Single particle state and symmetry

[andrex904]

Known that $$ [Q^a,p^\mu]=0 $$ where Q is a representation for the a-th generator of the algebra of group symmetry and p the 4-momentum, if we consider a single particle state, eigenstate of p

$$ p^\mu | \psi_A (k) > = k^\mu | \psi_A (k) > $$

then also $$ Q^a | \psi_A (k) > $$

is eigenstate of p. My text says that If we suppose that in our theory there aren't massless particle then the result state of the action of Q on the single particle state is in general linear combination of single particle states, because otherwise if i add another particle i change the invariant mass.
How i can derive this?

 

[eljose79]

First, let's define some terms for clarity. The generator Q is a mathematical object that represents a symmetry in the theory, and is associated with a particular group. The 4-momentum p is a physical quantity that describes the energy and momentum of a particle. The single particle state |ψA(k)> is an eigenstate of the 4-momentum with eigenvalue k.

Now, based on the given information, we can say that the operator Q^a commutes with the 4-momentum operator p^\mu. This means that they share a set of eigenstates, and if we apply Q^a to an eigenstate of p^\mu, we will get another eigenstate of p^\mu. In other words, if |ψA(k)> is an eigenstate of p^\mu, then Q^a|ψA(k)> is also an eigenstate of p^\mu.

Next, let's consider the statement that if there are no massless particles in the theory, then the result state of the action of Q on the single particle state is in general a linear combination of single particle states. This can be understood by considering the concept of invariant mass. Invariant mass is a property of a system that remains constant regardless of the reference frame in which it is measured. In other words, if we add another particle to the system, the invariant mass should not change.

Now, if we have a single particle state |ψA(k)> and we apply Q^a to it, we will get another state |ψB(k')>, where k' is the new 4-momentum after the action of Q^a. If we add another particle to the system, the invariant mass should not change. However, if the result state of Q^a is not a linear combination of single particle states, then the invariant mass of the system will change. This violates the principle of invariant mass and is not consistent with the assumption that there are no massless particles in the theory.

Therefore, in order for the theory to be consistent with the assumption of no massless particles, the result state of the action of Q^a on the single particle state must be a linear combination of single particle states. This allows the invariant mass of the system to remain constant when another particle is added.