Abstract
In the tensorial group field theory approach to quantum gravity, the theory is based on discrete building blocks and continuum spacetime is expected to emerge from their collective dynamics, possibly at criticality, via a phase transition. On a compact group of fixed volume this can be expected to be only possible in a large-volume or thermodynamic limit. Here we show how phase transitions are possible in TGFTs in two cases: a) considering the non-local group degrees of freedom on a non-compact Lie group instead of a compact one (or taking a large-volume limit of a compact group); b) in models including ℝ-valued local degrees of freedom (that can be interpreted as discrete scalar fields, often used in this context to provide a matter reference frame). After adapting the Landau-Ginzburg approach to this setting of mixed local/non-local degrees of freedom, we determine the critical dimension beyond which there is a Gaussian fixed point and a continuous phase transition which can be described by mean-field theory. This is an important step towards the realization of a phase transition to continuum spacetime in realistic TGFT models for quantum gravity.
Highlights
To (r − 1)-simplices and the states of the theory correspond to gluings of such (r −1)-simplicial building blocks of geometry
In the tensorial group field theory approach to quantum gravity, the theory is based on discrete building blocks and continuum spacetime is expected to emerge from their collective dynamics, possibly at criticality, via a phase transition
We show how phase transitions are possible in TGFTs in two cases: a) considering the non-local group degrees of freedom on a non-compact Lie group instead of a compact one; b) in models including Rvalued local degrees of freedom
Summary
We set up Landau-Ginzburg mean-field theory in the context of TGFTs including local directions as used to define relational frames. These are hybrid theories with local and non-local degrees of freedom which requires a modified regularization scheme and a careful discussion of the notion of correlation length, before the critical dimension can be deduced via the Ginzburg criterion thereafter
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