Derive Feynman Rules for Complex Scalar Field

Your name]In summary, the Feynman rules for a complex scalar field can be derived by considering the Lagrangian L=\partial_\mu\phi^\dagger\partial^\mu\phi +m^2\phi-\lambda/4 |\phi|^4. The propagator for the field is given by D_F(x-y)=-i\int \frac{d^4p}{(2\pi)^4} \frac{e^{-ip\cdot(x-y)}}{p^2-m^2+i\epsilon}, and the interaction term contributes to the Feynman rules as a vertex with a rule of -i\lambda for both particles and antiparticles. Additionally, charge must be conserved
  • #1
mahnamahna
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Homework Statement


Derive the Feynman rules for for a complex scalar field.

Homework Equations


[itex]L=\partial_\mu\phi^\dagger\partial^\mu\phi +m^2\phi-\lambda/4 |\phi|^4[/itex]

The Attempt at a Solution


I wrote the generating functional for the non-interacting theory
[itex]Z_0[J]=Z_0[0]exp(-\int d^4xd^4yJ^\dagger (x) J(y) D_F(x-y)[/itex]

And I think I can use this to calculate the correlation functions directly, I just don't understand exactly how the presence of antiparticles change the Feynman diagrams/rules. I guess charge has to be conserved at all vertices, but I don't explicitly see that condition (I see overall charge conservation). Is this the only change in the Feynman rules? The propagators for both seem the same, and each vertex still gives [itex]-i\lambda\int d^4z[/itex].
294Kl.png

These pictures contribute to different 4 point functions, but do they contribute the same term to their respective sums? Also, does the presence of anti particles change the calculation of symmetry factors?
 
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  • #2

Thank you for your question on deriving the Feynman rules for a complex scalar field. As you correctly pointed out, the presence of antiparticles does not significantly change the Feynman rules. The main difference is that the propagators for the particles and antiparticles have opposite signs in the exponential term of the generating functional.

To derive the Feynman rules, we can start with the Lagrangian you provided: L=\partial_\mu\phi^\dagger\partial^\mu\phi +m^2\phi-\lambda/4 |\phi|^4. From this, we can determine the propagator for the complex scalar field as:

D_F(x-y)=-i\int \frac{d^4p}{(2\pi)^4} \frac{e^{-ip\cdot(x-y)}}{p^2-m^2+i\epsilon}

Next, we can consider the interaction term in the Lagrangian, which is -\lambda/4 |\phi|^4. This term contributes to the Feynman rules as a vertex with 4 external lines, where each line represents a complex scalar field. The Feynman rule for this vertex is -i\lambda, and it is important to note that this rule applies to both particles and antiparticles.

Finally, to address your question about charge conservation, you are correct in noting that charge must be conserved at each vertex. This means that the sum of the charges of the incoming particles must equal the sum of the charges of the outgoing particles. In the case of a complex scalar field, the charge is given by the imaginary part of the field, so this must be conserved at each vertex.

To summarize, the main changes in the Feynman rules for a complex scalar field with the presence of antiparticles are the opposite sign of the propagator in the generating functional and the requirement of charge conservation at each vertex. I hope this helps in your derivation of the Feynman rules. Let me know if you have any further questions.

 

1. What is a complex scalar field in physics?

A complex scalar field is a theoretical concept in physics that describes the behavior of a particle with no spin and a non-zero mass. It is a mathematical representation of a physical system in which the particle's position and energy are described by a complex number.

2. What is the significance of Feynman rules in complex scalar field theory?

Feynman rules are a set of mathematical rules that allow us to calculate the probability of a particle interaction in complex scalar field theory. They provide a systematic way to represent and solve complex equations, making it easier to study and understand the behavior of particles in this field.

3. How do you derive Feynman rules for complex scalar field?

To derive Feynman rules for complex scalar field, we start with the Lagrangian density, which is a mathematical expression that describes the dynamics of a particle in the field. Then, using the path integral formalism, we calculate the transition amplitude for a particle to go from one state to another. Finally, we use this amplitude to derive Feynman rules, which are a graphical representation of the mathematical calculations.

4. Can Feynman rules be used to calculate any physical quantity in complex scalar field theory?

Yes, Feynman rules can be used to calculate any physical quantity in complex scalar field theory, such as scattering amplitudes and cross-sections. They provide a powerful and efficient method for calculating a wide range of physical quantities in this field.

5. What are the limitations of using Feynman rules in complex scalar field theory?

One limitation of using Feynman rules is that they can only be applied to perturbative calculations, which means they are only accurate for small interactions between particles. Additionally, Feynman rules may not be applicable in certain situations, such as when dealing with strong interactions or non-perturbative effects.

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