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Quantum Electrodynamics (Book): Difference between revisions
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In a previous edition of volume 4, the theory of strong and weak nuclear forces was covered, as is mentioned in the preface to the second edition. In hindsight, it wasn't possible to predict the path of these developments which continue today; This shouldn't reflect negatively on this QED volume since the basics methods of the electromagnetic field have not changed, and continuing to experimental applications such as quantum optics will not feel anything lost in this treatment. The present authors and likely Landau himself had the foresight to restrict focus on what could be completely understood, and explain: | In a previous edition of volume 4, the theory of strong and weak nuclear forces was covered, as is mentioned in the preface to the second edition. In hindsight, it wasn't possible to predict the path of these developments which continue today; This shouldn't reflect negatively on this quantum electrodynamics (QED) volume since the basics methods of the electromagnetic field have not changed, and continuing to experimental applications such as quantum optics will not feel anything lost in this treatment. Even at the time, QED was effectively a complete theory and maintains the highest accuracy predictions of any scientific theory in history. The same cannot be said of the excluded quantum gauge theories of the standard model. The present authors and likely Landau himself had the foresight to restrict focus on what could be completely understood, and explain: | ||
* what a photon is, polarization | * what a photon is, polarization | ||
* what a boson and fermion are, induced action by space-time symmetries | * what a boson and fermion are, induced action by space-time symmetries | ||
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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. 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, 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: | ||
* 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) | ||