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We present a quantum field theoretical derivation of the nondecay probability of an unstable particle with nonzero three-momentum

The study of the decay law is a fundamental part of quantum mechanics (QM). It is now theoretically [

An interesting question addresses the decay of an unstable particle with nonzero momentum

The purpose of this work is straightforward: we derive (

In this Introduction, we recall some basic and striking features connected to (

It is always instructive to investigate the exponential limit, in which the spectral function of the state

As a last point, it should be stressed that in this work we consider unstable states with a definite momentum

The paper is organized in this way: in Section

For completeness, we report here the “standard” derivation of (

We consider a system described by the Hamiltonian

Formally, the Hamiltonian can be written as

Let us now consider an unstable state

For the states of zero momentum, the Hamiltonian

Let us now consider an unstable state with definite momentum

The form of the Hamiltonian

We now turn to the nondecay amplitude of the state

Next, we consider a normalized unstable state

In principle, one could also start from the Hamiltonian

As a last comment of this section, we recall that the general nondecay probability of an arbitrary state

When a boost

Let us consider an unstable particle described by the scalar field

The (full) propagator of the state

We also recall that

The properties outlined above, although in general very important in specific calculations, turn out to be actually secondary to the proof that we present below, where only the formal expression of the propagator of (

As a next step, upon introducing the Mandelstam variable

Let us now consider the rest frame of the decaying particle:

Let us now consider the particle

The nondecay probability’s amplitude for a state

This expression can be manipulated by using (

The decay law of a moving unstable particle is an interesting subject that connects special relativity to QM and QFT. An important aspect is the validity of (

The main contribution of this paper has been the derivation of a quantum field theoretical proof of (

As discussed in the Introduction, there are interesting and peculiar consequences of (

There is no external data. Theoretical result is based on the author’s research.

The author declares that there are no conflicts of interest regarding the publication of this paper.

The author thanks S. Mrówczyński and G. Pagliara for useful discussions.