Bruinsma, Simen
(2022)
Higher linear algebraic quantum field theory.
PhD thesis, University of Nottingham.
Abstract
In this thesis several homotopical aspects of linear algebraic quantum field theory are treated. These homotopical aspects are crucial when formalizing quantum gauge theories in a way that fully respects their gauge symmetries. After preliminaries introducing the relevant elements of category theory, Lorentz geometry, algebraic quantum field theory, chain complexes, model categories and operads, the definition of semi-strict homotopy algebraic quantum field theory is given, weakening only the time-slice axiom.
Building on the work done in [BSW20], an operadic definition of general algebraic field theories is obtained. This allows for adjunctions between different kinds of field theories: a descent adjunction, related to local-to-global features of a field theory; a localization adjunction, related to the existence of dynamics for a field theory; and a canonical quantization adjunction between linear field theories and quantum field theories. The latter adjunction is shown to generalize to homotopy algebraic field theories, and to preserve weak equivalences. This yields a general machinery to produce linear homotopy quantum field theories.
After this, the construction of examples of homotopy algebraic quantum field theories is studied. From the input data of a field complex and an equation of motion, the solution complex is formed as a derived critical locus. This retrieves several features of the BV formalism: ghost fields, antifields and an antibracket, i.e. a canonical shifted Poisson structure on the solution complex. Crucially, it is found that using features of the Lorentzian geometry of spacetime, this shifted Poisson structure can be trivialized in two ways, yielding an unshifted Poisson structure on the solution complex and thus a homotopy algebraic linear field theory.
The canonical quantization functor then produces a linear homotopy algebraic quantum field theory. This is illustrated by two examples: Klein-Gordon theory, which is shown to be equivalent to the usual treatment in algebraic quantum field theory; and linear Yang-Mills theory, which is a first nontrivial example of a linear homotopy algebraic quantum field theory, and is not equivalent to any ordinary algebraic quantum field theory.
Finally, the issue of relative Cauchy evolution for linear homotopy algebraic quantum field theory is treated. Using the localization adjunction an equivalent perspective on relative Cauchy evolution for ordinary algebraic field theories is proposed, which is found to be more suitable for homotopy algebraic field theories since the weakening of the time-slice axiom turns out to severely complicate the usual approach. A rectification theorem is proven for linear observables, and a suitable Poisson structure is found on the strictified model. Combined with the homotopical properties of the linear quantization functor this allows for a well-defined notion of relative Cauchy evolution for linear homotopy algebraic quantum field theories, and it is shown that for the linear observables in such a theory this notion agrees with the naive approach of quasi-inverting the maps involved. The relative Cauchy evolution for the linear Yang-Mills model is then computed, and it is shown that the associated stress-energy tensor agrees with the usual Maxwell stress-energy tensor.
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