# Derivation of mass-lifetime relation in particle physics

How is the inverse relation between the lifetime and mass of a virtual particle derived?

For example, it is said that the strong force is short range because its force carrier is massive, and hence has a short lifetime, and hence cannot travel far.

Is this relation actually derived anywhere or is there just a tradition?

My own tentative (proposed) conclusions so far.

1. real particles do not have an inverse relation between lifetime and mass. For example, in the standard model the proton is stable, but in SU(5) it has a finite lifetime.

2. virtual particles have a questionable ontology, and only exist within Feynman diagrams calculations. For example, Schwinger theory works with the fields and has no virtual particles.

3. the uncertainty principle would suggest that the lifetime of a particle of a given mass should be at least a given time, rather than at most. The argument from the uncertainty principle is erroneous.

4. the above conclusions suggest that it is merely a cosmic coincidence that the uncertainty relation argument produces any correct results here.

5. The Fermi Golden Rule seems to be the only other viable alternative. This has more of the right feel, being based on the rate at which the coefficients of an expansion in a non stationary eigen basis change over time. However, all it derives is the initial rate of change for a step change in the potential. Hence, the result is only valid as determining a rate of transition if there is a virtual continuum of states. When this condition is satisfied in practical optics, the conclusions do indeed work.

• A rotation is a circle. Stretching it in time creates a sine wave. Real particles are waves. Virtual particles are circles. They don't move in time, but exist for one period of oscillation of their wave function. – safesphere Oct 24 '19 at 8:10
• A pair of virtual particles can be seen as a closed loop in spacetime, but in this context, my question is how is the diameter of the loop determined? – Ponder Stibbons Oct 28 '19 at 1:25
• Loops usually are higher order, not what I mean. I am saying the wavefunction of a single virtual particle is not a function of time. Imagine a rotating wheel that moves along its axis. A point on the wheel has a spiral trajectory. It is a wavefunction of a real particle moving in time. After 1 period of oscillations the function position has advanced in time like a spiral. Virtual particles stay in time. The wheel rotates, but doesn't move along the axis of time. After 1 period of oscillations the function is at the same point in time like a circle. The "diameter" is 1 period of oscillations. – safesphere Oct 28 '19 at 5:26
• Regarding your actual question, I've seen it before on this site. Have you tried searching? – safesphere Oct 28 '19 at 5:30
• I've heard this discussed in relation to the Yukawa potential being the time independent spherically symmetric solution to the Klein Gordon equation. I'm unclear how to flesh this into an answer currently though. – jacob1729 Oct 31 '19 at 23:31