Why some functional integral(in QTF theo)of a product equal product of two the integr

In summary, the conversation discusses section 11.4 of chapter 11 in the QTF theory book of Peskin & Schroeder. The authors explain how to compute the Effective Action by calculating a functional integral of the product of two exponentials, one being the "exact" Lagrangian and the other being the "counterterm" Lagrangian, using the same variable of integration. The person asking the question is confused about why the integral can be calculated by integrating the two exponentials separately, and wonders why the product of two exponentials is equal to the two integrals of each exponential. They mention equations (11.57), (11.59), and (11.62) in chapter 11 as reference. The person
  • #1
ndung200790
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Please teach me this:
In section 11.4 chapter 11 of QTF theory book of Peskin&Schroeder,computing Effective Action,they calculate a functional integral of product of two exponentials of ''exact'' Lagrangian and ''counterterm'' Lagrangian with the same variable of integral(value of field).I do not understand why they can calculate the integral by integrating separately the two exponentials.Why the integral of the product of two exponentials equals the two integrals of each exponential?(That is (11.57),(11.59),(11.62) chapter 11).
Thank you very much for your kind helping.
 
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  • #2


Now,I am thinking that it is possible,because the freely choosing the value of counterterms.Is it correct?
 
  • #3


Sorry,I had misunderstood the authors,because the product of the two integrals adding connected diagrams is equal the functional integral of product of the two factors.
 

1. Why is the functional integral of a product equal to the product of two integrals in Quantum Field Theory?

The functional integral in Quantum Field Theory is a mathematical tool used to calculate the probability amplitude of a particle's path. In simple terms, it is a sum over all possible paths that a particle can take. When dealing with a product of two quantities, the functional integral can be split into two separate integrals, one for each quantity. This is because the integral is a linear operator, meaning it follows the distributive property, allowing for the product to be split into two separate integrals.

2. How does this relate to the Feynman path integral formulation?

The functional integral in Quantum Field Theory is closely related to the Feynman path integral formulation. In fact, the Feynman path integral can be derived from the functional integral by discretizing the path into smaller time intervals. Therefore, the relationship between the two is that the functional integral is a generalization of the Feynman path integral to continuous time.

3. Does this apply to all types of integrals?

No, the relationship between the functional integral and the product of two integrals only applies to certain types of integrals. More specifically, it applies to Gaussian integrals, which are integrals that can be solved using the Gaussian function. Other types of integrals, such as line integrals or surface integrals, do not follow this relationship.

4. Can this concept be extended to more than two integrals?

Yes, the concept of splitting a functional integral into multiple integrals can be extended to more than two integrals. This is known as the Wick's theorem, which states that any product of an even number of Gaussian integrals can be rewritten as a single Gaussian integral. This allows for more complex calculations to be simplified and solved using the functional integral method.

5. Are there any limitations to using the functional integral in Quantum Field Theory?

While the functional integral is a powerful tool in Quantum Field Theory, it does have some limitations. One limitation is that it only works for systems that can be described by a Lagrangian or Hamiltonian function. Additionally, it may not be applicable to systems with strong interactions or in non-equilibrium states. Therefore, it is important for scientists to understand the limitations of the functional integral and when it is appropriate to use in their calculations.

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