# How to demonstrate the asymptotic charateristic of perturbative QTF Theo?

• ndung200790
In summary: Please teach me this:Why the perturbative QTF Theory is an asymptotic theory,because in the asymptotic expansion,the error at N order is a ''infinity small'' of order the (N+1)th term (meaning O(term(N+1)).So I wonder why we know the error at order N in perturbative QTF Theory is of approximation of (N+1)th term in the series.Thank you very much in advance.
ndung200790
Why the perturbative QTF Theory is an asymptotic theory,because in the asymptotic expansion,the error at N order is a ''infinity small'' of order the (N+1)th term (meaning O(term(N+1)).So I wonder why we know the error at order N in perturbative QTF Theory is of approximation of (N+1)th term in the series.
Thank you very much in advanced.

ndung200790 said:
Why the perturbative QTF Theory is an asymptotic theory,because in the asymptotic expansion,the error at N order is a ''infinity small'' of order the (N+1)th term (meaning O(term(N+1)).So I wonder why we know the error at order N in perturbative QTF Theory is of approximation of (N+1)th term in the series.

It is known in special cases. In general, one just hopes. The only guarantee is that
''if the expansion parameter is sufficiently small'' the error is of the first neglected order.
But usually there are no guarantees at all that for the expansion parameter of interest, this will be the case.

ndung200790 said:
Why the perturbative QTF Theory is an asymptotic theory,because in the asymptotic expansion,the error at N order is a ''infinity small'' of order the (N+1)th term (meaning O(term(N+1)).So I wonder why we know the error at order N in perturbative QTF Theory is of approximation of (N+1)th term in the series.
Thank you very much in advanced.

You must do the calculation for the integral describing each type of process you find within a specific quantum field theory. For example, in quantum electrodynamics, you set up all the kinds of Feynman diagram that correspond to a S-matrix perturbated above tree level: this shows you that you have fundamentally 5 types of different processes only. Three of them only are a problem: vertex correction, photon self-energy and electron self-energy: they all give integrals which diverges. What does it mean exactly? Well, when you calculate the taylor expansion, you find out that what is supposedly giving you smalller and smaller terms give you terms that are becoming extremely huge and that explode to infinity. Renormalization is teh process by which we mathematically get rid off that problem. It is very well defined mathematically, and has a physical interpretaion: basically, it is equivalent to the rescaling of wavefunction renormalization appearing in the propagators involved (fermionic/electromagntic), as well as a rescaling of the mass of partices involved.

I mean the ''asymptotic'' (or the divergent series) after renormalization for S-matrix.

## 1. What is the asymptotic characteristic of perturbative QTF theory?

The asymptotic characteristic of perturbative QTF (Quantum Theory of Fields) is a mathematical property that describes the behavior of a system as its energy or momentum approaches infinity. In perturbative QTF, this characteristic is used to study the behavior of quantum fields in the high-energy limit.

## 2. How is the asymptotic characteristic of perturbative QTF theory demonstrated?

The asymptotic characteristic of perturbative QTF theory is demonstrated through the use of mathematical techniques such as Feynman diagrams, which allow for the calculation of scattering amplitudes in the high-energy limit. These amplitudes can then be compared to experimental data to verify the predictions of the theory.

## 3. Why is it important to demonstrate the asymptotic characteristic of perturbative QTF theory?

Demonstrating the asymptotic characteristic of perturbative QTF theory is important because it allows us to understand the behavior of quantum fields at extremely high energies, which are not accessible through experimental means. This can provide insight into fundamental aspects of particle physics and help us refine our understanding of the underlying laws of nature.

## 4. What are some challenges in demonstrating the asymptotic characteristic of perturbative QTF theory?

One of the main challenges in demonstrating the asymptotic characteristic of perturbative QTF theory is the complexity of the mathematical calculations involved. These calculations can become increasingly difficult as the number of particles and interactions in a system increases. Additionally, experimental data may not always be precise enough to accurately verify the predictions of the theory.

## 5. How does the asymptotic characteristic of perturbative QTF theory relate to other theories in physics?

The asymptotic characteristic of perturbative QTF theory is closely related to other important concepts in theoretical physics, such as renormalization and the renormalization group. These ideas are used to study the behavior of physical systems at different energy scales and can help us understand the connections between different theories, such as quantum mechanics and general relativity.

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