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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 [[Statistical Physics part 2 - quantum theory (Book)| 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 [[Statistical Physics part 2 - quantum theory (Book)| volume 9, condensed matter physics]]. | ||
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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, or equivalently directly compute the expectation values of these operators with path integrals using the classical field Lagrangian. 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. The representation theory arguments in Woit's and Mackey's books are used by Weinberg to pin down particle species and interactions, determining much of the basic structure of the relativistic theory from first principles alone. This is the single biggest practical step in how the theory is viewed by physicists since its inception by Jordan, Born, and Heisenberg. 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, or equivalently directly compute the expectation values of these operators with path integrals using the classical field Lagrangian. Classical gauge theory has been used to further describe dynamical properties of the quantum theory, famously such as Weinberg and Salam's Electroweak theory and Anderson-Higgs' symmetry breaking. At the quantum level, we measure complex 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) | ||
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As Costello puts it, QFT based on Lagrangians of fields (and correlator techniques) is the most fundamental. C*-algebra QFT has been used to describe information theoretic aspects of QFT, even near black holes, but yields few concrete techniques in the way of relevant QFT. TQFTs skirt formulating the analytic content of what a QFT is, focusing instead on their emergent topological properties, but goes even further from real physics. They are however a novel topological invariant, so more resources on TQFT will appear here under further algebraic topology. | As Costello puts it, QFT based on Lagrangians of fields (and correlator techniques) is the most fundamental. C*-algebra QFT has been used to describe information theoretic aspects of QFT, even near black holes, but yields few concrete techniques in the way of relevant QFT. TQFTs skirt formulating the analytic content of what a QFT is, focusing instead on their emergent topological properties, but goes even further from real physics. They are however a novel topological invariant, so more resources on TQFT will appear here under further algebraic topology. | ||
Once some structural understanding of many basic examples of QFTs was achieved, starting with the S-matrix, the "bootstrap" philosophy began where one algebraically specified the relations between observables and their symmetries out of principle. This leads to the perspective of there being a space of QFTs, where CFTs (conformal field theories) are realized as special fixed points of a flow - much like as with phase transitions in statistical mechanics. | Once some structural understanding of many basic examples of QFTs was achieved, starting with the S-matrix, the "bootstrap" philosophy began where one algebraically specified the relations between observables and their symmetries out of principle. This leads to the perspective of there being a space of QFTs, where CFTs (conformal field theories) are realized as special fixed points of a flow - much like as with phase transitions in statistical mechanics. Alternatively, other physicists try to determine the source of the analytic properties of the S-matrix leading them to vast simplifications in the computations of amplitudes by circumventing their expression as space-integrals. | ||