We study the decay law for a moving unstable particle. The usual time-dilatation formula states that the decay width for an unstable state moving with a momentum $p$ and mass $M$ is $\tildeΓ_{p}=ΓM/\sqrt{p^{2}+M^{2}}$ with $Γ$ being the decay width in the rest frame. In agreement with previous studies, we show that in the context of QM as well as QFT this equation is \textit{not} correct provided that the quantum measurement is performed in a reference frame in which the unstable particle has momentum $p$ (note, a momentum eigenstate is \textit{not} a velocity eigenstate in QM). We then give, to our knowledge for the first time, an analytic expression of an improved formula and we show that the deviation from $\tildeΓ_{p}$ has a maximum for $p/M=\sqrt{2/3},$ but is typically \textit{very} small. Then, the result can be easily generalized to a momentum wave packet and also to an arbitrary initial state. Here, we give a very general expression of the non-decay probability. As a next step, we show that care is needed when one makes a boost of an unstable state with zero momentum/velocity: namely, the resulting state has zero overlap with the elements of the basis of unstable states (it is already decayed!). However, when considering a spread in velocity, one finds again that $\tilde{Γ}_{p}$ is typically a very good approximation. The study of a velocity wave-packet represents an interesting subject which constitutes one of the main outcomes of the present manuscript.\textbf{ }In the end, it should be stressed that there is no whatsoever breaking of special relativity, but as usual in QM, one should specify which kind of measurement on which kind of state is performed.We study the decay law for a moving unstable particle. The usual time-dilatation formula states that the decay width for an unstable state moving with a momentum $p$ and mass $M$ is $\tildeΓ_{p}=ΓM/\sqrt{p^{2}+M^{2}}$ with $Γ$ being the decay width in the rest frame. In agreement with previous studies, we show that in the context of QM as well as QFT this equation is \textit{not} correct provided that the quantum measurement is performed in a reference frame in which the unstable particle has momentum $p$ (note, a momentum eigenstate is \textit{not} a velocity eigenstate in QM). We then give, to our knowledge for the first time, an analytic expression of an improved formula and we show that the deviation from $\tildeΓ_{p}$ has a maximum for $p/M=\sqrt{2/3},$ but is typically \textit{very} small. Then, the result can be easily generalized to a momentum wave packet and also to an arbitrary initial state. Here, we give a very general expression of the non-decay probability. As a next step, we show that care is needed when one makes a boost of an unstable state with zero momentum/velocity: namely, the resulting state has zero overlap with the elements of the basis of unstable states (it is already decayed!). However, when considering a spread in velocity, one finds again that $\tilde{Γ}_{p}$ is typically a very good approximation. The study of a velocity wave-packet represents an interesting subject which constitutes one of the main outcomes of the present manuscript.\textbf{ }In the end, it should be stressed that there is no whatsoever breaking of special relativity, but as usual in QM, one should specify which kind of measurement on which kind of state is performed.

arXiv:1512.00232Acta Physica Polonica BJan 1, 201617 citations

MANY SUCCESSIVE MEASUREMENTS AND EXPONENTIAL DECAY

K. Urbanowski

A multiple repeated measurement of the unstable state evolving in time is discussed. We show that there is no decay in the limit of continuous measurements and there is a purely exponential decay law in other cases. We also give formulae for the effective decay width in the cases of very short, intermediate, relatively long and very long time intervals separating successive measurements. Our results are compared with the Weisskopf-Wigner formula for weak decays. Based on our model of a measurement, we show that the decay width does not depend on properties of the real detectors in practice.A multiple repeated measurement of the unstable state evolving in time is discussed. We show that there is no decay in the limit of continuous measurements and there is a purely exponential decay law in other cases. We also give formulae for the effective decay width in the cases of very short, intermediate, relatively long and very long time intervals separating successive measurements. Our results are compared with the Weisskopf-Wigner formula for weak decays. Based on our model of a measurement, we show that the decay width does not depend on properties of the real detectors in practice.

International Journal of Modern Physics AMar 10, 199111 citations

Logarithmic decays of unstable states

Filippo Giraldi

The European Physical Journal DJan 1, 201524 citations

Energy uncertainty of the final state of a decay process

Francesco Giacosa

We derive the expression for the energy uncertainty of the final state of a decay of an unstable quantum state prepared at initial time $t=0$ . This expression is a function of the time $t$ at which a measurement is performed to determine if the state has decayed and, if yes, in which one of the infinitely many possible final states. For large times the energy spread is, as expected, given by the decay width $\ensuremath{\Gamma}$ of the initial unstable state. However, if the measurement of the final state is performed at a time $t$ comparable to (or smaller than) the mean lifetime of the state $1/\ensuremath{\Gamma}$, then the uncertainty in the energy of the final state is much larger than the decay width $\ensuremath{\Gamma}$. Namely, for short times an uncertainty of type $1/t$ dominates, while at long times the usual spread $\ensuremath{\Gamma}$ is recovered. Then we turn to a generic two-body decay process and describe the energy uncertainty of each of the two outgoing particles. We apply these formulas to the two-body decays of neutral and charged pions and to the spontaneous emission process of an excited atom. As the last step, we study a case in which the nonexponential decay is realized and show that for short times eventual asymmetric terms are enhanced in the spectrum.We derive the expression for the energy uncertainty of the final state of a decay of an unstable quantum state prepared at initial time $t=0$. This expression is a function of the time $t$ at which a measurement is performed to determine if the state has decayed and, if yes, in which one of the infinitely many possible final states. For large times the energy spread is, as expected, given by the decay width $\ensuremath{\Gamma}$ of the initial unstable state. However, if the measurement of the final state is performed at a time $t$ comparable to (or smaller than) the mean lifetime of the state $1/\ensuremath{\Gamma}$, then the uncertainty in the energy of the final state is much larger than the decay width $\ensuremath{\Gamma}$. Namely, for short times an uncertainty of type $1/t$ dominates, while at long times the usual spread $\ensuremath{\Gamma}$ is recovered. Then we turn to a generic two-body decay process and describe the energy uncertainty of each of the two outgoing particles. We apply these formulas to the two-body decays of neutral and charged pions and to the spontaneous emission process of an excited atom. As the last step, we study a case in which the nonexponential decay is realized and show that for short times eventual asymmetric terms are enhanced in the spectrum.

arXiv:1305.4467Physical Review ANov 27, 201317 citations

Symmetry breaking in the decay of unstable states

P. T. Arecchi, Antonio Politi

Lettere al nuovo cimento della societa italiana di fisica/Lettere al nuovo cimentoApr 1, 19806 citations

We study the decay law for a moving unstable particle. The usual time-dilatation formula states that the decay width for an unstable state moving with a momentum

p

and mass

M

\tildeΓ_{p}=ΓM/\sqrt{p^{2}+M^{2}}

with

Γ

being the decay width in the rest frame. In agreement with previous studies, we show that in the context of QM as well as QFT this equation is \textit{not} correct provided that the quantum measurement is performed in a reference frame in which the unstable particle has momentum

p

(note, a momentum eigenstate is \textit{not} a velocity eigenstate in QM). We then give, to our knowledge for the first time, an analytic expression of an improved formula and we show that the deviation from

\tildeΓ_{p}

has a maximum for

p/M=\sqrt{2/3},

but is typically \textit{very} small. Then, the result can be easily generalized to a momentum wave packet and also to an arbitrary initial state. Here, we give a very general expression of the non-decay probability. As a next step, we show that care is needed when one makes a boost of an unstable state with zero momentum/velocity: namely, the resulting state has zero overlap with the elements of the basis of unstable states (it is already decayed!). However, when considering a spread in velocity, one finds again that

\tilde{Γ}_{p}

is typically a very good approximation. The study of a velocity wave-packet represents an interesting subject which constitutes one of the main outcomes of the present manuscript.\textbf{ }In the end, it should be stressed that there is no whatsoever breaking of special relativity, but as usual in QM, one should specify which kind of measurement on which kind of state is performed.

We derive the expression for the energy uncertainty of the final state of a decay of an unstable quantum state prepared at initial time

t=0

. This expression is a function of the time

t

at which a measurement is performed to determine if the state has decayed and, if yes, in which one of the infinitely many possible final states. For large times the energy spread is, as expected, given by the decay width $\ensuremath{\Gamma}$ of the initial unstable state. However, if the measurement of the final state is performed at a time

t

comparable to (or smaller than) the mean lifetime of the state $1/\ensuremath{\Gamma}$, then the uncertainty in the energy of the final state is much larger than the decay width $\ensuremath{\Gamma}$. Namely, for short times an uncertainty of type

1/t

dominates, while at long times the usual spread $\ensuremath{\Gamma}$ is recovered. Then we turn to a generic two-body decay process and describe the energy uncertainty of each of the two outgoing particles. We apply these formulas to the two-body decays of neutral and charged pions and to the spontaneous emission process of an excited atom. As the last step, we study a case in which the nonexponential decay is realized and show that for short times eventual asymmetric terms are enhanced in the spectrum.