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arXiv:1803.10192 · Entropy

The Definition of Entropy for Quantum Unstable Systems: A View-Point Based on the Properties of Gamow States

Osvaldo Civitarese, Manuel Gadella

Published Mar 28, 2018• 7 citations
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Abstract

In this paper, we review the concept of entropy in connection with the description of quantum unstable systems. We revise the conventional definition of entropy due to Boltzmann and extend it so as to include the presence of complex-energy states. After introducing a generalized basis of states which includes resonances, and working with amplitudes instead of probabilities, we found an expression for the entropy which exhibits real and imaginary components. We discuss the meaning of the imaginary part of the entropy on the basis of the similarities existing between thermal and time evolutions.

DOI
10.3390/e20040231

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Abstract In the standard quantum–field–theoretic description of unstable particles, exponential decay with width is encoded by promoting the energy (or mass) to a complex quantity . This prescription works extremely well in practice, but its imaginary part has no direct counterpart in the usual real–valued notion of energy, and it raises a natural question: where does the imaginary unit really belong? In this letter we show that the same physics can be reformulated in a complex–time framework in which the rest mass remains real and the decay is absorbed into a complex phase associated with the time direction. For a stable relativistic particle, the proper–time action can be written so that the mass controls the rate of phase rotation along a complex extension of time. For a quasi–stable state with width , the standard replacement is equivalently realized by a complex phase whose real part encodes the mass–controlled oscillation and whose imaginary part encodes the exponential damping. As a concrete example we apply this construction to the Higgs boson, whose mass arises from the Higgs vacuum expectation value while its finite lifetime is generated by interactions. In the complex–time picture these two aspects appear as the real and imaginary components of a single geometric object, providing a unified way to relate mass, width and the arrow of time without complexifying the energy spectrum itself. Running Title Unstable States in Complex Time Key Words unstable quantum states; complex time; complex energy; Higgs boson; vacuum structure

•Dec 7, 2025•0 citations
In this paper, we review the concept of entropy in connection with the description of quantum unstable systems. We revise the conventional definition of entropy due to Boltzmann and extend it so as to include the presence of complex-energy states. After introducing a generalized basis of states which includes resonances, and working with amplitudes instead of probabilities, we found an expression for the entropy which exhibits real and imaginary components. We discuss the meaning of the imaginary part of the entropy on the basis of the similarities existing between thermal and time evolutions.
In this paper, we review the concept of entropy in connection with the description of quantum unstable systems. We revise the conventional definition of entropy due to Boltzmann and extend it so as to include the presence of complex-energy states. After introducing a generalized basis of states which includes resonances, and working with amplitudes instead of probabilities, we~found an expression for the entropy which exhibits real and imaginary components. We discuss the meaning of the imaginary part of the entropy on the basis of the similarities existing between thermal and time evolutions.
We review some results in the theory of non-relativistic quantum unstable systems. We account for the most important definitions of quantum resonances that we identify with unstable quantum systems. Then, we recall the properties and construction of Gamow states as vectors in some extensions of Hilbert spaces, called Rigged Hilbert Spaces. Gamow states account for the purely exponential decaying part of a resonance; the experimental exponential decay for long periods of time physically characterizes a resonance. We briefly discuss one of the most usual models for resonances: the Friedrichs model. Using an algebraic formalism for states and observables, we show that Gamow states cannot be pure states or mixtures from a standard view point. We discuss some additional properties of Gamow states, such as the possibility of obtaining mean values of certain observables on Gamow states. A modification of the time evolution law for the linear space spanned by Gamow shows that some non-commuting observables on this space become commuting for large values of time. We apply Gamow states for a possible explanation of the Loschmidt echo.
It is well known that a Shannon-based definition of information entropy leads in the classical case to the Boltzmann entropy. It is tempting to regard the von Neumann entropy as the corresponding quantum-mechanical definition. But the latter is problematic from the quantum information point of view. Consequently, we introduce a new definition of entropy that reflects the inherent uncertainty of quantum-mechanical states. We derive for it an explicit expression, and discuss some of its general properties. We distinguish between the minimum uncertainty entropy of pure states, and the excess statistical entropy of mixtures.
Abstract Entropy has played an essential role in the history of physics. Its mathematical definition and applications have changed over time till today. In this paper, we first review the historical evolution of these various points of view, from the thermodynamic definition to information entropy from Shannon in classical physics, up to the modern concept of Neumann’s quantum entropy. As a specific example, we consider entanglement entropy and compare the phase space approach in classical physics to the Hilbert space approach in quantum physics in simple model systems. We derive a general expression for the entanglement entropy of fermions and bosons in arbitrary partitions of Hilbert space, valid beyond the thermodynamic limit. Next, we compare thermodynamic heat engines with quantum heat engines. Finally, we proceed to the more general concept of quantum (computational) complexity and argue, using the concept of entanglement entropy, that the Heisenberg time in classically chaotic systems coincides with the time when maximal complexity is reached in the quantum case for systems with all–all interactions.
Abstract It is common knowledge that the Wigner function of a quantum state may admit negative values, so that it cannot be viewed as a genuine probability density. Here, we examine the difficulty in finding an entropy-like functional in phase space that extends to negative Wigner functions and then advocate the merits of defining a complex-valued entropy associated with any Wigner function. This quantity, which we call thecomplex Wigner entropy, is defined via the analytic continuation of Shannon’s differential entropy of the Wigner function in the complex plane. We show that the complex Wigner entropy enjoys interesting properties, especially its real and imaginary parts are both invariant under Gaussian unitaries (displacements, rotations, and squeezing in phase space). Its real part is physically relevant when considering the evolution of the Wigner function under a Gaussian convolution, while its imaginary part is simply proportional to the negative volume of the Wigner function. Finally, we define the complex-valued Fisher information of any Wigner function, which is linked (via an extended de Bruijn’s identity) to the time derivative of the complex Wigner entropy when the state undergoes Gaussian additive noise. Overall, it is anticipated that the complex plane yields a proper framework for analyzing the entropic properties of quasiprobability distributions in phase space.
Abstract In the standard quantum–field–theoretic description of unstable particles, exponential decay with width is encoded by promoting the energy (or mass) to a complex quantity . This prescription works extremely well in practice, but its imaginary part has no direct counterpart in the usual real–valued notion of energy, and it raises a natural question: where does the imaginary unit really belong? In this letter we show that the same physics can be reformulated in a complex–time framework in which the rest mass remains real and the decay is absorbed into a complex phase associated with the time direction. For a stable relativistic particle, the proper–time action can be written so that the mass controls the rate of phase rotation along a complex extension of time. For a quasi–stable state with width , the standard replacement is equivalently realized by a complex phase whose real part encodes the mass–controlled oscillation and whose imaginary part encodes the exponential damping. As a concrete example we apply this construction to the Higgs boson, whose mass arises from the Higgs vacuum expectation value while its finite lifetime is generated by interactions. In the complex–time picture these two aspects appear as the real and imaginary components of a single geometric object, providing a unified way to relate mass, width and the arrow of time without complexifying the energy spectrum itself. Running Title Unstable States in Complex Time Key Words unstable quantum states; complex time; complex energy; Higgs boson; vacuum structure