##CP^N## model in Peskin & Schroeder problem 13.3

In summary, a better way to expand the log is to use the Taylor series expansion, which will give us a more accurate result.
  • #1
Manu_
12
1
Homework Statement
Hello,

In Peskin & Schroeder exercise 13.3 question d, it is asked to perform an expansion of the term
$$iS=−N.tr[log(−D2−λ)]+ig2∫d2xλ$$

where ##Dμ=(∂μ+iAμ)##, and ##λ##,##N## and ##g## numbers. The expansion should be made around ##A_μ=0##, and we should use this result to prove the expansion is proportional to the vacuum polarization of massive scalar fields. In momentum space, the log can be written as

$$∫ddx(2π)dlog(k2+A2−λ)$$
Relevant Equations
$$iS=−N.tr[log(−D2−λ)]+ig2∫d2xλ$$


$$∫ddx(2π)dlog(k2+A2−λ)$$
My naive attempt to expand the log was##log(k2+A2−λ)=log[(k2−λ)(1+A2(k2−λ))]=log(k2−λ)+log(1+A2(k2−λ))≈log(k2−λ)+A2(k2−λ)##but it did not help me so far since the second term vanishes. Can someone point me to the right direction?
 
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  • #2
A better way to expand the log is to use the Taylor series expansion. The Taylor series of log(1+x) is##log(1+x) = x - \frac{x^2}{2} + \frac{x^3}{3} - \frac{x^4}{4} + \frac{x^5}{5} - \dots##Therefore, ##log(k^2 - \lambda) + A^2(k^2 - \lambda) \approx log(k^2 - \lambda) + A^2(k^2 - \lambda) - \frac{A^4(k^2 - \lambda)^2}{2} + \frac{A^6(k^2 - \lambda)^3}{3} - \dots##
 

Related to ##CP^N## model in Peskin & Schroeder problem 13.3

1. What is the ##CP^N## model in Peskin & Schroeder problem 13.3?

The ##CP^N## model is a quantum field theory that describes the interactions of particles in a space with a curved geometry known as a complex projective space. It is a generalization of the well-known quantum chromodynamics (QCD) model and is often used in theoretical physics to study the behavior of particles in non-Euclidean spaces.

2. What is the physical significance of studying the ##CP^N## model?

Studying the ##CP^N## model allows us to gain insight into the behavior of particles in curved spaces, which is essential for understanding the fundamental laws of nature. It also has connections to several other areas of physics, including string theory and cosmology.

3. What are the main features of the ##CP^N## model?

The main features of the ##CP^N## model include chiral symmetry breaking, topological solitons, and confinement. Chiral symmetry breaking refers to the spontaneous breaking of a continuous symmetry in the theory, while topological solitons are non-perturbative solutions to the theory's equations of motion. Confinement is the phenomenon in which particles are confined to certain regions of space due to the strong interactions between them.

4. How is the ##CP^N## model solved in Peskin & Schroeder problem 13.3?

In problem 13.3, the ##CP^N## model is solved using the method of perturbation theory, which involves expanding the solution in a series of increasingly small corrections. This allows us to calculate the behavior of the system to higher and higher orders of accuracy.

5. What are some potential applications of the ##CP^N## model?

The ##CP^N## model has numerous potential applications in physics, including in the study of phase transitions, the behavior of particles in extreme environments such as black holes, and the structure of the early universe. It also has connections to condensed matter physics and may have implications for the development of new materials and technologies.

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