Why Must a Particle Decay into Two or More Photons to Conserve 4-Momentum?

[thenewbosco]

Explain using 4-momenta, how if a particle of mass M decays into photons, it must decay into two or more photons. Does your explanation still hold if the particle is moving at high speeds while it decays?

I can see if the particle is at rest and decays how it would have to decay into two or more to conserve the 3 momentum part of 4 momentum, that is the two photons travel in opposite directions.

If the massive particle is moving, why must it decay into at least two photons? the speed is c, regardless for the photons, and since the massive particle must be traveling less than c, the resultant photons must be such that the various components cancel to leave the original velocity? is this correct? or is there some other angle i have not looked at?

thanks

 

[quasar987]

Don't speculate and hypothesize. Use the equations you know.

To orient you in the right direction, consider the following: "Suppose the particle decays into only one photon. Can momentum and energy be both conserved? ($E^2=(pc)^2+(m_0c^2)^2$)"

Another way to approach the question is simply by "noticing" that a particle moving in one referencial is at rest in another. And you've shown that if the particle is at rest, then it cannot decay in just one photon. So it must decay in 2 no matter the frame of reference.

 

[kyle8921]

I can confirm that the concept of relativistic momentum is an important aspect of understanding particle decay. In order to fully explain why a particle of mass M must decay into two or more photons, we need to use the concept of 4-momenta.

The 4-momentum of a particle is a four-dimensional vector that contains information about the particle's energy and momentum. It is given by the equation p = (E, p), where E is the energy of the particle and p is its three-dimensional momentum.

When a particle decays, it must conserve both energy and momentum. This means that the total 4-momentum before the decay must be equal to the total 4-momentum after the decay. In the case of a particle decaying into photons, the total 4-momentum before the decay is equal to the 4-momentum of the initial particle, while the total 4-momentum after the decay is equal to the sum of the 4-momenta of the individual photons.

Now, if we consider the case of a particle moving at high speeds, its 4-momentum will be different from that of a particle at rest. This is because the energy and momentum of a particle are dependent on its velocity. As the particle's velocity increases, so does its energy and momentum.

Therefore, in order for the total 4-momentum to be conserved after the decay, the individual photons must have a higher energy and momentum compared to the photons produced from a particle at rest. This is why a particle moving at high speeds must decay into two or more photons in order to conserve 4-momentum.

In conclusion, the explanation for why a particle of mass M must decay into two or more photons still holds even if the particle is moving at high speeds. This is because the concept of 4-momenta takes into account the energy and momentum of a particle, which can change with its velocity.