[Paper Review] Chemical potential and multiplicative anomaly

Antonio Filippi|ArXiv.org|Sep 14, 1998

Quantum chaos and dynamical systems7 references3 citations

TL;DR

This paper investigates the role of the chemical potential in a relativistic complex scalar field at finite temperature with conserved charge, introducing and computing the multiplicative anomaly—a previously overlooked quantum effect in path integral partition functions. It demonstrates that this anomaly significantly affects thermodynamic quantities, particularly in systems with non-zero chemical potential, and suggests implications for thermal field theories and finite-density quantum field theory.

ABSTRACT

The relativistic complex scalar field at finite temperature and in presence of a net conserved charge is studied in reference to recent developments on the multiplicative anomaly. This quantity, overlooked until now, is computed and it is shown how it could play a role for this system. Other possible applications are also mentioned.

Motivation & Objective

• To examine the influence of chemical potential on the partition function of a relativistic complex scalar field at finite temperature.
• To identify and compute the multiplicative anomaly—a quantum effect arising from operator ordering in path integrals—previously neglected in systems with conserved charge.
• To assess the physical implications of this anomaly in thermal quantum field theories with non-zero chemical potential.
• To explore potential applications in finite-density field theory and thermal field theory phenomenology.
• To clarify the role of the chemical potential in modifying the structure of the effective action and thermodynamic potential.

Proposed method

• Formalism of finite-temperature quantum field theory is applied to a relativistic complex scalar field with a conserved U(1) charge.
• The partition function is derived using the path integral approach, incorporating the chemical potential as a source for the conserved current.
• The multiplicative anomaly is computed via zeta function regularization of the determinant of the field operator in the presence of chemical potential.
• The anomaly is identified as a discrepancy in the product of determinants due to non-commutative ordering of operators in the effective action.
• The analysis uses dimensional regularization and analytic continuation techniques to handle divergences and extract finite physical contributions.
• Thermodynamic quantities such as the free energy are derived from the regularized partition function, including corrections from the anomaly.

Experimental results

Research questions

• RQ1 How does the presence of a chemical potential affect the structure of the partition function in a relativistic scalar field theory?
• RQ2 What is the origin and magnitude of the multiplicative anomaly in systems with conserved charge and finite temperature?
• RQ3 How does the multiplicative anomaly modify the effective action and thermodynamic potential in such systems?
• RQ4 What are the observable consequences of the multiplicative anomaly in thermal field theories with non-zero chemical potential?
• RQ5 Can the multiplicative anomaly be consistently incorporated into existing frameworks of finite-density quantum field theory?

Key findings

• The multiplicative anomaly is explicitly computed and found to be non-zero in the presence of chemical potential, indicating a breakdown of naive factorization of determinants.
• The anomaly contributes to the thermodynamic potential and modifies the effective action in a way that depends on the chemical potential and temperature.
• The effect is most significant in regimes where the chemical potential is comparable to the thermal scale, suggesting relevance for dense systems.
• The anomaly arises due to the non-commutative nature of the field operators in the path integral when the chemical potential is introduced.
• The result implies that standard treatments of finite-density field theories may miss important quantum corrections if the multiplicative anomaly is ignored.
• The paper suggests that the anomaly could play a role in understanding phase transitions and critical phenomena in systems with conserved charges.