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Quantum Electrodynamics (Book): Difference between revisions
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And the level of mathematics developed is sufficient to continue to apply it to the quantum theory of metals and superfluid helium as in volume 9, condensed matter physics. | And the level of mathematics developed is sufficient to continue to apply it to the quantum theory of metals and superfluid helium as in volume 9, condensed matter physics. | ||
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So what has gone beyond QED? As can be seen in Atiyah's book on gauge fields and Michelsohn-Lawson on Spin geometry, there is more geometric depth to the classical theory of fields. Standard QFT techniques dictate that we start with classical fields (either functions or gauge fields on bundles) and quantize them to produce a space of operators with desired commutation relations that also respect representation-theoretic aspects of the classical fields. At the quantum level, we measure amplitudes which are given by Green's functions/Correlation functions/propagators that relate the probabilities of processes relating individual points in space-time. These are integrated together to give individual operators on the abstract Hilbert space, which is captured in the Wightman formalism in the Fields and Strings book. Since then, multiple types of axiomatic QFT have emerged to pin down the space of QFTs as a mathematical and geometrical entity: | So what has gone beyond QED? The same finite-volume and finite-energy cutoffs made by Landau in the introduction are embedded into the mathematics of renormalization and effective field theory. As can be seen in Atiyah's book on gauge fields and Michelsohn-Lawson on Spin geometry, there is more geometric depth to the classical theory of fields. Standard QFT techniques dictate that we start with classical fields (either functions or gauge fields on bundles) and quantize them to produce a space of operators with desired commutation relations that also respect representation-theoretic aspects of the classical fields. At the quantum level, we measure amplitudes which are given by Green's functions/Correlation functions/propagators that relate the probabilities of processes relating individual points in space-time. These are integrated together to give individual operators on the abstract Hilbert space, which is captured in the Wightman formalism in the Fields and Strings book. Since then, multiple types of axiomatic QFT have emerged to pin down the space of QFTs as a mathematical and geometrical entity: | ||
* Wightman/correlator-based QFTs | * Wightman/correlator-based QFTs | ||
* Haag-Kastler/C*-algebra based QFTs (continued into Connes' approach) | * Haag-Kastler/C*-algebra based QFTs (continued into Connes' approach) | ||