Symmetry: The Formalism

Sukhwinder Singh

In this thesis we extend the formalism of tensor network algorithms to incorporate global internal symmetries. We describe how to both numerically protect the symmetry and exploit it for computational gain in tensor network simulations. Our formalism is independent of the details of a specific tensor network decomposition since the symmetry constraints are imposed at the level of individual tensors. Moreover, the formalism can be applied to a wide spectrum of physical symmetries described by any discrete or continuous group that is compact and reducible. We describe in detail the implementation of the conservation of total particle number (U(1) symmetry) and of total angular momentum (SU(2) symmetry). Our formalism can also be readily generalized to incorporate more exotic symmetries such as conservation of total charge in anyonic systems.

Patrick Suppes, Mario Zanotti

Luc Lathouwers

For pt.I see ibid., vol.15, no.9, p.2785-79 (1982). The implications of intrinsic symmetries on the structure of rotational secular equations in the angular momentum projection formalism are studied for both the exact and approximate versions. The symmetry group of the rigid rotor, D 2 , is used to illustrate formal results.

Felix M. Haehl, R. Loganayagam, Mukund Rangamani

We review the Schwinger-Keldysh, or in-in, formalism for studying quantum dynamics of systems out-of-equilibrium. The main motivation is to rephrase well known facts in the subject in a mathematically elegant setting, by exhibiting a set of BRST symmetries inherent in the construction. We show how these fundamental symmetries can be made manifest by working in a superspace formalism. We argue that this rephrasing is extremely efficacious in understanding low energy dynamics following the usual renormalization group approach, for the BRST symmetries are robust under integrating out degrees of freedom. In addition we discuss potential generalizations of the formalism that allow us to compute out-of-time-order correlation functions that have been the focus of recent attention in the context of chaos and scrambling. We also outline a set of problems ranging from stochastic dynamics, hydrodynamics, dynamics of entanglement in QFTs, and the physics of black holes and cosmology, where we believe this framework could play a crucial role in demystifying various confusions. Our companion paper [1] describes in greater detail the mathematical framework embodying the topological symmetries we uncover here.