- The paper demonstrates that introducing a background scalar field alters the momentum operator in the Schrödinger equation, affecting wavefunctions without changing observable energy levels.
- It shows that for a particle in a potential well, Lorentz symmetry violation influences the wavefunctions while the energy eigenvalues remain invariant.
- The study confirms that despite modifications to the quantum harmonic oscillator wavefunctions due to LSV, the energy levels are preserved, indicating robust physical symmetry.
Effects of a Background Scalar Field Induced by Lorentz Symmetry Violation on Non-Relativistic Quantum Mechanics
This paper explores the extension of the Standard Model of particle physics, specifically focusing on the Lorentz Symmetry Violation (LSV) and its implications in Non-Relativistic Quantum Mechanics (NRQM). The work is rooted in the foundational efforts by Colladay and Kostelecky, which proposed a framework allowing for Lorentz symmetry violations through the introduction of a background tensor field. This paper explores this concept further by exploring the effects of an induced background scalar field.
Theoretical Framework
The Standard Model Extension (SME) framework incorporates all possible Lorentz violating terms from non-zero expectation values of Lorentz tensors acting as spacetime background fields. This leads to a modification in particle properties, such as energy dispersion relations and field equations, which is particularly relevant in both ultrarelativistic and non-relativistic regimes. The paper addresses a gap in the literature by exploring the non-relativistic sector of SME, investigating how these background fields impact the Schrödinger equation, a cornerstone of quantum mechanics.
Core Investigation
The paper analyzes extensions to the Schrödinger Equation inspired by the SME in various scenarios, including the quantum harmonic oscillator and a particle confined within a potential well. Lorentz symmetry violations are modeled through modifications to the momentum operator, leading to a Schrödinger equation that lacks parity invariance. Through meticulous theoretical derivations, it is shown that Lorentz violating terms can be removed by redefining wavefunctions, thereby restoring symmetry in observable quantum states.
Key Findings
- Modified Schrödinger Equation: The introduction of a background constant scalar field affects the formulation of the momentum operator. This results in modified Schrödinger equations whose solutions highlight the effects of Lorentz symmetry violations at the wavefunction level, though not in the observable spectrum of physical systems.
- Particle in a Box: For a particle in a potential well, it was demonstrated that while the Lorentz symmetry violation influences the wavefunctions, the energy eigenvalues remain unchanged, suggesting the invariance of observable physical phenomena to such violations.
- Quantum Harmonic Oscillator: Similar observations were made for the quantum harmonic oscillator, where although the wavefunctions are affected by LSV, the energy levels are preserved. This indicates a robustness in physical observables against these theoretical modifications.
Implications and Future Work
The findings suggest that while Lorentz symmetry violations manifest in the mathematical descriptions of wavefunctions, they do not alter key quantum mechanical observables like energy levels and normalization constants in the systems studied. This raises questions on the experimental detectability of such symmetry violations at low energy scales, posing challenges for current experimental setups to provide constraints on SME parameters.
Future work may extend these analyses to higher-dimensional systems, aiming to uncover potential experimental scenarios where LSV could produce detectable effects, thereby providing empirical guidance to refine theoretical models. The exploration of alternative quantum systems and interactions, possibly incorporating relativistic corrections, could also offer further insights into the implications of Lorentz symmetry breaking in quantum mechanics.