Abstract
In a recent proposal we applied methods from constructive QFT to derive a Hamiltonian Renormalization Group in order to employ it ultimately for canonical quantum gravity. The proposal was successfully tested for free scalar fields and thus a natural next step is to test it for free gauge theories. This can be done in the framework of reduced phase space quantization which allows using techniques developed earlier for scalar field theories. In addition, in canonical quantum gravity one works in representations that support holonomy operators which are ill defined in the Fock representation of say Maxwell or Proca theory. Thus, we consider toy models that have both features, i.e. which employ Fock representations in which holonomy operators are well-defined. We adapt the coarse graining maps considered for scalar fields to those theories for free vector bosons. It turns out that the corresponding fixed pointed theories can be found analytically.
Highlights
The construction of interacting four-dimensional Quantum Field Theories (QFTs) is an interesting and fundamentally important problem in modern physics
Considering approaches toward Quantum Gravity, it motivated proposals where the discretization of space(-time) was assumed to be fundamental (Loll, 1998; Giesel and Thiemann, 2007; Bahr and Dittrich, 2009; Bahr and Dittrich, 2009; Dupuis et al, 2012; Loll, 2019). This allowed to make a wide range of predictions by performing computations using established tools from LatticeGauge Theories (LGT), see for example (Kogut and Susskind, 1975; Bahr et al, 2017; Dapor and Liegener, 2018; Glaser and Steinhaus, 2019; Han and Liu, 2020). As it is not yet experimentally supported whether these discrete structures are fundamental, one can independently ask if they can be understood as coarse graining of some underlying continuum QFT and––the construction of such a QFT is in itself an aspirational goal
We present a possible strategy to extend the framework of direct Hamiltonian renormalization developed in (Lang et al, 2018a; Lang et al, 2018b; Lang et al, 2018c; Lang et al, 2018d) to Abelian gauge theories via reduced phase space quantization
Summary
The construction of interacting four-dimensional Quantum Field Theories (QFTs) is an interesting and fundamentally important problem in modern physics. Considering approaches toward Quantum Gravity, it motivated proposals where the discretization of space(-time) was assumed to be fundamental (Loll, 1998; Giesel and Thiemann, 2007; Bahr and Dittrich, 2009; Bahr and Dittrich, 2009; Dupuis et al, 2012; Loll, 2019) This allowed to make a wide range of predictions by performing computations using established tools from LGT, see for example (Kogut and Susskind, 1975; Bahr et al, 2017; Dapor and Liegener, 2018; Glaser and Steinhaus, 2019; Han and Liu, 2020). We test both of them, where we study a gauge-variant version of the toy model from the previous section– not relying on a reduced phase space quantization This model features a Proca like mass term and higher powers of the Laplacian in order that holonomy operators be well-defined in the Fock space defined by that Hamiltonian.
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