Analytic Continuation: Definition & Uses in QFT

Thread starter touqra
Start date Feb 19, 2007
Tags

Analytic continuation

In summary, Analytic continuation in quantum field theory is a mathematical technique used to extend the domain of a function beyond its originally defined region. This allows for the evaluation of the function at values that were previously not accessible, providing a more complete understanding of its behavior. It is primarily used to evaluate complex integrals and sums in calculations, and has many applications such as in the calculation of scattering amplitudes and the study of phase transitions. Some benefits of using analytic continuation include more accurate and precise results, and the ability to connect seemingly unrelated physical systems. However, there are also limitations and challenges associated with its use, such as relying on assumptions and approximations, and requiring a strong mathematical background. Careful consideration must be given to its validity and

Feb 19, 2007

touqra

What is analytic continuation? Seems to be used often in QFT.

Physics news on Phys.org

Feb 19, 2007

HallsofIvy

Science Advisor

Homework Helper

Actually, it more mathematics than physics- If a function is analytic on an open subset of the complex numbers, then there is a unique way to define it outside that set in such a way that it is analytic for all complex numbers.

Related to Analytic Continuation: Definition & Uses in QFT

1. What is analytic continuation in the context of quantum field theory?

Analytic continuation in quantum field theory is a mathematical technique used to extend the domain of a function beyond its originally defined region. This allows for the evaluation of the function at values that were previously not accessible, providing a more complete understanding of its behavior.

2. How is analytic continuation used in quantum field theory?

In quantum field theory, analytic continuation is primarily used to evaluate complex integrals and sums that arise in calculations. It allows for the manipulation of these expressions in a mathematically rigorous way, ultimately leading to more accurate and precise results.

3. What are some applications of analytic continuation in quantum field theory?

One of the main applications of analytic continuation in quantum field theory is in the calculation of scattering amplitudes. By extending the domain of these amplitudes, physicists are able to make predictions about the behavior of particles at higher energies than can be directly measured.

Additionally, analytic continuation is used in the study of phase transitions and critical phenomena in quantum field theory, as well as in the development of renormalization techniques.

4. What are the benefits of using analytic continuation in quantum field theory?

The use of analytic continuation in quantum field theory allows for calculations to be performed in a more mathematically rigorous manner, leading to more accurate and precise results. It also allows for the exploration of physical phenomena that would otherwise be inaccessible.

Furthermore, analytic continuation allows for the connection between seemingly unrelated physical systems, providing a deeper understanding of the underlying mathematical structures at play.

5. Are there any limitations or challenges associated with analytic continuation in quantum field theory?

One limitation of analytic continuation in quantum field theory is that it relies on certain assumptions and approximations, which may not always hold true in real-world scenarios. Additionally, the techniques involved can be quite complex and require a strong mathematical background to fully understand and apply.

Moreover, there may be cases where analytic continuation is not possible or yields incorrect results, requiring the use of alternative methods. As with any mathematical tool, it is important to carefully assess its validity and applicability in a given situation.