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Unstable states in a model of nonrelativistic quantum electrodynamics: rate of decay, regeneration by decay products, sojourn time and irreversibility

Walter Felipe Wreszinski

Published Oct 21, 2019• 0 citations
OpenAlex

Abstract

We consider a family of states in a model of nonrelativistic quantum electrodynamics, with positive Hamiltonian $H$. For a given initial state $\Psi$, the return probability amplitude $R_{\Psi}(t) = (\Psi, \exp(-iHt) \Psi)$ may be written for positive times, as the sum of an exponentially decaying term and a correction $O(\frac{1}{t})$, for large times $t$ and small coupling constants (Theorem 4.1). The correction term is seen to be related both to the positivity of $H$ and to the existence of the virtual process of regeneration of the decaying state from the decay products, which is shown to be essentially quantum field theoretic, i.e., not present in nonrelativistic Schroedinger quantum mechanics. Some implications of this fact are analysed from the point of view of a general picture of irreversibility and the arrow of in quantum field theory. Finally, we make a first application of a time-energy uncertainty theorem to a quantum field theoretic model, in order to find a lower bound to the energy fluctuation in the state $\Psi$ (Theorem 5.2). In the process, it is also suggested that the time of sojourn $\tau_{H}(\Psi)= \int_{0}^{\infty} |R_{\Psi}(t)|^{2}dt$ is the most natural quantity to consider in connection with the decay of unstable atoms or particles: it is proved to coincide with the the average lifetime of the decaying state, a standard quantity in quantum probability. No use is made of complex energies associated to analytic continuations of the resolvent operator to unphysical Riemann sheets.

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Journal of the Physical Society of Japan•Jan 1, 2003•2 citations

Quantum transitions and dressed unstable states

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We consider the problem of the meaning of quantum unstable states including their dressing. According to both Dirac and Heitler this problem has not been solved in the usual formulation of quantum mechanics. A precise definition of excited states is still needed to describe quantum transitions. We use our formulation given in terms of density matrices outside the Hilbert space. We obtain a dressed unstable state for the Friedrichs model, which is the simplest model that incorporates both bare and dressed quantum states. The excited unstable state is derived from the stable states through analytic continuation. It is given by an irreducible density matrix with broken time symmetry. It can be expressed by a superposition of Gamow density operators. The main difference from previous studies is that excited states are not factorizable into wave functions. The dressed unstable state satisfies all the criteria that we can expect: it has a real average energy and a nonvanishing trace. The average energy differs from Green's function energy by a small effect starting with fourth order in the coupling constant. Our state decays following a Markovian equation. There are no deviations from exponential decay neither for short nor for long times, as is the case for the bare state. The dressed state satisfies an uncertainty relation between energy and lifetime. We can also define dressed photon states and describe how the energy of the excited state is transmitted to the photons. There is another very important aspect: deviations from exponential decay would be in contradiction with indiscernibility as one could define, e.g., old mesons and young mesons according to their lifetime. This problem is solved by showing that quantum transitions are the result of two processes: a dressing process, discussed in a previous publication, and a decay process, which is much slower for electrodynamic systems. During the dressing process the unstable state is prepared. Then the dressed state decays in a purely exponential way. In the Hilbert space the two processes are not separated. Therefore it is not astonishing that we obtain for the unstable dressed state an irreducible density matrix outside the Liouville-Hilbert-space. This is a limit of Hilbert space states that are arbitrarily close to the decaying state. There are experiments that could verify our proposal. A typical one would be the study of the line shape, which is due to the superposition of the short-time process and the long-time process. The long-time process taken separately leads to a much sharper line shape, and avoids the divergence of the fluctuation predicted by the Lorentz line shape.

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An eect generated by the nonexponential behavior of the survival amplitude of an unstable state in the long time region is considered. In 1957 Khalfin proved that this amplitude tends to zero as t ! 1 more slowly than any exponential function oft. This can be described in terms of the time-dependent decay rate (t) which, when considered with the Khalfin result, means that this (t) is not a constant for large t but that it tends to zero as t!1. We find that a similar conclusion can be drawn for a large class of models of unstable states for a quantity, which can be interpreted as the ”instantaneous energy” of the unstable state. This energy should be much smaller for suitably larger values of t than when t is of the order of the lifetime of the considered state. Within a given model we show that the energy corrections in the long (t ! 1) and relatively short (lifetime of the state) time regions, are dierent. This is a purely quantum mechanical eect. It is hypothesized that there is a possibility to detect this eect by analyzing the spectra of distant astrophysical objects. The above property of unstable states may influence the measured values of astrophysical and cosmological parameters.

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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

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On Time in Quantum Mechanics

Nicola Vona

Although time measurements are routinely performed in laboratories, their theoretical description is still an open problem. Correspondingly, the status of the energy-time uncertainty relation is unsettled. In the first part of this work the necessity of positive operator valued measures (POVM) as descriptions of every quantum experiment is reviewed, as well as the suggestive role played by the probability current in time measurements. Furthermore, it is shown that no POVM exists, which approximately agrees with the probability current on a very natural set of wave functions; nevertheless, the choice of the set is crucial, and on more restrictive sets the probability current does provide a good arrival time prediction. Some ideas to experimentally detect quantum effects in time measurements are discussed. In the second part of the work the energy-time uncertainty relation is considered, in particular for a model of alpha decay for which the variance of the energy can be calculated explicitly, and the variance of time can be estimated. This estimate is tight for systems with long lifetimes, in which case the uncertainty relation is shown to be satisfied. Also the linewidth-lifetime relation is shown to hold, but contrary to the common expectation, it is found that the two relations behave independently, and therefore it is not possible to interpret one as a consequence of the other. To perform the mentioned analysis quantitative scattering estimates are necessary. To this end, bounds of the form $\|1_Re^{-iHt}ψ\|_2^2 \leq C t^{-3}$ have been derived, where $ψ$ denotes the initial state, $H$ the Hamiltonian, $R$ a positive constant, and $C$ is explicitly known. As intermediate step, bounds on the derivatives of the $S$-matrix in the form $\|1_K S^{(n)}\|_\infty \leq C_{n,K} $ have been established, with $n=1,2,3$, and the constants $C_{n,K}$ explicitly known.

arXiv:1403.2496•Jan 1, 2014•2 citations
We consider a family of states in a model of nonrelativistic quantum electrodynamics, with positive Hamiltonian HHH. For a given initial state Ψ\PsiΨ, the return probability amplitude RΨ(t)=(Ψ,exp⁡(−iHt)Ψ)R_{\Psi}(t) = (\Psi, \exp(-iHt) \Psi)RΨ​(t)=(Ψ,exp(−iHt)Ψ) may be written for positive times, as the sum of an exponentially decaying term and a correction O(1t)O(\frac{1}{t})O(t1​), for large times ttt and small coupling constants (Theorem 4.1). The correction term is seen to be related both to the positivity of HHH and to the existence of the virtual process of regeneration of the decaying state from the decay products, which is shown to be essentially quantum field theoretic, i.e., not present in nonrelativistic Schroedinger quantum mechanics. Some implications of this fact are analysed from the point of view of a general picture of irreversibility and the arrow of in quantum field theory. Finally, we make a first application of a time-energy uncertainty theorem to a quantum field theoretic model, in order to find a lower bound to the energy fluctuation in the state Ψ\PsiΨ (Theorem 5.2). In the process, it is also suggested that the time of sojourn τH(Ψ)=∫0∞∣RΨ(t)∣2dt\tau_{H}(\Psi)= \int_{0}^{\infty} |R_{\Psi}(t)|^{2}dtτH​(Ψ)=∫0∞​∣RΨ​(t)∣2dt is the most natural quantity to consider in connection with the decay of unstable atoms or particles: it is proved to coincide with the the average lifetime of the decaying state, a standard quantity in quantum probability. No use is made of complex energies associated to analytic continuations of the resolvent operator to unphysical Riemann sheets.
The problem of the meaning of quantum unstable states including their dressing is considered. The formulation given in terms of density matrices outside the Hilbert space is used. A dressed unstable state for the Friedrichs model, which is the simplest model that incorporates both bare and dressed quantum states is obtained. Due to resonance singularities that appear in the frequency denominators, quantum unstable systems are categorized as non-integrable systems in the sense of Poincaré. The excited unstable state is derived from the stable states through a suitable analytic continuation of the denominators. It is given by an irreducible density matrix with broken time-symmetry. Our state decays following a Markovian equation. There are no deviations from exponential decay neither for short nor for long times, as is the case for the bare state. The dressed state satisfies an uncertainty relation between energy and lifetime. There are experiments that could verify our proposal. A typical one would be the study of the line shape, which is due to the superposition of the short time process and the long time process. The long time process taken separately leads to a much sharper line shape, and avoids the divergence of the fluctuation predicted by the Lorentz line shape.
We consider the problem of the meaning of quantum unstable states including their dressing. According to both Dirac and Heitler this problem has not been solved in the usual formulation of quantum mechanics. A precise definition of excited states is still needed to describe quantum transitions. We use our formulation given in terms of density matrices outside the Hilbert space. We obtain a dressed unstable state for the Friedrichs model, which is the simplest model that incorporates both bare and dressed quantum states. The excited unstable state is derived from the stable states through analytic continuation. It is given by an irreducible density matrix with broken time symmetry. It can be expressed by a superposition of Gamow density operators. The main difference from previous studies is that excited states are not factorizable into wave functions. The dressed unstable state satisfies all the criteria that we can expect: it has a real average energy and a nonvanishing trace. The average energy differs from Green's function energy by a small effect starting with fourth order in the coupling constant. Our state decays following a Markovian equation. There are no deviations from exponential decay neither for short nor for long times, as is the case for the bare state. The dressed state satisfies an uncertainty relation between energy and lifetime. We can also define dressed photon states and describe how the energy of the excited state is transmitted to the photons. There is another very important aspect: deviations from exponential decay would be in contradiction with indiscernibility as one could define, e.g., old mesons and young mesons according to their lifetime. This problem is solved by showing that quantum transitions are the result of two processes: a dressing process, discussed in a previous publication, and a decay process, which is much slower for electrodynamic systems. During the dressing process the unstable state is prepared. Then the dressed state decays in a purely exponential way. In the Hilbert space the two processes are not separated. Therefore it is not astonishing that we obtain for the unstable dressed state an irreducible density matrix outside the Liouville-Hilbert-space. This is a limit of Hilbert space states that are arbitrarily close to the decaying state. There are experiments that could verify our proposal. A typical one would be the study of the line shape, which is due to the superposition of the short-time process and the long-time process. The long-time process taken separately leads to a much sharper line shape, and avoids the divergence of the fluctuation predicted by the Lorentz line shape.
An eect generated by the nonexponential behavior of the survival amplitude of an unstable state in the long time region is considered. In 1957 Khalfin proved that this amplitude tends to zero as t ! 1 more slowly than any exponential function oft. This can be described in terms of the time-dependent decay rate (t) which, when considered with the Khalfin result, means that this (t) is not a constant for large t but that it tends to zero as t!1. We find that a similar conclusion can be drawn for a large class of models of unstable states for a quantity, which can be interpreted as the ”instantaneous energy” of the unstable state. This energy should be much smaller for suitably larger values of t than when t is of the order of the lifetime of the considered state. Within a given model we show that the energy corrections in the long (t ! 1) and relatively short (lifetime of the state) time regions, are dierent. This is a purely quantum mechanical eect. It is hypothesized that there is a possibility to detect this eect by analyzing the spectra of distant astrophysical objects. The above property of unstable states may influence the measured values of astrophysical and cosmological parameters.
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
We reexamine the problem of particle decay within the Hamiltonian formalism. By deforming contours of integration, the survival amplitude is expressed as a sum of purely exponential contributions arising from the simple poles of the resolvent on the second sheet plus a background integral along a complex contour $\ensuremath{\Gamma}$ running below the location of the poles. We observe that the time dependence of the survival amplitude in the small-time region is strongly correlated to the asymptotic behavior of the energy spectrum of the system; we compute the small-time behavior of the survival amplitude for a wide variety of asymptotic behaviors. In the special case of the Lee model, using a formal procedure of analytic continuation, we show that a complete set of complex energy eigenvectors of the Hamiltonian can be associated with the poles of the resolvent and the background contour $\ensuremath{\Gamma}$. These poles and points along $\ensuremath{\Gamma}$ correspond to the discrete and the continuum states, respectively. In this context, each unstable particle is associated with a well-defined object, which is a discrete generalized eigenstate of the Hamiltonian having a complex eigenvalue, with its real and negative imaginary parts being the mass and half-width of the particle, respectively. Finally, we briefly discuss the analytic continuation of the scattering amplitude within this generalized scheme, and note the appearance of "redundant poles" which do not correspond to discrete solutions of the modified eigenvalue problem.
Although time measurements are routinely performed in laboratories, their theoretical description is still an open problem. Correspondingly, the status of the energy-time uncertainty relation is unsettled. In the first part of this work the necessity of positive operator valued measures (POVM) as descriptions of every quantum experiment is reviewed, as well as the suggestive role played by the probability current in time measurements. Furthermore, it is shown that no POVM exists, which approximately agrees with the probability current on a very natural set of wave functions; nevertheless, the choice of the set is crucial, and on more restrictive sets the probability current does provide a good arrival time prediction. Some ideas to experimentally detect quantum effects in time measurements are discussed. In the second part of the work the energy-time uncertainty relation is considered, in particular for a model of alpha decay for which the variance of the energy can be calculated explicitly, and the variance of time can be estimated. This estimate is tight for systems with long lifetimes, in which case the uncertainty relation is shown to be satisfied. Also the linewidth-lifetime relation is shown to hold, but contrary to the common expectation, it is found that the two relations behave independently, and therefore it is not possible to interpret one as a consequence of the other. To perform the mentioned analysis quantitative scattering estimates are necessary. To this end, bounds of the form ∥1Re−iHtψ∥22≤Ct−3\|1_Re^{-iHt}ψ\|_2^2 \leq C t^{-3}∥1R​e−iHtψ∥22​≤Ct−3 have been derived, where ψψψ denotes the initial state, HHH the Hamiltonian, RRR a positive constant, and CCC is explicitly known. As intermediate step, bounds on the derivatives of the SSS-matrix in the form $\|1_K S^{(n)}\|_\infty \leq C_{n,K} $ have been established, with n=1,2,3n=1,2,3n=1,2,3, and the constants Cn,KC_{n,K}Cn,K​ explicitly known.