Propagator of a Scalar Field via Path Integrals

In summary, the conversation discusses the derivation of the propagator of a scalar field as presented in page 291 of Peskin and Schroeder. The main confusion was with the functional derivative with respect to ##J(x_2)## and how it relates to the term within the brackets in the final result. With the use of the product rule and the fact that ##\frac{\delta J(y)}{\delta J(x)} = \delta^4(x-y)##, it becomes clear how the result is obtained. The ##\frac{Z[J]}{Z_0}## term is also clarified as simply being ##\text{exp}[-\frac{1}{2}\int d^4 x \; d
  • #1
Wledig
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I don't understand a step in the derivation of the propagator of a scalar field as presented in page 291 of Peskin and Schroeder. How do we go from:
$$-\frac{\delta}{\delta J(x_1)} \frac{\delta}{\delta J(x_2)} \text{exp}[-\frac{1}{2} \int d^4 x \; d^4 y \; J(x) D_F (x-y) J(y)]|_{J=0}$$
To:
$$-\frac{\delta}{\delta J(x_1)} [ -\frac{1}{2} \int d^4 y \; D_F (x_2-y)J(y) - \frac{1}{2} \int d^4 x \; J(x) D_F (x-x_2)]\frac{Z[J]}{Z_0} |_{J=0} \; \; \;\text{?}$$
Why is the functional derivative with respect to ##J(x_2)## equal to the term within the brackets above?
 
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  • #2
Well, do you know how to evaluate the following derivatives?
$$\frac{\delta (F[J]G[J])}{\delta J(x)}$$
$$\frac{\delta J(y)}{\delta J(x)}$$
Where ##F##, ##G## are functionals and ##J## is a function. Knowing this is almost immediat to obtain Peskin's result.
 
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  • #3
I see now. I just had to use the product rule and the fact that:
$$\frac{\delta J(y)}{\delta J(x)} = \delta^4(x-y)$$
What really confused me was the ##\frac{Z[J]}{Z_0}## term appearing out of nowhere. But on second look that's just:
$$\frac{Z[J]}{Z_0} = \text{exp}[-\frac{1}{2}\int d^4 x \; d^4 y \; J(x)D_F(x-y) J(y)]$$
Thanks for clearing things out, sorry for the silly question.
 
  • #4
Exact, very good
 

1. What is the propagator of a scalar field via path integrals?

The propagator of a scalar field via path integrals is a mathematical tool used in quantum field theory to calculate the probability amplitude for a particle to travel from one point to another in a given time. It takes into account all possible paths that the particle could take, and the result is a complex-valued function that describes the dynamics of the field.

2. How is the propagator of a scalar field calculated?

The propagator of a scalar field is calculated using a path integral, which is a mathematical technique that sums up all possible paths that a particle can take between two points. The path integral is then evaluated using a Lagrangian, which is a mathematical function that describes the dynamics of the field. The resulting propagator is a complex-valued function that depends on the initial and final positions of the particle, as well as the time interval.

3. What is the significance of the propagator of a scalar field?

The propagator of a scalar field is significant because it provides a way to calculate the probability amplitude for a particle to travel between two points in a given time. This is important in quantum field theory, as it allows us to make predictions about the behavior of particles and their interactions. It also helps us to understand the dynamics of the field and how it evolves over time.

4. What are the applications of the propagator of a scalar field?

The propagator of a scalar field has many applications in theoretical physics, particularly in quantum field theory and particle physics. It is used to calculate scattering amplitudes, decay rates, and other properties of particles. It is also used in the study of quantum field theories in curved spacetime, such as in cosmology and black hole physics.

5. Are there any limitations to using the propagator of a scalar field?

One limitation of the propagator of a scalar field is that it is only applicable to free fields, where the particles do not interact with each other. In cases where interactions between particles are present, more advanced techniques are needed to calculate the dynamics of the field. Additionally, the path integral approach can be quite mathematically complex, making it difficult to apply in certain situations. However, it remains a powerful tool in theoretical physics and has been successfully used to make predictions and explain experimental results.

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