# Relation between symmetry, charges and one-particle states

• alphaone
In summary, symmetry in relation to one-particle states refers to the idea that the system remains unchanged under certain transformations and the properties of the system are independent of its orientation or position. Charges play a crucial role in determining the type of symmetries allowed in a system. Symmetry can be used as a tool to predict the properties of one-particle states. The relation between symmetry and one-particle states is studied through theoretical models and experimental techniques. Symmetries in one-particle states can be broken under certain conditions, such as external influences like electric or magnetic fields.

#### 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.

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.

## 1. What is the meaning of symmetry in relation to one-particle states?

Symmetry refers to the idea that a physical system remains unchanged under certain transformations, such as rotations or reflections. In the context of one-particle states, symmetry refers to the fact that the properties of the system should be the same regardless of its orientation or position in space.

## 2. How do charges affect the symmetry of one-particle states?

Charges play a crucial role in determining the symmetry of one-particle states. The type and magnitude of charges present in a system can dictate the type of symmetries allowed and affect the behavior of the system under certain transformations.

## 3. Can symmetry be used to predict the properties of one-particle states?

Yes, symmetry can be a powerful tool in predicting the properties of one-particle states. By identifying the symmetries present in a system, scientists can make predictions about the behavior and properties of the system, such as its energy levels and allowed transitions.

## 4. How is the relation between symmetry and one-particle states studied?

The relation between symmetry and one-particle states is studied through various theoretical and experimental methods. Theoretical models, such as group theory, are used to identify the symmetries present in a system. Experimental techniques, such as spectroscopy, can also provide information about the symmetries of one-particle states.

## 5. Can symmetries in one-particle states be broken?

Yes, symmetries in one-particle states can be broken under certain conditions. This can occur when the system is subjected to external influences, such as an electric or magnetic field, which can disrupt the symmetries and change the properties of the system.