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Analyzing Heisenberg-Robertson (HR) and Schr\"{o}dinger uncertainty relations we found, that there can exist large set of states of the quantum system under considerations, for which the lower bound of the product of the standard deviations of a pair of non-commuting observables, $A$ and $B$, is zero. These states are not eigenstates of either the observable $A$ or $B$. We have also shown that the so-called "sum uncertainty relations" also do not provide any information about lower bounds on the standard deviations calculated for these states., Comment: 12 pages, Neww comments and results added

IC 417 is in the Galactic Plane, and likely part of the Aur OB2 association; it is ~2 kpc away. Stock 8 is one of the densest cluster constituents; off of it to the East, there is a 'Nebulous Stream' (NS) that is dramatic in the infrared (IR). We have assembled a list of literature-identified young stellar objects (YSOs), new candidate YSOs from the NS, and new candidate YSOs from IR excesses. We vetted this list via inspection of the images, spectral energy distributions (SEDs), and color-color/color-magnitude diagrams. We placed the 710 surviving YSOs and candidate YSOs in ranked bins, nearly two-thirds of which have more than 20 points defining their SEDs. The lowest-ranked bins include stars that are confused, or likely carbon stars. There are 503 in the higher-ranked bins; half are SED Class III, and $\sim$40\% are SED Class II. Our results agree with the literature in that we find that the NS and Stock 8 are at about the same distance as each other (and as the rest of the YSOs), and that the NS is the youngest region, with Stock 8 a little older. We do not find any evidence for an age spread within the NS, consistent with the idea that the star formation trigger came from the north. We do not find that the other literature-identified clusters here are as young as either the NS or Stock 8; at best they are older than Stock 8, and they may not all be legitimate clusters., Comment: Accepted by AAS Journals

We try to find conditions, the fulfillment of which allows a universe born in a metastable false vacuum state to survive and not to collapse. The conditions found are in the form of inequalities linking the depending on time $t$ instantaneous decay rate ${\it\Gamma}(t)$ of the false vacuum state and the Hubble parameter $H(t)$. Properties of the decay rate of a quantum metastable states are discussed and then the possible solutions of the conditions found are analyzed and discussed. Within the model considered it is shown that a universe born in the metastable vacuum state has a very high chance of surviving until very late times if the lifetime, $\tau_{0}^{F}$, of the metastable false vacuum state is much shorter, than the duration of the inflation process. Our analysis shows that the instability of the electroweak vacuum does not have to result in the tragic fate of our Universe leading to its death., Comment: 13 pages, 7 figures; some typos corrected, comments and references added, accepted by the European Physical Journal C. arXiv admin note: text overlap with arXiv:2110.11957

The cosmological constant $\Lambda$ is a measure of the energy density of the vacuum. Therefore properties of the energy of the system in the metastable vacuum state reflect properties of $\Lambda = \Lambda(t)$. We analyze properties of the energy, $E(t)$, of a general quantum system in the metastable state in various phases of the decay process: In the exponential phase, in the transition phase between the exponential decay and the later phase, where decay law as a function of time $t$ is in the form of powers of $1/t$, and also in this last phase. We found that this energy having an approximate value resulting from the Weisskopf--Wigner theory in the exponential decay phase is reduced very fast in the transition phase to its asymptotic value $E(t) \simeq E_{min} + \alpha_{2}/t^{2}+\ldots$ in the late last phase of the decay process. (Here $E_{min}$ is the minimal energy of the system). This quantum mechanism reduces the energy of the system in the unstable state by a dozen or even several dozen orders or more. We show that if to assume that a universe was born in metastable false vacuum state then according to this quantum mechanism the cosmological constant $\Lambda$ can have a very great value resulting from the quantum field theory calculations in the early universe in the inflationary era, $\Lambda \simeq \Lambda_{qft}$, and then it can later be quickly reduced to the very, very small values., Comment: 39 pages, 9 figures; new comments added, some typos corrected, references added

Analyzing general uncertainty relations one can find that there can exist such pairs of non-commuting observables $A$ and $B$ and such vectors that the lower bound for the product of standard deviations $\Delta A$ and $\Delta B$ calculated for these vectors is zero: $\Delta A\,\cdot\,\Delta B \geq 0$. Here we discuss examples of such cases and some other inconsistencies which can be found performing a rigorous analysis of the uncertainty relations in some special cases. As an illustration of such cases matrices $(2\times 2)$ and $(3 \times 3)$ and the position--momentum uncertainty relation for a quantum particle in the box are considered. The status of the uncertainty relation in $\cal PT$--symmetric quantum theory and the problems associated with it are also studied., Comment: 25, pages

The Heisenberg and Mandelstam-Tamm time-energy uncertainty relations are analyzed. The conlusion resulting from this analysis is that within the Quantum Mechanics of Schr\"{o}dinger and von Neumann, the status of these relations can not be considered as the same as the status of the position-momentum uncertainty relations, which are rigorous. The conclusion is that the time--energy uncertainty relations can not be considered as universally valid., Comment: 16 pages, talk given at 3rd Jagiellonian Symposium on Fundamental and Applied Subatomic Physics, 2019, Krakow, Poland. arXiv admin note: substantial text overlap with arXiv:1810.11462

We investigate a cosmological model in which dark energy identified with the vacuum energy which is running and decaying. In this model vacuum is metastable and decays into a bare (true) vacuum. This decaying process has a quantum nature and is described by tools of the quantum decay theory of unstable systems. We have found formulas for an asymptotic behavior of the energy density of dark energy in the form of a series of inverse powers of the cosmological time. We investigate the dynamics of FRW models using dynamical system methods as well as searching for exact solutions. From dynamical analysis we obtain different evolutional scenarios admissible for all initial conditions. For the interpretation of the dynamical evolution caused by the decay of the quantum vacuum we study the thermodynamics of the apparent horizon of the model as well as the evolution of the temperature. For the early Universe, we found that the quantum effects modified the evolution of the temperature of the Universe. In our model the adiabatic approximation is valid and the quantum vacuum decay occurs with an adequate unknown particle which constitutes quantum vacuum. We argue that the late-time evolution of metastable energy is the holographic dark energy., Comment: 38 pages, 14 figures; v3: accepted for publication in JCAP

We analyze general uncertainty relations and we show that there can exist such pairs of non--commuting observables $A$ and $B$ and such vectors that the lower bound for the product of standard deviations $\Delta A$ and $\Delta B$ calculated for these vectors is zero: $\Delta A\,\cdot\,\Delta B \geq 0$. We show also that for some pairs of non--commuting observables the sets of vectors for which $\Delta A\,\cdot\,\Delta B \geq 0$ can be complete (total). The Heisenberg, $\Delta t \,\cdot\, \Delta E \geq \hbar/2$, and Mandelstam--Tamm (MT), $ \tau_{A}\,\cdot \,\Delta E \geq \hbar/2$, time--energy uncertainty relations ($\tau_{A}$ is the characteristic time for the observable $A$) are analyzed too. We show that the interpretation $\tau_{A} = \infty$ for eigenvectors of a Hamiltonian $H$ does not follow from the rigorous analysis of MT relation. We show also that contrary to the position--momentum uncertainty relation, the validity of the MT relation is limited: It does not hold on complete sets of eigenvectors of $A$ and $H$., Comment: 16 pages, new results and comments added, accepted for Modern Physics Letters A

We consider a cosmology with decaying metastable dark energy and assume that a decay process of this metastable dark energy is a quantum decay process. Such an assumption implies among others that the evolution of the Universe is irreversible and violates the time reversal symmetry. We show that if to replace the cosmological time $t$ appearing in the equation describing the evolution of the Universe by the Hubble cosmological scale time, then we obtain time dependent $\Lambda (t)$ in the form of the series of even powers of the Hubble parameter $H$: $\Lambda (t) = \Lambda (H)$. Out special attention is focused on radioactive like exponential form of the decay process of the dark energy and on the consequences of this type decay., Comment: RevTeX 4-1, 18 pages, 5 figures

We consider a model of a unstable state defined by the truncated Breit-Wigner energy density distribution function. An analytical form of the survival amplitude $a(t)$ of the state considered is found. Our attention is focused on the late time properties of $a(t)$ and on effects generated by the non--exponential behavior of this amplitude in the late time region: In 1957 Khalfin proved that this amplitude tends to zero as $t$ goes to the infinity more slowly than any exponential function of $t$. This effect can be described using a time-dependent decay rate $\gamma(t)$ and then the Khalfin result means that this $\gamma(t)$ is not a constant but at late times it tends to zero as $t$ goes to the infinity. It appears that the energy $E(t)$ of the unstable state behaves similarly: It tends to the minimal energy $E_{min}$ of the system as $t \to \infty$. Within the model considered we find two first leading time dependent elements of late time asymptotic expansions of $E(t)$ and $\gamma (t)$. We discuss also possible implications of such a late time asymptotic properties of $E(t)$ and $\gamma (t)$ and cases where these properties may manifest themselves., Comment: 24 pages, 4 figures, typos corrected, commnts added

We analyze properties of unstable systems at rest and in motion., Comment: 8 pages, 4 Figures, Talk presented at "2-nd Jagiellonian Symposium on Fundamental and Applied Subatomic Physics", June 4 --- 9, 2017, Krak\'ow, Poland

We derive the Shafieloo, Hazra, Sahni and Starobinsky (SHSS) phenomenological formula for the radioactive-like decay of metastable dark energy directly from the quantum mechanics principles. For this aim we use the Fock-Krylov theory of quantum unstable states. We obtain deeper insight on the decay process as having three basic phases: the phase of radioactive decay, the next phase of damping oscillations, and finally the phase of power law decaying. We consider the cosmological model with matter and dark energy in the form of decaying metastable dark energy and study its dynamics in the framework of non-conservative cosmology with an interacting term determined by the running cosmological parameter. We study cosmological implications of metastable dark energy and estimate the characteristic time of ending of the radioactive-like decay epoch as 22296 of the present age of the Universe. We also confront the model with astronomical data which show that the model is in good agreement with the observations. Our general conclusion is that we are living in the epoch of the radioactive-like decay of metastable dark energy which is a relict of the quantum age of the Universe., Comment: v2: 14 pages, 7 figures, change of title, new discussion of crossover time and new result on characteristic time of ending of radioactive-like decay epoch

We study the cosmology with the running dark energy. The parametrization of dark energy with the respect to the redshift is derived from the first principles of quantum mechanics. Energy density of dark energy is obtained from the quantum process of transition from the false vacuum state to the true vacuum state. This is the class of the extended interacting $\Lambda$CDM models. We consider the energy density of dark energy parametrization $\rho_\text{de}(t)$, which follows from the Breit-Wigner energy distribution function which is used to model the quantum unstable systems. The idea that properties of the process of the quantum mechanical decay of unstable states can help to understand the properties of the observed universe was formulated by Krauss and Dent and this idea was used in our considerations. In the cosmological model with the mentioned parametrization there is an energy transfer between the dark matter and dark energy. In such a evolutional scenario the universe is starting from the false vacuum state and going to the true vacuum state of the present day universe. We find that the intermediate regime during the passage from false to true vacuum states takes place. The intensity of the analyzed process is measured by a parameter $\alpha$. For the small value of $\alpha$ ($0<\alpha <0.4$) this intermediate (quantum) regime is characterized by an oscillatory behavior of the density of dark energy while the for $\alpha > 0.4$ the density of the dark energy simply jumps down. In both cases (independent from the parameter $\alpha$) the today value of density of dark energy is reached at the value of $0.7$. We estimate the cosmological parameters for this model with visible and dark matter. This model becomes in good agreement with the astronomical data and is practically indistinguishable from $\Lambda$CDM model., Comment: 14 pages, 16 figures

We analyze properties of unstable vacuum states from the point of view of the quantum theory. In the literature one can find some suggestions that some of false (unstable) vacuum states may survive up to times when their survival probability has a non-exponential form. At asymptotically late times the survival probability as a function of time $t$ has an inverse power--like form. We show that at this time region the energy of the false vacuum states tends to the energy of the true vacuum state as $1/t^{2}$ for $t \to \infty$. This means that the energy density in the unstable vacuum state should have analogous properties and hence the cosmological constant $\Lambda = \Lambda (t)$ too. The conclusion is that $\Lambda$ in the Universe with the unstable vacuum should have a form of the sum of the "bare" cosmological constant and of the term of a type $1/t^{2}$: $\Lambda(t) \equiv \Lambda_{bare} + d/ t^{2}$ (where $\Lambda_{bare}$ is the cosmological constant for the Universe with the true vacuum)., Comment: 14 pages, 2 figures, accepted for Theoretical and Mathematical Physics. arXiv admin note: substantial text overlap with arXiv:1304.2796

Results of theoretical studies of the quantum unstable systems caused that there are rather widespread belief that a universal feature od the quantum decay process is the presence of three time regimes of the decay process: the early time (initial) leading to the Quantum Zeno (or Anti Zeno) Effects, "exponential" (or "canonical") described by the decay law of the exponential form, and late time characterized by the decay law having inverse--power law form. Based on the fundamental principles of the quantum theory we give the proof that there is no time interval in which the survival probability (decay law) could be a decreasing function of time of the purely exponential form but even at the "exponential" regime the decay curve is oscillatory modulated with a smaller or a large amplitude of oscillations depending on parameters of the model considered., Comment: LaTeX 2e,13 pages, 3 figures

We study properties of moving relativistic quantum unstable systems. We show that in contrast to the properties of classical particles and quantum stable objects the velocity of moving freely relativistic quantum unstable systems can not be constant in time. We show that this new quantum effect results from the fundamental principles of the quantum theory and physics: It is a consequence of the principle of conservation of energy and of the fact that the mass of the quantum unstable system is not defined. This effect can affect the form of the decay law of moving relativistic quantum unstable systems., Comment: LaTeX2e, 11 pages, new coments and references added

Estimates of the Higgs and top quark masses, $m_{H} \simeq 125.10 \pm 0.14$ [GeV] and $m_{t} \simeq 172.76 \pm 0.30$ [GeV] based on the experimental results place the Standard Model in the region of the metastable vacuum. A consequence of the metastability of the Higgs vacuum is that it should induce the decay of the electroweak vacuum in the early Universe with catastrophic consequences. It may happen that certain universes were lucky enough to survive the time of canonical decay, that is the exponential decay, and live longer. This means that it is reasonable to analyze conditions allowing for that. We analyze properties of an ensemble of Universes with unstable vacua considered as an ensemble of unstable systems from the point of view of the quantum theory of unstable states. We found some symmetry relations for quantities characterizing the metastable state. We also found a relation linking the decay rate, ${\it\Gamma}$ of the metastable vacuum state with the Hubble parameter $H(t)$, which may help to explain why a universe with an unstable vacuum that lives longer then the canonical decay times need not decay., Comment: LaTeX2e, 31 pages, 2 figures, new comments and references added

Results presented in a recent paper "Which is the Quantum Decay Law of Relativistic particles?", arXiv: 1412.3346v2 [quant--ph]], are analyzed. We show that approximations used therein to derive the main final formula for the survival probability of finding a moving unstable particle to be undecayed at time $t$ force this particle to almost stop moving, that is that, in fact, the derived formula is approximately valid only for $\gamma \cong 1$, where $\gamma = 1/\sqrt{1-\beta^{2}}$ and $\beta = v/c$, or in other words, for the velocity $v \simeq 0$., Comment: 8 pages, additional analysis and comments added

Within the quantum mechanical treatment of the decay problem one finds that at late times $t$ the survival probability of an unstable state cannot have the form of an exponentially decreasing function of time $t$ but it has an inverse power-like form. This is a general property of unstable states following from basic principles of quantum theory. The consequence of this property is that in the case of false vacuum states the cosmological constant becomes dependent on time: $\Lambda - \Lambda_{\text{bare}}\equiv \Lambda(t) -\Lambda_{\text{bare}} \sim 1/t^{2}$. We construct the cosmological model with decaying vacuum energy density and matter for solving the cosmological constant problem and the coincidence problem. We show the equivalence of the proposed decaying false vacuum cosmology with the $\Lambda(t)$ cosmologies (the $\Lambda(t)$CDM models). The cosmological implications of the model of decaying vacuum energy (dark energy) are discussed. We constrain the parameters of the model with decaying vacuum using astronomical data. For this aim we use the observation of distant supernovae of type Ia, measurements of $H(z)$, BAO, CMB and others. The model analyzed is in good agreement with observation data and explain a small value of the cosmological constant today., Comment: 9 pages, 4 figures. Talk given at Seventh International Workshop DICE 2014: Spacetime -- Matter -- Quantum Mechanics ... news on missing links, Castiglioncello (Tuscany, Italy), September 15--19, 2014

A new quantum effect connected with the late time behavior of decaying states is described and its possible observational consequences are analyzed: It is shown that charged unstable particles as well as neutral unstable particles with non--zero magnetic moment which live sufficiently long may emit electromagnetic radiation. This mechanism is due to the nonclassical behavior of unstable particles at late times (at the post exponential time region). Analyzing the transition times region between exponential and non-exponential form of the survival amplitude it is found that the instantaneous energy of the unstable particle can take very large values, much larger than the energy of this state at times from the exponential time region. Based on the results obtained for the model considered, it is shown that this new purely quantum mechanical effect may be responsible for causing unstable particles produced by astrophysical sources and moving with relativistic velocities to emit electromagnetic--, $X$-- or $\gamma$--rays at some time intervals from the transition time regions., Comment: 8 pages, 3 figures, Talk given at "Seventh International Workshop DICE 2014: Spacetime -- Matter -- Quantum Mechanics ... news on missing links", Castiglioncello (Tuscany, Italy), September 15 -- 19, 2014

We study the survival probability of moving relativistic unstable particles with definite momentum $\vec{p} \neq 0$. The amplitude of the survival probability of these particles is calculated using its integral representation. We found decay curves of such particles for the quantum mechanical models considered. These model studies show that late time deviations of the survival probability of these particles from the exponential form of the decay law, that is the transition times region between exponential and non-expo\-nen\-tial form of the survival probability, should occur much earlier than it follows from the classical standard approach resolving itself into replacing time $t$ by $t/\gamma$ (where $\gamma$ is the relativistic Lorentz factor) in the formula for the survival probability and that the survival probabilities should tend to zero as $t\rightarrow \infty$ much slower than one would expect using classical time dilation relation. Here we show also that for some physically admissible models of unstable states the computed decay curves of the moving particles have fluctuating form at relatively short times including times of order of the lifetime., Comment: 24 pages, 7 figures, LaTeX2e, new sections and comments added, Acta Physica Polonica B - in press

Late time properties of moving relativistic particles are studied. Within the proper relativistic treatment of the problem we find decay curves of such particles and we show that late time deviations of the survival probability of these particles from the exponential form of the decay law, that is the transition times region between exponential and non-expo\-nen\-tial form of the survival amplitude, occur much earlier than it follows from the classical standard approach boiled down to replace time $t$ by $t/\gamma_{L}$ (where $\gamma_{L}$ is the relativistic Lorentz factor) in the formula for the survival probability. The consequence is that fluctuations of the corresponding decay curves can appear much earlier and much more unstable particles have a chance to survive up to these times or later. It is also shown that fluctuations of the instantaneous energy of the moving unstable particles has a similar form as the fluctuations in the particle rest frame but they are seen by the observer in his rest system much earlier than one could expect replacing $t$ by $t/\gamma_{L}$ in the corresponding expressions for this energy and that the amplitude of these fluctuations can be even larger than it follows from the standard approach. All these effects seems to be important when interpreting some accelerator experiments with high energy unstable particles and the like (possible connections of these effects with GSI anomaly are analyzed) and some results of astrophysical observations., Comment: 15 pages, 3 figures

Effective Hamiltonians governing the time evolution in a subspace of unstable states can be found using more or less accurate approximations. A convenient tool for deriving them is the evolution equation for a subspace of state space sometime called the Krolikowski-Rzewuski (KR) equation. KR equation results from the Schr\"{o}dinger equation for the total system under considerations. We will discuss properties of approximate effective Hamiltonians derived using KR equation for $n$--particle, two particle and for one particle subspaces. In a general case these affective Hamiltonians depend on time $t$. We show that at times much longer than times at which the exponential decay take place the real part of the exact effective Hamiltonian for the one particle subsystem (that is the instantaneous energy) tends to the minimal energy of the total system when $t \rightarrow \infty$ whereas the imaginary part of this effective Hamiltonian tends to the zero as $t\rightarrow \infty$., Comment: 12 pages, 2 figures

Recent LHC results concerning the mass of the Higgs boson indicate that the vacuum in our Universe may be unstable. We analyze properties of unstable vacuum states from the point of view of the quantum theory of unstable states. From the literature it is known that some of false vacuum states may survive up to times when their survival probability has a non-exponential form. At times much latter than the transition time, when contributions to the survival probability of its exponential and non-exponential parts are comparable, the survival probability as a function of time $t$ has an inverse power-like form. We show that at this time region the instantaneous energy of the false vacuum states tends to the energy of the true vacuum state as $1/t^{2}$ for $t \to \infty$. Properties of the instantaneous energy at transition times are also analyzed for a given model. It is shown that at this time region large and rapid fluctuations of the instantaneous energy take place. This suggests analogous behavior of the cosmological constant at these time regions., Comment: 13 pages, 4 figures, a talk given at 25th Recontres de Blois: Particle Physics and Cosmology, Blois, May 26 - 31, 2013. arXiv admin note: substantial text overlap with arXiv:1304.2796

Properties of unstable false vacuum states are analyzed from the point of view of the quantum theory of unstable states. Some of false vacuum states survive up to times when their survival probability has a non-exponential form. At times much latter than the transition time, when contributions to the survival probability of its exponential and non-exponential parts are comparable, the survival probability as a function of time t has an inverse power-like form. We show that at this time region the instantaneous energy of the false vacuum states tends to the energy of the true vacuum state as $1/t^{2}$ for $t \to \infty$., Comment: 4 pages, 2 figures, based on a talk given at Multiverse and Fundamental Cosmology (Multicosmofun'12), Szczecin, Poland, 10-14 September 2012. arXiv admin note: text overlap with arXiv:1006.2417 by other authors

Not all astrophysical mechanisms of the emission of electromagnetic radiation including $X$-- and $\gamma$-- rays coming from the space are clear. We find that charged unstable particles as well as neutral unstable particles with non--zero magnetic moment which live sufficiently long may emit electromagnetic radiation. This new mechanism is connected with the properties of unstable particles at the post exponential time region. Analyzing the transition time region between exponential and non-exponential form of the survival amplitude it is found that the instantaneous energy of the unstable particle can take very large values, much larger than the energy of this state for times from the exponential time region. Basing on the results obtained for the model considered, it is shown that this purely quantum mechanical effect may be responsible for causing unstable particles to emit electromagnetic--, $X$-- or $\gamma$--rays at some time intervals from the transition time regions., Comment: 15 pages, 5 figures

Comments on the paper "Late Time Behavior of false Vacuum Decay: Possible Implications for Cosmology and Metastable Inflating States" [arXiv:0711.1821] by L. M. Krauss and J. Dent are presented and the possible behavior of the unstable false vacuum at very late times, where deviations from the exponential decay law become to be dominant is analyzed., Comment: 1 page

An effect generated by the nonexponential behavior of the survival amplitude of an unstable state in the long time region is considered. We find that the instantaneous energy of the unstable state for a large class of models of unstable states tends to the minimal energy of the system ${\cal E}_{min}$ as $t\rightarrow\infty$ which is much smaller than the energy of this state for $t$ of the order of the lifetime of the considered state. Analyzing the transition time region between exponential and non-exponential form of the survival amplitude we find that the instantaneous energy of the considered unstable state can take large values, much larger than the energy of this state for $t$ from the exponential time region. Taking into account results obtained for a model considered, it is hypothesized that this purely quantum mechanical effect may be responsible for the properties of broad resonances such as $\sigma$ meson as well as having astrophysical and cosmological consequences., Comment: 19 pages, 9 figures, LateX2e

An effect 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 $t\to\infty$ more slowly than any exponential function of $t$. For a time-dependent decay rate $\gamma(t)$ Khalfin's result means that this $\gamma(t)$ is not a constant for large $t$ but that it tends to zero as $t\to\infty$. We find that a similar conclusion can be drawn for the instantaneous energy of the unstable state for a large class of models of unstable states: This energy tends to the minimal energy of the system ${\cal E}_{min}$ as $t\to\infty$ which is much smaller than the energy of this state for $t$ of the order of the lifetime of the considered state. Analyzing the transition time region between exponential and non-exponential form of the survival amplitude we find that the instantaneous energy of a considered unstable state can take large values, much larger than the energy of this state for $t$ from the exponential time region. Taking into account results obtained for a model considered, it is hypothesized that this purely quantum mechanical effect may be responsible for the properties of broad resonances such as $\sigma$ meson as well as having astrophysical and cosmological consequences., Comment: 21 pages, 9 figures, Talk given at the "14th Lomonosov Conference on Elementary Particle Physics", Moscow, Russia, August 19 -- 25, 2009; 1 Figure added, typos corrected

We consider an effect generated by the nonexponential behavior of the survival amplitude of an unstable state in the long time region: In 1957 Khalfin proved that this amplitude tends to zero as $t$ goes to the infinity more slowly than any exponential function of $t$. This effect can be described in terms of time-dependent decay rate $\gamma(t)$ and then the Khalfin result means that this $\gamma(t)$ is not a constant for long times but that it tends to zero as $t$ goes to the infinity. It appears that a similar conclusion can be drawn for the energy of the unstable state for a large class of models of unstable particles: This energy should be much smaller for suitably long times $t$ than the energy of this state for $t$ of the order of the lifetime of the considered state. Within a given model we show that the energy corrections in the long ($t \to \infty$) and relatively short (lifetime of the state) time regions, are different. It is shown that these corrections decrease to ${\cal E} = {\cal E}_{min} < {\cal E}_{\phi}$ as $t \to \infty$, where ${\cal E}_{\phi}$ is the energy of the system in the state $|\phi>$ measured at times $t \sim \tau_{\phi}= \frac{\hbar}{\gamma}$. This is a purely quantum mechanical effect. It is hypothesized that there is a possibility to detect this effect by analyzing the spectra of distant astrophysical objects. The above property of unstable states may influence the measured values of astrophysical and cosmological parameters., Comment: 13 pages, talk prepared for the Conference "Problems of Practical Cosmology"

An effect generated by the nonexponential behavior of the survival amplitude of an unstable state at the long time region is considered. It is known that this amplitude tends to zero as $t$ goes to the infinity more slowly than any exponential function of $t$. Using methods of asymptotic analysis we find the asymptotic form of this amplitude in the long time region in a general model independent case. We find that the long time behavior of this amplitude affects the form of the energy of unstable states: This energy should be much smaller for suitably long times $t$ than the energy of this state for $t$ of the order of the lifetime of the considered unstable state., Comment: 12 pages, new comments and references added, accepted for publication in The European Physical Journal D

We analyze the proof of the Khalfin Theorem for neutral meson complex. The consequences of this Theorem are discussed: using this Theorem we find, eg., that diagonal matrix elements of the exact effective Hamiltonian for the neutral meson complex can not be equal if CPT symmetry holds and CP symmetry is violated. The Properties of time evolution governed by a time--independent effective Hamiltonian acting in the neutral mesons subspace of states are considered. By means of the Khalfin's Theorem we show that if such Hamiltonian is time--independent then the evolution operator for the total system containing neutral meson complex can not be a unitary operator. Within a given specific model we examine numerically the Khalfin's Theorem. We show for this model in a graphic form how the Khalfin's Theorem works. We also show for this model how the difference of the mentioned diagonal matrix elements of the effective Hamiltonian varies in time., Comment: 26 pages, 3 figures, talk given at "13th Lomonosov Conference on Elementary Particle Physics", Moscow, August 23 -- 29, 2007

The proof of the Khalfin Theorem for neutral meson complex is analyzed. It is shown that the unitarity of the time evolution operator for the total system under considerations assures that the Khalfin's Theorem holds. The consequences of this Theorem for the neutral mesons system are discussed: it is shown, eg., that diagonal matrix elements of the exact effective Hamiltonian for the neutral meson complex can not be equal if CPT symmetry holds and CP symmetry is violated. Properties of time evolution governed by a time--independent effective Hamiltonian acting in the neutral mesons subspace of states are considered. Using the Khalfin's Theorem it is shown that if such Hamiltonian is time--independent then the evolution operator for the total system containing the neutral meson complex can not be a unitary operator. It is shown graphically for a given specific model how the Khalfin's Theorem works. It is also shown for this model how the difference of the mentioned diagonal matrix elements of the effective Hamiltonian varies in time., Comment: LaTeX2e, 24 pages, 3 figures, typos corrected

By analyzing the survival probability amplitude of an unstable state we show that the energy corrections to this state in the long ($t \to \infty$) and relatively short (lifetime of the state) time regions, are different. It is shown that in the considered model the above corrections decrease to zero as $t \to \infty$. It is hypothesized that this property could be detected by analyzing the spectra of distant astrophysical objects. The above property of unstable states may influence the measured values of possible deviations of the fine structure constant $\alpha$ as well as other astrophysical and cosmological parameters., Comment: LaTeX2e, 16 pages, new comments added

The general properties of the effective Hamiltonian for neutral meson system improved by L.A. Khalfin in 1980 are studied. It is shown that contrary to the standard result of the Lee--Oehme--Yang (LOY) theory, the diagonal matrix elements of this effective Hamiltonian can not be equal in a CPT invariant system. It is also shown that the scalar product of short, $|K_{S}>$, and long, $|K_{L}>$, living superpositions of neutral kaons can not be real when CPT symmetry is conserved in the system under considerations whereas within the LOY theory such a scalar product is real., Comment: LaTeX2e, 25 pages, new comment and references added

This paper contains a detailed analysis of the properties of the scalar product of short and long living superpositions of neutral $|K_{0}>$, $|{\bar{K}}_{0}>$ mesons. It is shown for the exact effective Hamiltonian for neutral meson subsystem that the scalar product of its eigenvectors, which correspond with these short and long living superpositions, can not be real with the assumption of CPT conserved and CP violated. The standard conclusion obtained within the Lee-Oehme-Yang theory of neutral kaons is that in this case such a product should be real., Comment: LaTeX2e, 21 pages, new comments nad references added

General properties of eigenvectors and eigenvalues for an effective Hamiltonian governing time evolution in a two state subspace of the state space of the total system under consideration are discussed. The Lee, Oehme and Yang (LOY) theory of time evolution in such a subspace and CP-- and CPT--symmetry properties of the LOY effective Hamiltonian are considered. The CPT transformation properties of the exact effective Hamiltonian for two state subspace are discussed also. Using the Khalfin Theorem we show that contrary to the properties of the LOY theory, the diagonal matrix elements of the exact effective Hamiltonian governing the time evolution in the subspace of states of an unstable particle and its antiparticle need not be equal at for $t > t_{0}$ ($t_{0}$ is the instant of creation of the pair) when the total system under consideration is CPT invariant but CP noninvariant. Using the more accurate approximation than the LOY one we show that the interpretation of the tests measuring the difference between the $K_{0}$ mass and the ${\bar K}_{0}$ mass as the CPT--symmetry test is wrong., Comment: LaTeX2e, 47 pages, invited paper for the book "Progress in Experimental High Energy Physics Research", (Nova Science)

We start from a discussion of the general form and general CP-- and CPT-- transformation properties of the Lee--Oehme--Yang (LOY) effective Hamiltonian for the neutral kaon complex. Next we show that there exists an approximation which is more accurate than the LOY, and which leads to an effective Hamiltonian whose diagonal matrix elements posses CPT transformation properties, which differ from those of the LOY effective Hamiltonian. These properties of the mentioned effective Hamiltonians are compared with the properties of the exact effective Hamiltonian for the neutral kaon complex. We show that the diagonal matrix elements of the exact effective Hamiltonian governing the time evolution in the subspace of states of an unstable particle and its antiparticle need not be equal at for $t > t_{0}$ ($t_{0}$ is the instant of creation of the pair) when the total system under consideration is CPT invariant but CP noninvariant. The unusual consequence of this result is that, contrary to the properties of stable particles, the masses of the unstable particle "1" and its antiparticle "2" need not be equal for $t \gg t_{0}$ in the case of preserved CPT and violated CP symmetries., Comment: LaTex2e, 18 pages, Talk given at the conference on "Symmetry in Nonlinear Mathematical Physics", June 23 -- 29, 2003, Institute of Mathematics, National Academy of Sciences of Ukraine, Kiev, Ukraine; typos corrected

The Bell--Steinberger relation is analyzed. The questionable points of the standard derivation of this relation are discussed. It is shown that the use of a more accurate approximation than the one usually used in the derivation of this relation can lead to corrections to the right hand side of the standard Bell--Steinberger relation., Comment: LaTex2e, 17 pages, some typos corrected and references added

We begin with a discussion of the general form and general CP-- and CPT-- transformation properties of the Lee--Oehme--Yang (LOY) effective Hamiltonian for the neutral kaon complex. Next, the properties of the exact effective Hamiltonian for this complex are discussed. Using the Khalfin Theorem we show that the diagonal matrix elements of the effective Hamiltonian governing the time evolution in the subspace of states of an unstable particle and its antiparticle need not be equal at for t > t_0 (t_0 is the instant of creation of the pair) when the total system under consideration is CPT invariant but CP noninvariant. The unusual consequence of this result is that, contrary to the properties of stable particles, the masses of the unstable particle "1" and its antiparticle "2" need not be equal for t >> t_0 in the case of preserved CPT and violated CP symmetries. We also show that there exists an approximation which is more accurate than the LOY, and which leads to an effective Hamiltonian whose diagonal matrix elements posses properties consistent with the conclusions for the exact effective Hamiltonian described above., Comment: LaTex2e, 17 pages