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Register a new userBoundaries in relativistic quantum field theory
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Added by
Karl-Henning Rehren
Publication date
2016
fields
Physics
and research's language is
English
Authors
Karl-Henning Rehren
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Boundary conditions in relativistic QFT can be classified by deep results in the theory of braided or modular tensor categories.
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These third-year lecture notes are designed for a 1-semester course in topological quantum field theory (TQFT). Assumed background in mathematics and physics are only standard second-year subjects: multivariable calculus, introduction to quantum mechanics and basic electromagnetism. Keywords: quantum mechanics/field theory, path integral, Hodge decomposition, Chern-Simons and Yang-Mills gauge theories, conformal field theory
We construct examples of Functorial Quantum Field Theories in the Riemannian setting by quantizing free massive bosons.
We study transport properties of discrete quantum dynamical systems on the lattice, in particular Coined Quantum Walks and the Chalker--Coddington model. We prove existence of a non trivial charge transport and that the absolutely continuous spectrum covers the whole unit circle under mild assumptions. For Quantum Walks we exhibit explicit constructions of coins which imply existence of stable directed quantum currents along classical curves. The results are of topological nature and independent of the details of the model.
A consistent, local coordinate formulation of covariant Hamiltonian field theory is presented. Whereas the covariant canonical field equations are equivalent to the Euler-Lagrange field equations, the covariant canonical transformation theory offers more general means for defining mappings that preserve the form of the field equations than the usual Lagrangian description. It is proved that Poisson brackets, Lagrange brackets, and canonical 2-forms exist that are invariant under canonical transformations of the fields. The technique to derive transformation rules for the fields from generating functions is demonstrated by means of various examples. In particular, it is shown that the infinitesimal canonical transformation furnishes the most general form of Noethers theorem. We furthermore specify the generating function of an infinitesimal space-time step that conforms to the field equations.
Applied to field theory, the familiar symplectic technique leads to instantaneous Hamiltonian formalism on an infinite-dimensional phase space. A true Hamiltonian partner of first order Lagrangian theory on fibre bundles $Yto X$ is covariant Hamiltonian formalism in different variants, where momenta correspond to derivatives of fields relative to all coordinates on $X$. We follow polysymplectic (PS) Hamiltonian formalism on a Legendre bundle over $Y$ provided with a polysymplectic $TX$-valued form. If $X=mathbb R$, this is a case of time-dependent non-relativistic mechanics. PS Hamiltonian formalism is equivalent to the Lagrangian one if Lagrangians are hyperregular. A non-regular Lagrangian however leads to constraints and requires a set of associated Hamiltonians. We state comprehensive relations between Lagrangian and PS Hamiltonian theories in a case of semiregular and almost regular Lagrangians. Quadratic Lagrangian and PS Hamiltonian systems, e.g. Yang - Mills gauge theory are studied in detail. Quantum PS Hamiltonian field theory can be developed in the frameworks both of familiar functional integral quantization and quantization of the PS bracket.
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