Certain aspects of Lagrangian and Hamiltonian dynamics are investigated within the framework of an extended complex phase space approach, characterized by the transformation and by the presence of the imaginary time variable in the complex variable in a local chart. It is an application of imaginary time which is essential in connecting quantum mechanics with statistical physics. We argue that the novel complexified approach enhances the system of dynamical equations obtained in the sense that the new derived equations appear as certain combinations of former equations. Furthermore, it was shown that the physics we experience in the real time is somewhat different from what is experienced in imaginary region. Further consequences are discussed in some details.Certain aspects of Lagrangian and Hamiltonian dynamics are investigated within the framework of an extended complex phase space approach, characterized by the transformation and by the presence of the imaginary time variable in the complex variable in a local chart. It is an application of imaginary time which is essential in connecting quantum mechanics with statistical physics. We argue that the novel complexified approach enhances the system of dynamical equations obtained in the sense that the new derived equations appear as certain combinations of former equations. Furthermore, it was shown that the physics we experience in the real time is somewhat different from what is experienced in imaginary region. Further consequences are discussed in some details.

Journal of Applied AnalysisJan 1, 201213 citations

Imaginary-Time Mpemba Effect in Quantum Many-Body Systems

Wei-Xuan Chang, Shuai Yin, Shi‐Xin Zhang +1 more

Various exotic phenomena emerge in nonequilibrium quantum many-body systems. The Mpemba effect, denoting the situation where a hot system freezes faster than a colder one, is a counterintuitive nonequilibrium phenomenon that has attracted enduring interest for more than half a century. In this Letter, we unveil a novel phenomenon of the Mpemba effect in the imaginary-time relaxation dynamics in quantum many-body systems, termed as the imaginary-time Mpemba effect (ITME). We propose a mechanism for the ITME rooted in the initial state's eigenbasis expansion, and we demonstrate its emergence in representative quantum lattice models. Using sign-free quantum Monte Carlo (QMC) simulations, we unequivocally establish the existence of the ITME across diverse interacting quantum systems. Crucially, given the widespread use of imaginary-time dynamics for ground-state calculations, the discovery of the ITME potentially provides a new pathway to accelerate quantum many-body simulations, particularly for QMC methods hindered by the sign problem.Various exotic phenomena emerge in nonequilibrium quantum many-body systems. The Mpemba effect, denoting the situation where a hot system freezes faster than a colder one, is a counterintuitive nonequilibrium phenomenon that has attracted enduring interest for more than half a century. In this Letter, we unveil a novel phenomenon of the Mpemba effect in the imaginary-time relaxation dynamics in quantum many-body systems, termed as the imaginary-time Mpemba effect (ITME). We propose a mechanism for the ITME rooted in the initial state's eigenbasis expansion, and we demonstrate its emergence in representative quantum lattice models. Using sign-free quantum Monte Carlo (QMC) simulations, we unequivocally establish the existence of the ITME across diverse interacting quantum systems. Crucially, given the widespread use of imaginary-time dynamics for ground-state calculations, the discovery of the ITME potentially provides a new pathway to accelerate quantum many-body simulations, particularly for QMC methods hindered by the sign problem.

arXiv:2409.06547Physical Review LettersFeb 12, 20267 citations

Time Evolution in Quantum Mechanics

CRC Press eBooksSep 25, 20090 citations

Imaginary Time and the Arrow of Time: Interplay Between Relativity and Quantum Mechanics

Wen-Xiang Chen

The existence and directionality of time are fundamental issues in modern physics. Special relativity is based on two principles: the invariance of physical laws in all inertial frames and the constancy of the speed of light in vacuum\cite{SRPostulates}. These principles lead naturally to a four‑dimensional spacetime with a distinct time coordinate. In quantum mechanics and quantum field theory the imaginary unit~ $i$ appears essentially in dynamical equations such as the Schrödinger equation and in path‑integral formulations. Mathematically, multiplying real time by~ $i$ —a so‑called Wick rotation—transforms the Minkowski spacetime metric into a Euclidean metric\cite{WickRotation}. This paper analyses how the requirement of imaginary numbers in quantum theory and the postulates of relativity together imply the existence of a one‑way time axis associated with the imaginary direction. We review the role of imaginary time in relativistic geometry and quantum dynamics, discuss the thermodynamic arrow of time and its link to entropy increase\cite{ArrowOfTime}, and present theoretical considerations supporting the claim that time exists and is oriented along a one‑directional imaginary axis.The existence and directionality of time are fundamental issues in modern physics. Special relativity is based on two principles: the invariance of physical laws in all inertial frames and the constancy of the speed of light in vacuum\cite{SRPostulates}. These principles lead naturally to a four‑dimensional spacetime with a distinct time coordinate. In quantum mechanics and quantum field theory the imaginary unit~$i$ appears essentially in dynamical equations such as the Schrödinger equation and in path‑integral formulations. Mathematically, multiplying real time by~$i$—a so‑called Wick rotation—transforms the Minkowski spacetime metric into a Euclidean metric\cite{WickRotation}. This paper analyses how the requirement of imaginary numbers in quantum theory and the postulates of relativity together imply the existence of a one‑way time axis associated with the imaginary direction. We review the role of imaginary time in relativistic geometry and quantum dynamics, discuss the thermodynamic arrow of time and its link to entropy increase\cite{ArrowOfTime}, and present theoretical considerations supporting the claim that time exists and is oriented along a one‑directional imaginary axis.

Aug 11, 20250 citations

Formalism for Time-Dependent Quantum Dynamics

Royal Society of Chemistry eBooksJul 3, 20150 citations

i

appears essentially in dynamical equations such as the Schrödinger equation and in path‑integral formulations. Mathematically, multiplying real time by~

i

—a so‑called Wick rotation—transforms the Minkowski spacetime metric into a Euclidean metric\cite{WickRotation}. This paper analyses how the requirement of imaginary numbers in quantum theory and the postulates of relativity together imply the existence of a one‑way time axis associated with the imaginary direction. We review the role of imaginary time in relativistic geometry and quantum dynamics, discuss the thermodynamic arrow of time and its link to entropy increase\cite{ArrowOfTime}, and present theoretical considerations supporting the claim that time exists and is oriented along a one‑directional imaginary axis.