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Algebra of Operators and Observables in the Algebraic Theory of Quantum Fields and Their Generalization in the Presence of Gravity

Sadeghian, Mohammad Amin | 2024

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  1. Type of Document: M.Sc. Thesis
  2. Language: Farsi
  3. Document No: 57612 (04)
  4. University: Sharif University of Technology
  5. Department: Physics
  6. Advisor(s): Faraji Astaneh, Amin
  7. Abstract:
  8. In recent years, a series of deep connections have been found between the ideas that used to be in the Algebraic Theory of Quantum Fields and the important problems of High Energy and Fundamental Physics. In the 1960s and later, some people who were mostly closer to mathematicians were trying to formulate the basics and principles of Quantum Field Theory in mathematically precise way. Many and various efforts have been made in this direction. One of these important efforts has been the Algebraic Theory of Quantum Fields. The main motivation in the Algebraic Theory of Quantum Fields is to focus on the operators and observables of the theory and study the algebraic structure between them instead of focusing on the specific Hilbert space and defining states on it. also instead of focusing on global information, we pay attention to local information and algebras. The motivation of this work is that the physical observer in reality only has access to local observations and measurements, and this issue is naturally institutionalized in the structure of the Algebraic Theory of Quantum Fields. The literature and works that were done in this field, because they were mostly by mathematicians, had a more mathematical aspects and were not related to the questions and concerns of physicists. But recently, due to the work started by Witten a few years ago and now being followed by other people, many connections have been found between ideas that used to exist mostly in the community of mathematicians and issues such as the physics of black holes, information paradox, and quantum gravity. If we want to point out the main ideas and motives in more detail, we can see that in the theory of quantum fields, the algebraic structure of operators or observables forms something called von Neumann type III algebras. This kind of algebra against all natural intuitions that we have from ordinary and finite dimensional quantum mechanics. In this type of algebra, it can be shown that the decomposition of the Hilbert space into tensor products is wrong, and also, in the strict sense, no density matrix or entropy can be defined for this algebra in formal ways. In the past, a number of famous mathematical theorems, including the Tomita-Taksaki theorem, were invented, which give us the tools to study these new situation. Although we cannot define any density matrix, we can somehow calculate entropy and other physical quantities in this case with new tools, such as modular operators. The existence of entropy with universal divergence in the quantum field theory tells us that this divergence should not have anything to do with a particular physical state or Hilbert space, but should be a more general feature of the theory that lies in the algebraic structure of the operators. An interesting thing happens when we add gravity to the theory of quantum fields. In this case, our algebra will find a better and closer behavior to ordinary quantum mechanics from the chaotic behavior that it used to have, and it will change to type II algebras. In this type of algebra, the definition of the density and entropy matrix is done as before, but it still has differences with ordinary quantum mechanics. In summary, in the first 3 chapters, we describe the general and mathematical structure of the algebraic theory of quantum fields. We explain its main theorems and important results. Also, we will rewrite the whole basis of conventional quantum information theory in chapter 3 in the language of algebraic theory. We can see that in this case the concepts and tools are more easily generalized to infinite dimensions. The main work starts from chapter 4 onwards, when we draw our attention to the physical aspects and results of the contents of the first 3 chapters. In chapter 4, we will get acquainted with some basic theorems in the theory of quantum fields and explain why the theory of quantum fields has a chaotic algebraic behavior and it is not possible to decompose space into tensor multiplication. I also introduce and use tools such as Tomita-Taksaki theorem and relative entropy in field theory in this chapter. Finally, we apply these to a specific example, namely Minkowski space and Rindler regions, and we see that it is consistent with the results we expect.. In chapter 5, we see that the tools we have learned so far, help us to define the thermodynamic limit of quantum statistical mechanics precisely, and we point out the problems that exist in its conventional definition. Also, in this chapter, we mention the connection between quantum statistical mechanics and the theory of quantum fields in curved space. We see that the algebra defined on both has a similar structure with the same divergences. The main ideas are expressed in the last chapter, which adds gravity to the story. We see that in the presence of gravity, because the space-time regions fluctuate, we can no longer attribute the algebra of operators to the space-time regions. We propose an algebra of operators along an observer’s worldline and this make more sense both physically and mathematically. also, one of the most important points that we see, at least in the example of De Sitter space, is that the observer must be part of the theory in order to reach well-behaved results. That is, in the presence of gravity, we cannot consider the observer as an external observer (like ordinary quantum mechanics) who measures the system, but we must include the observer himself as a part of the theory, at least in a minimal way. We see that if we do this, the chaotic algebras that we had in the theory of quantum fields, will become well-behaved algebras for which entropy and density matrix can be defined. Finally, we define the algebra available to the observer on the observer’s world-line as the Background-Independent algebra for quantum gravity. We propose the Hartley- Hawking state as the state with the maximum entropy over all spacetimes, which gives us a universal definition of the entropy of a physical observer. The sum of all these results together, gives us a new insight into the physical phenomena that we previously knew in another language and gives us a new proposal to solve the most important problems of theoretical physics, including the information paradox or quantum gravity
  9. Keywords:
  10. Von Neumann Algebra ; Entropy ; Quantum Gravity ; Observables Algebra ; Operators Algebra ; Quantum Field Theory

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