Yukawa Interaction: Constraints and Transformations

[Surrealist]

I have been trying to independently learn quantum field theory, and I've been stumped on a few points...

Suppose I had an interacting Lagrangian with the following form:

L = H(x) PsiBar(x) (a + i b gamma5) Psi(x)

where:

H(x) is a field in which the Hermiticity condition is imposed,
Psi(x) and PsiBar(x) are the field operators,
a and b are constants,
and gamma5 is the axial "gamma 5" matrix

Exactly what constraints would the Hermiticity condition impose?

Also, how would this expression transform under C, P, T, CP and CPT?

Thanks.

Surrealist

 

[eljose79]

Scientist:

Hi there,

First of all, congratulations on taking on the challenge of learning quantum field theory independently! It can be a difficult subject, but with determination and persistence, I'm sure you'll make progress.

To answer your questions, let's start with the Hermiticity condition. In quantum field theory, the Hermiticity condition requires that the Hamiltonian (H) is a Hermitian operator, meaning that it is equal to its own conjugate transpose. This condition ensures that the Hamiltonian is self-adjoint and allows for the conservation of probability in quantum systems.

In your specific case, the Hermiticity condition would impose constraints on the field H(x) to ensure that the Lagrangian is Hermitian. This would involve making sure that the coefficients of the field H(x) are real and that the field itself is self-adjoint.

Moving on to the transformation properties, let's consider each one individually:

- C (charge conjugation): This transformation changes particles into their antiparticles and vice versa. In your Lagrangian, the C transformation would change Psi(x) into PsiBar(x) and vice versa, leaving the rest of the expression unchanged.

- P (parity): This transformation changes left-handed particles into right-handed particles and vice versa. In your Lagrangian, the P transformation would change the gamma5 matrix into -gamma5, and the rest of the expression would remain unchanged.

- T (time reversal): This transformation changes the direction of time in a system. In your Lagrangian, the T transformation would change the sign of all time-dependent terms, and the rest of the expression would remain unchanged.

- CP (combined charge conjugation and parity): This transformation combines the effects of C and P. In your Lagrangian, the CP transformation would change Psi(x) into PsiBar(x) and vice versa, as well as change the gamma5 matrix into -gamma5.

- CPT (combined charge conjugation, parity, and time reversal): This transformation combines the effects of C, P, and T. In your Lagrangian, the CPT transformation would change the sign of all time-dependent terms, as well as change Psi(x) into PsiBar(x) and vice versa, and change the gamma5 matrix into -gamma5.

I hope this helps clarify the constraints and transformation properties for your Lagrangian. Keep up the good work in your independent studies!

 

[denian]

I must say that your question is quite intriguing and complex. The Yukawa interaction, also known as the strong interaction, is a fundamental force in quantum field theory that describes the interactions between particles that make up the atomic nucleus. It is mediated by the exchange of particles called mesons, specifically the pion. This interaction is crucial in understanding the stability and structure of matter.

Now, let's tackle your first question regarding the Hermiticity condition. In this context, the Hermiticity condition refers to the requirement that the Lagrangian is invariant under the Hermitian conjugate transformation. This means that the Lagrangian must remain unchanged when the fields are replaced by their complex conjugates. In your expression, this would imply that the field H(x) must also be Hermitian, meaning it is equal to its own complex conjugate. This constraint ensures that the Lagrangian is self-consistent and maintains the unitarity of the theory.

Moving on to your second question, the expression you have provided is a combination of the Higgs field, fermion fields, and the axial gamma matrix. Therefore, it would transform differently under C, P, T, CP, and CPT transformations. For example, under charge conjugation (C), the fields would transform as Psi(x) -> PsiBar(x) and PsiBar(x) -> Psi(x), while the Higgs field and the gamma matrix would remain unchanged. Similarly, under parity (P) transformation, the fields would transform as Psi(x) -> Psi(-x) and PsiBar(x) -> PsiBar(-x), with the Higgs field and gamma matrix remaining unchanged. The transformations under time reversal (T), charge-parity (CP), and charge-parity-time (CPT) would also involve a combination of these transformations.

I hope this response has provided some insight into your questions. Quantum field theory is a complex and fascinating field of study, and I encourage you to continue your independent learning journey. Keep exploring and questioning, as that is the essence of scientific inquiry. Best of luck!