Relation between symmetry, charges and one-particle states

[alphaone]

Hi,
I recently saw an author make the following statement:
If the symmetry leaves the 1-particle states invariant then its associated charge Q acts on the 1-particle states |p> such that Q|p> = 0
This statement is nontrivial to me, so if anybody could show me how it works please let me know.

 

[alphaone]

Figured this out myself already, so no reply needed.

 

[denian]

I am happy to provide some clarification on this statement. The relationship between symmetry, charges, and one-particle states is an important concept in quantum mechanics.

Firstly, let's define what we mean by symmetry. In physics, symmetry refers to the invariance of a system under certain transformations. These transformations can be spatial, temporal, or related to other properties such as charge or spin. Symmetry is a fundamental concept in physics and plays a crucial role in understanding the behavior of physical systems.

Now, let's consider the one-particle states mentioned in the statement. These are quantum states that describe the properties of a single particle, such as its position, momentum, and spin. In quantum mechanics, these states are represented by wavefunctions, and they can be transformed by applying operators to them.

One important type of operator in quantum mechanics is the charge operator, denoted by Q in the statement. This operator represents the charge of a particle and is associated with a conserved quantity in a system. In other words, the total charge of a system remains constant under certain transformations.

The statement suggests that if a symmetry leaves the one-particle states invariant, then the charge operator Q acts on these states in a specific way. Specifically, it acts as a "zero operator," meaning that it does not change the state at all. This is because the symmetry of the system ensures that the one-particle states are already in an eigenstate (a state that is unchanged by the operator) of the charge operator.

To understand this better, let's consider an example. Imagine a system that is symmetric under rotations, meaning that its properties remain unchanged when rotated around a certain axis. In this case, the one-particle states would also be symmetric under rotations. This implies that the charge operator Q, which represents the charge of a particle, would also be invariant under rotations. Therefore, when we apply Q to a one-particle state, it will remain unchanged.

In summary, the statement is saying that if a symmetry leaves the one-particle states invariant, then the associated charge operator will act as a zero operator on these states. This is a consequence of the fact that the one-particle states are already in an eigenstate of the charge operator. I hope this explanation helps clarify the relationship between symmetry, charges, and one-particle states.