Fermionic field

Abstract: The interaction between a scalar field and a set of n fermion fields in three space dimensions is investigated by decomposing the total Hamiltonian H into a sum of two terms: H = H/sub qcl/ + H/sub corr/, where H/sub qcl/ denotes the quasiclassical part and H/sub corr/ the quantum correction. General theorems are given for H/sub qcl/ concerning the existence of soliton solutions, the general properties of such solutions, and the condition under which the lowest energy state of H/sub qcl/ is a soliton solution, not the usual plane-wave solution. The effects of the quantum-correction term H/sub corr/ are examined. It is shown that the quasiclassical solution is a good approximation to the quantum solution over a wide range of the coupling constant. The approximation becomes very good when the fermion number N is large. Even for small N (2 or 3) and weak coupling, the quasiclassical solution remains a fairly good approximation. In the strong-coupling region and for arbitrary N, the quasiclassical approximation becomes again very good, at least when the fermions are nonrelativistic. The question whether the relativistic quantum field theory has a strong-coupling limit or not is not resolved.

Abstract: We define and calculate versions of complexity for free fermionic quantum field theories in 1 + 1 and 3 + 1 dimensions, adopting Nielsen's geodesic perspective in the space of circuits. We do this both by discretizing and identifying appropriate classes of Bogoliubov-Valatin transformations, and also directly in the continuum by defining squeezing operators and their generalizations. As a closely related problem, we consider cMERA tensor networks for fermions: viewing them as paths in circuit space, we compute their path lengths. Certain ambiguities that arise in some of these results because of cutoff dependence are discussed.

Abstract: We study circuit complexity for free fermionic field theories and Gaussian states. Our definition of circuit complexity is based on the notion of geodesic distance on the Lie group of special orthogonal transformations equipped with a right-invariant metric. After analyzing the differences and similarities to bosonic circuit complexity, we develop a comprehensive mathematical framework to compute circuit complexity between arbitrary fermionic Gaussian states. We apply this framework to the free Dirac field in four dimensions where we compute the circuit complexity of the Dirac ground state with respect to several classes of spatially unentangled reference states. Moreover, we show that our methods can also be applied to compute the complexity of excited energy eigenstates of the free Dirac field. Finally, we discuss the relation of our results to alternative approaches based on the Fubini-Study metric, the relevance to holography and possible extensions.

Abstract: We extend the recently developed kinematical framework for diffeomorphism-invariant theories of connections for compact gauge groups to the case of a diffeomorphism-invariant quantum field theory which includes, besides connections, also fermions and Higgs fields. This framework is appropriate for coupling matter to quantum gravity. The presence of diffeomorphism invariance forces us to choose a representation which is a rather non-Fock-like one: the elementary excitations of the connection are along open or closed strings, while those of the fermions or Higgs fields are at the end points of the string. Nevertheless we are able to promote the classical reality conditions to quantum adjointness relations which, in turn, uniquely fixes the gauge- and diffeomorphism-invariant probability measure that underlies the Hilbert space. Most of the fermionic part of this work is independent of the recent preprint by Baez and Krasnov and earlier work by Rovelli and Morales-Tecotl because we use new canonical fermionic variables, so-called Grassman-valued half-densities, which enable us to solve the difficult fermionic adjointness relations.