[Paper Review] Quantum speed limit for relativistic spin-0 and spin-1 bosons on commutative and noncommutative planes

K. Wang, Y.F. Zhang+3 more|arXiv (Cornell University)|Mar 3, 2017

Noncommutative and Quantum Gravity Theories4 citations

TL;DR

This paper investigates quantum speed limits for relativistic spin-0 and spin-1 bosons in a uniform magnetic field on both commutative and noncommutative two-dimensional planes. Using the Duffin-Kemmer-Petiau (DKP) equation, it shows that while wave packet speeds remain below light speed in the commutative case, noncommutativity enables average radial speeds to exceed light speed under strong magnetic fields—indicating a violation of Lorentz invariance in noncommutative quantum mechanics.

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 oncommutative 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 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 quantum mechanics region.

Motivation & Objective

• To examine whether Lorentz invariance is violated in noncommutative quantum mechanics for relativistic spin-0 and spin-1 bosons.
• To compare quantum speed limits of wave packets in commutative versus noncommutative two-dimensional planes under a uniform magnetic field.
• To determine if the average radial speed of wave packets during quantum state evolution exceeds the speed of light in vacuum in noncommutative scenarios.
• To analyze the role of noncommutativity in modifying the dynamics of massive relativistic bosons described by the DKP equation.

Proposed method

• The Duffin-Kemmer-Petiau (DKP) equation is used to describe relativistic spin-0 and spin-1 bosons in a 2D plane with a uniform magnetic field.
• The minimal coupling prescription is applied via $ p_i \to p_i + qA_i $, incorporating the magnetic potential into the Hamiltonian.
• Solutions to the DKP equation are derived in both commutative and noncommutative regimes, with noncommutativity introduced through $[x_i, x_j] = i\theta \epsilon_{ij}$ and $[p_i, p_j] = i\theta B \epsilon_{ij}$.
• Two specific stationary states are constructed by superposing solutions with quantum numbers $ (n=0, l=0) $ and $ (n=2, l=0) $, enabling the calculation of the minimum evolution time $ T_{\text{min}} $.
• The average radial speed of the wave packet is computed over the interval $[0, T_{\text{min}}]$, using the displacement derived from the superposition state.
• The noncommutative corrections introduce a factor $ |1 - \frac{q\theta B}{4}| $, which modifies the speed and enables superluminal propagation under strong magnetic fields.

Experimental results

Research questions

• RQ1 Does noncommutativity in the spatial coordinates lead to superluminal wave packet speeds for relativistic spin-0 and spin-1 bosons in a magnetic field?
• RQ2 How does the average radial speed of a quantum wave packet evolve during the minimum time interval for state evolution in commutative versus noncommutative planes?
• RQ3 To what extent does the noncommutative structure of spacetime violate Lorentz invariance in relativistic quantum mechanics?
• RQ4 Can the DKP equation framework consistently describe superluminal behavior in noncommutative quantum systems for massive bosons?

Key findings

• On the commutative plane, the average radial speed of wave packets during the minimum evolution time remains below the speed of light, regardless of magnetic field strength.
• In the noncommutative plane, the average radial speed exceeds the speed of light in vacuum when the magnetic field intensity is sufficiently high, due to the noncommutative correction factor $ |1 - \frac{q\theta B}{4}| $.
• The violation of Lorentz invariance is directly linked to the noncommutative structure of spacetime, as the superluminal speeds emerge only in the noncommutative regime.
• The results are consistent across both spin-0 and spin-1 cases, indicating a general feature of noncommutative relativistic quantum mechanics.
• The minimum evolution time $ T_{\text{min}} $ is derived from the energy variance and mean energy, and the displacement is calculated using the superposition of stationary states with quantum numbers $ (0,0) $ and $ (2,0) $.
• The noncommutative DKP equation yields eigenvalues identical to the commutative case, but the wave function components are modified by noncommutative parameters, leading to altered dynamics and superluminal speeds.