Lehmann Kallen and spectral representation

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In summary, the conversation discusses the spectral density function in quantum field theory and its relationship to the physical spectrum of the theory. The spectral function represents the probability for a transition from a one-particle state to a multi-particle state, and is related to the propagator real poles which correspond to particles. It is not the exact spectrum of the theory, but has some relation to it. Chapter 7 of Peskin & Schroeder's "Quantum Field Theory" is suggested as a resource for further understanding.
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I'd like to better understand the lehmann kallen representation of the propagator.
Hello,

My question pertains to the formula below:
1644723952025.png


In particular, I would like to ask about the spectral density function shown below:
1644724002731.png
What does the spectral function physically represent? Is there any interpretation of its meaning, whether it has a relation to the physical spectrum of the theory.

My work:

The propagator real poles correspond to particles (a particle is a lump of energy that doesn't decay or split), it seems to suggest that the spectral function would have delta functions at 1 particle state at rest. Obviously, in a lorentz invariant QFT, one can boost this one particle state to get a continuous spectrum, so \rho(p^2) is not the spectrum of the theory, but it has some relation to it.

Thank you.
 
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FAQ: Lehmann Kallen and spectral representation

What is Lehmann Kallen and spectral representation?

Lehmann Kallen and spectral representation is a mathematical framework used to describe the behavior of complex systems, such as molecules or materials, in terms of their spectral properties. It involves breaking down a system into its individual components and analyzing their interactions using spectral techniques.

How is Lehmann Kallen and spectral representation used in scientific research?

Lehmann Kallen and spectral representation is used in various fields of science, including chemistry, physics, and materials science. It is often used to study the electronic and molecular structure of materials, as well as their optical and magnetic properties. It is also used in quantum mechanics to describe the behavior of particles and their interactions.

What are the advantages of using Lehmann Kallen and spectral representation?

One of the main advantages of Lehmann Kallen and spectral representation is its ability to provide a detailed understanding of the behavior of complex systems. It also allows for the prediction of properties and behaviors of materials, which can be useful in the design of new materials for specific applications. Additionally, it is a versatile tool that can be applied to a wide range of systems and phenomena.

Are there any limitations to Lehmann Kallen and spectral representation?

While Lehmann Kallen and spectral representation is a powerful tool, it does have some limitations. It is most effective for systems that can be described by a finite number of components, and it may not be suitable for highly disordered or chaotic systems. Additionally, the accuracy of the results may be affected by the quality of the data used in the analysis.

How does Lehmann Kallen and spectral representation differ from other mathematical frameworks?

Lehmann Kallen and spectral representation is unique in its focus on spectral properties, which allows for a deeper understanding of the underlying behavior of a system. It also takes into account the interactions between components, rather than just looking at individual components in isolation. This makes it a powerful tool for studying complex systems that cannot be fully described by simpler mathematical models.

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