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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