Coherent States in Classical Field Theory: A Classical Analogue Approach
This paper explores the emergence of coherent states within the framework of classical field theory, drawing an intriguing parallel with their quantum counterparts. The authors effectively leverage the algebraic approach to Poisson brackets and the symmetries inherent in classical field theory, facilitating the translation of quantum principles into a classical context. The classical phase space plays a pivotal role in this endeavor, offering a viable platform for the construction of coherent states.
The foundation of this research lies in the examination of one-dimensional nonrelativistic complex scalar field theory. This theory is characterized by a Lagrangian density incorporating potential local self-interactions, which introduces nonlinearity into the system. The paper devotes significant attention to defining a position variable, noted as X, that is canonically conjugate to momentum P, mirroring quantum mechanics constructs. The construction of the classical phase space framework is fundamental to realizing this canonical position, facilitating the establishment of coherent states analogues in classical settings.
The authors extend the annihilation operator concept from quantum mechanics into the classical domain, defining the classical analogue as ac. This progression enables them to derive the classical counterpart to quantum coherent states, wherein specific field configurations become eigenstates of ac. Such classical states are represented mathematically in a manner akin to quantum coherent states, elucidated via a consistent algebraic framework.
Moreover, this exploration is not limited to harmonic oscillator coherent states. The paper identifies classical analogues of various quantum entities, including cat states, kitten or compass states, and photon-added coherent states. The methodology utilized further demonstrates that other wavepacket forms, such as the self-accelerating Airy wavepackets, also possess classical counterparts. These Airy wavepackets traditionally align with Perelomov coherent states in quantum mechanics, yet find relevance in the classical interpretation through coherent field configurations predicated on Galilean eigenvalue problems.
Theoretical implications suggest a profound interrelation between quantum and classical constructs, emphasizing the continuity and compatibility of physical principles across different realms. Practically, the findings could stimulate advancements in field theoretical applications, potentially influencing areas like optical coherence and wavepacket propagation, where classical analogues may serve as tangible models for experimental and computational pursuits.
In conclusion, this paper presents a compelling case for the existence of coherent states in classical field theory, skillfully bridging canonical quantum concepts to classical frameworks. This work paves the way for further investigations into classical counterparts of quantum states, potentially enriching the understanding of coherence and nonlinearity in both fields. Future explorations might consider extending these principles to more complex or higher-dimensional field theories, additionally examining inter-theoretical applications or experiments that exploit the established classical-quantum correlation.
