Kohn-Sham vs Gell-Mann & Low correspondence

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In summary, the Kohn-Sham and Gell-Mann & Low theorems both provide methods for mapping an interacting condensed matter system onto a non-interacting system, but in different ways and using different language. The Kohn-Sham approach uses density functional theory and a one-electron density to calculate properties, while the Gell-Mann & Low approach uses the one-electron Green function in second quantized solid-state physics. The main difference is that the Kohn-Sham approach introduces an additional potential to match the densities of the two systems, while the Gell-Mann & Low approach does not. The choice of which approach to use depends on the specific needs and goals of the study.
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thephystudent
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Both the Kohn-Sham theorem and the Gell-Mann & Low theorem provide a way to map an interacting condensed matter system on a non-interacting system, each in their own language. The Kohn-Sham approach does so for density functional theory, calculating the properties of a system from the one-electron density. On the other hand, the Gell-Mann & Low approach is used in second quantized solid-state physics where the one-electron Green function is the important object.

Despite the difference in language, is there some deeper connection between the two approaches? And what determines which one of the two languages is the one that is used?
 
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I think there is a decisive difference between the two approaches: In DFT, there is an extra potential introduced to make the density of the non-interacting system coincide with the density of the interacting system. There is no such potential in the Gellmann-Low approach and the densities of the interacting and non-interacting system are generically different.
 
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1. What is the difference between Kohn-Sham and Gell-Mann & Low correspondence?

Kohn-Sham and Gell-Mann & Low correspondence are two different theoretical frameworks used in the study of many-body quantum systems. Kohn-Sham theory is a density functional theory approach, while Gell-Mann & Low correspondence is a renormalization group approach. Both aim to simplify the complex many-body problem, but they use different methods to do so.

2. Which approach is more commonly used in research?

Kohn-Sham theory is more commonly used in research, particularly in the field of condensed matter physics. This is because it is computationally less demanding and can be applied to a wider range of systems. Gell-Mann & Low correspondence, on the other hand, is more commonly used in particle physics and high-energy physics.

3. Can Kohn-Sham and Gell-Mann & Low correspondence be used together?

Yes, it is possible to use both approaches together. In fact, some researchers have developed hybrid methods that combine elements of both Kohn-Sham and Gell-Mann & Low correspondence. This allows for a more accurate and efficient description of many-body systems.

4. What are the main limitations of Kohn-Sham and Gell-Mann & Low correspondence?

Kohn-Sham theory relies on the assumption that the electronic density is the only relevant quantity for describing the system, which may not always be the case. Gell-Mann & Low correspondence, on the other hand, is limited by the complexity of the renormalization group equations, which can be difficult to solve for large systems.

5. How do Kohn-Sham and Gell-Mann & Low correspondence contribute to our understanding of quantum systems?

Both Kohn-Sham and Gell-Mann & Low correspondence provide valuable insights into the behavior of many-body quantum systems. Kohn-Sham theory has been successful in predicting properties of materials, while Gell-Mann & Low correspondence has helped to explain the behavior of particles in high-energy physics. Together, they contribute to our overall understanding of quantum systems and their complex behavior.

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