Charge Conjugation and Internal Symmetry Representations

In summary, for the QFT problem stated in Tom Bank's book, it suffices to show that the representation of the group G for scalar fields is not complex, which can be done by proving that R_S^\dagger=U^\dagger R_SU, where U is a unitary matrix. This can be further understood by looking at the details on pages 50-52 of the book.
  • #1
erccarls
1
0
Hi All,
I am trying to work through a QFT problem for independent study and I can't quite get my head around it. It is 5.16 from Tom Bank's book (http://www.nucleares.unam.mx/~Alberto/apuntes/banks.pdf) which goes as follows:

"Show that charge conjugation symmetry implies that the representation of the internal symmetry group G is real or pseudo-real." (I think we only need to deal with scalar fields here but I don't know that it matters.)

The book linked to above has more details on pages 50-52. I am am pretty confused but I think that I need to show that the representation [itex]R_S[/itex] is not complex if C-symmetry exists. That is to show that [itex]R_s^\dagger=U^\dagger R_s U[/itex]. (i.e. unitary equivalence) is implied by C-symmetry on scalar fields.

Thanks in advance for any help.

-Eric
 
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  • #2
To show that charge conjugation symmetry implies the representation of the internal symmetry group G is real or pseudo-real, you can use the following steps: 1. First, recall that charge conjugation (C) symmetry is a symmetry of the Lagrangian which implies that the transition matrix elements of antiparticles and particles must be the same. 2. This implies that under a C-transformation, a scalar field should transform as follows: \phi(x) \rightarrow \phi^*(-x).3. Then, consider the representation of G on the scalar field. That is, for any element g in G,\phi(x) \rightarrow R_S(g)\phi(x).4. Using the fact that G is an internal symmetry group, if we apply the C-symmetry transformation to the scalar field, it should also obey the group transformation i.e. \phi^*(-x) \rightarrow R_S(g)\phi^*(-x).5. But this can only be true if R_S(g) is real (or pseudo-real).Therefore, we have shown that charge conjugation symmetry implies that the representation of the internal symmetry group G is real or pseudo-real.
 

FAQ: Charge Conjugation and Internal Symmetry Representations

1. What is charge conjugation in particle physics?

Charge conjugation is a mathematical operation that describes the transformation of a particle into its antiparticle. This transformation involves changing the sign of the particle's charge and other quantum numbers, such as baryon number and lepton number.

2. How is charge conjugation related to internal symmetry representations?

In the context of particle physics, internal symmetry refers to the symmetries of a system that are not related to its spatial orientation or external forces. Charge conjugation is a type of internal symmetry that transforms a particle into its antiparticle and vice versa. This symmetry is described by a mathematical representation known as the charge conjugation operator.

3. What are the implications of charge conjugation in particle physics?

Charge conjugation has important implications for the behavior and interactions of particles. For example, in theories such as the Standard Model, particles and antiparticles have opposite charges and therefore experience different forces. Charge conjugation plays a crucial role in understanding the behavior of matter and antimatter in the universe.

4. Can charge conjugation be used to create new particles?

No, charge conjugation is not a physical process that can create new particles. It is a mathematical operation that describes the transformation of a particle into its antiparticle. In order to create new particles, other physical processes such as particle collisions or decays are needed.

5. How is charge conjugation experimentally observed?

Charge conjugation is experimentally observed through the detection of particles and antiparticles. This is done by observing their decay products and comparing them to theoretical predictions based on charge conjugation symmetry. In addition, experiments such as the Large Hadron Collider have been able to directly produce and observe antiparticles, providing further evidence for charge conjugation in particle physics.

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