[Paper Review] Coherent States for the Deformed Algebras

V. Sunilkumar, Bindu A. Bambah+2 more|ArXiv.org|May 5, 1999

Cold Atom Physics and Bose-Einstein Condensates2 references3 citations

TL;DR

This paper presents a unified framework for constructing coherent states in deformed algebras—such as quadratic, Higgs, and q-deformed algebras—by identifying canonical conjugate operators to the annihilation operators. The method maps deformed algebras to Lie algebras, enabling the systematic derivation of Perelomov-type generalized coherent states, offering a consistent approach for physical systems with non-standard commutation relations.

ABSTRACT

We provide a unified approach for finding the coherent states of various deformed algebras, including quadratic, Higgs and q-deformed algebras, which are relevant for many physical problems. For the non-compact cases, coherent states, which are the eigenstates of the respective annihilation operators, are constructed by finding the canonical conjugates of these operators. We give a general procedure to map these deformed algebras to appropriate Lie algebras. Generalized coherent states, in the Perelomov sense, follow from this construction.

Motivation & Objective

• To develop a general method for constructing coherent states in various deformed algebras relevant to quantum physics.
• To address the lack of a systematic approach for coherent states in non-standard algebras such as quadratic, Higgs, and q-deformed types.
• To establish a mapping between deformed algebras and Lie algebras to facilitate the construction of generalized coherent states.
• To extend the Perelomov formalism to non-compact deformed algebras through canonical conjugate operators.
• To provide a unified framework applicable to physical systems with modified commutation relations.

Proposed method

• The authors identify the canonical conjugate operators to the annihilation operators in deformed algebras to define coherent states as eigenstates of the annihilation operator.
• A general procedure is developed to map deformed algebras to appropriate Lie algebras, enabling the use of standard group-theoretic methods.
• The construction follows the Perelomov approach, generating generalized coherent states via group action on a highest-weight state.
• The method applies to non-compact algebras, including those with quadratic and q-deformed commutation relations.
• The formalism is applied to specific cases: quadratic, Higgs, and q-deformed algebras, demonstrating consistency across different deformation types.
• The approach relies on algebraic techniques from quantum groups and Lie algebra theory to ensure mathematical consistency.

Experimental results

Research questions

• RQ1 How can coherent states be consistently defined for deformed algebras such as quadratic, Higgs, and q-deformed types?
• RQ2 What is the general procedure to map deformed algebras to Lie algebras to enable coherent state construction?
• RQ3 Can the Perelomov formalism be extended to non-compact deformed algebras through canonical conjugate operators?
• RQ4 What is the role of the canonical conjugate in defining eigenstates of the annihilation operator in deformed settings?
• RQ5 How does the unified framework ensure consistency across different types of deformed algebras?

Key findings

• The paper successfully constructs coherent states as eigenstates of the annihilation operator by identifying their canonical conjugates in deformed algebras.
• A general mapping procedure from deformed algebras to Lie algebras is established, enabling systematic state construction.
• Generalized coherent states in the Perelomov sense are derived for non-compact deformed algebras, including q-deformed and Higgs types.
• The method is consistent across different deformation types, demonstrating broad applicability to physical systems with modified commutation relations.
• The framework provides a unified approach that avoids ad hoc constructions, enhancing theoretical coherence and predictive power.
• The results are applicable to quantum systems with non-standard symmetries, such as those in quantum optics and quantum gravity models.