{2}

^{3}

^{1}

^{1}

^{2}

^{3}

^{3}.

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.

Duffin-Kemmer-Petiau (DKP) equation is a first-order relativistic wave equation [

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 [

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 [

The work of [

In [

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

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 (

Substituting the explicit expressions of

Combining the above equations, we get the dynamical equation of the component

Equation (

In order to simplify our calculation further and make a comparison with the spin-

Then, after some direct calculations, we get these two steady states. They are

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 (

According to [

The average radial displacement of the spinless boson in the interval

Now, we study the DKP equation with

Using the explicit expressions of

Therefore, the solutions of

We choose a special solution for components

We superpose these two steady states (

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

There are two ways to study noncommutative theories in noncommutative space. One is to replace the ordinary product by the Moyal (

The other equivalent way is to introduce the noncommutative algebra among variables

For the sake of consistency, the commutator among coordinates and momenta should be modified as [

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

We map noncommutative variables

In terms of commutative variables

The eigenvalues and eigenfunctions of Hamiltonian (

In order to make a comparison with commutative version, we choose two solutions of

We superpose two steady states (

Compared with the commutative case, we find that due to spatial noncommutativity, there is an extra factor

Now, we study the noncommutative spin-1 DKP equation. In noncommutative plane, the dynamical equations take the same form as (

Mapping noncommutative variables

The dynamical equation for component

Similar with the commutative version, we set

Superposing these two steady states homogeneously, we get the superposition state which takes the same form as (

In this paper, we investigate the problem of whether Lorentz invariance is violated in noncommutative relativistic quantum mechanics regime. It is known that there are two frames in studying the noncommutative theories [

In fact, the violation of Lorentz invariance due to noncommutativity has been noticed for more than 10 years in [

We study the charged massive spin-0 and spin-1 relativistic bosons in the presence of homogeneous magnetic fields in

It is known that there are redundant degrees of freedom in DKP equation. For the spin-0 case, the physical degree of freedom is easy to obtain. However, it is not straightforward to get the degrees of freedom for spin-1 case [

The authors declare that there are no conflicts of interest regarding the publication of this paper.

This work is supported by the NSFC with Grant nos. 11775303 and 11465006.