Tensor network methods are playing a central role in multiple disciplines of modern quantum physics, such as condensed matter physics, quantum information and quantum gravity. Based on renormalization group (RG) ideas, they provide a large class of variational ansatz for quantum many-body systems. There are various tensor network structures which are suited for different systems based on their entanglement structures, such as matrix product states (MPS), projected entangled pair states (PEPS), multi-scale entanglement renormalization ansatz (MERA). Numerous techniques involving coarse-graining tensor networks have been developed, such as tensor renormalization group (TRG), tensor network renormalization (TNR) and its variants. \n \nFirst, we study the generalization of MERA to continuous systems, or cMERA, which is expected to become a powerful variational ansatz for the ground state of strongly interacting quantum field theories. We investigate, in the simpler context of Gaussian cMERA for free theories, the extent to which the cMERA state
ketPsicM with finite UV cut-off
Lambda can capture the spacetime symmetries of the ground state
ketPsi. For a free boson conformal field theory (CFT) in 1+1 dimensions as a concrete example, we build a quasi-local unitary transformation
V that maps
ketPsi into
ketPsicM and show two main results. (i) Any spacetime symmetry of the ground state
ketPsi is also mapped by
V into a spacetime symmetry of the cMERA
ketPsicM. However, while in the CFT the stress-energy tensor
Tmunu(x) (in terms of which all the spacetime symmetry generators are expressed) is local, the corresponding cMERA stress-energy tensor
TmunucM(x)=VTmunu(x)Vdagger is quasi-local. (ii) From the cMERA, we can extract quasi-local scaling operators
OalphacM(x) characterized by the exact same scaling dimensions
Deltaalpha, conformal spins
salpha, operator product expansion coefficients
Calphabetagamma, and central charge
c as the original CFT. We argue that these results should also apply to interacting theories. \n \nSecond we extend TNR to field theories in the continuum. A short-distance length scale
1/Lambda is introduced in the continuum partition function by smearing the fields. The resulting object is still defined in the continuum but has no fluctuations at distances shorter than
1/Lambda. An infinitesimal coarse-graining step is then generated by the combined action of a \\textit{rescaling} operator
mathfrakL and a \\textit{disentangling} operator
mathfrakK that implements a quasilocal field redefinition. As demonstrated for a free boson in two dimensions, continuous TNR exactly preserves translation and rotation symmetries and can generate a proper RG flow. Moreover, from a critical fixed point of this RG flow one can then extract the conformal data of the underlying conformal field theory. \n \nLeaving behind tensor network methods, we explore the universal structure that emerges from the eigenvectors of the reduced density matrix (RDM) for 1+1 dimensional critical lattice systems. Using the Ising model as an example, we demonstrate that the large-weight eigenvectors are related to the low-lying states of the underlying CFT with suitable boundary conditions by a conformal map, and they transform accordingly under the Virasoro generators. \n \nFinally, using the quantum renormalization group (QRG), we derive the bulk geometry that emerges in the holographic dual of the fermionic U(
N) vector model at a nonzero charge density. The obstruction that prohibits the metallic state from being smoothly deformable to the direct product state under the renormalization group flow gives rise to a horizon at a finite radial coordinate in the bulk. The region outside the horizon is described by the Lifshitz geometry with a higher-spin hair determined by microscopic details of the boundary theory. On the other hand, the interior of the horizon is not described by any Riemannian manifold, as it exhibits an algebraic non-locality. The non-local structure inside the horizon carries the information on the shape of the filled Fermi sea.
Quantum field theory accurately describes the world on the finest scales to which we have empirical access. There has been significant disagreement, however,about which mathematical structures ought to be taken as constitutive of the theory, and thus over which structures should serve as the basis for its interpretation. Perturbative methods allow for successful empirical prediction but require mathematical manipulations that are at odds with the canonical approach to interpreting physical theories that has been passed down from the logical positivists. Axiomatic characterizations of the theory, on the other hand, have not been shown to admit empirically interesting models. This dissertation shows how to understand the empirical success of quantum field theory by reconsidering widely held commitments about how physical meaning accrues to mathematical structure.
A nonperturbative lattice regularization of chiral fermions and bosons with anomaly-free chiral symmetry
G in
1+1D spacetime is proposed. More precisely, we ask ``whether there is a local short-range quantum Hamiltonian with a finite Hilbert space for a finite system realizing on-site symmetry
G defined on a 1D spatial lattice with continuous time, such that its low-energy physics produces a
1+1D anomaly-free chiral matter theory of symmetry
G?'' In particular, we propose that the chiral fermion theory with chiral U(1) ${3}_{L}\text{\ensuremath{-}}{5}_{R}\text{\ensuremath{-}}{4}_{L}\text{\ensuremath{-}}{0}_{R}$ symmetry, with two left-moving fermions of charge 3 and 4, and two right-moving fermions of charge 5 and 0 at IR low energy, can emerge from a 1D UV spatial lattice with a chiral U(1) symmetry, if we include properly designed multi-fermion interactions with intermediate strength (i.e., the dimensionless coupling constant is naturally order 1). In general, we propose that any
1+1D U(1)-anomaly-free chiral matter theory can be defined as a finite system on a 1D lattice with on-site symmetry by using a quantum Hamiltonian with continuous time, but without suffering from Nielsen-Ninomiya theorem's fermion doubling, if we include properly designed interactions between matter fields. We propose how to design such interactions by looking for extra anomaly-free symmetries via bosonization/fermionization. We comment on the new ingredients and the differences of ours compared to Ginsparg-Wilson fermion, Eichten-Preskill and Chen-Giedt-Poppitz (CGP) models, and suggest modifying CGP model to have successful mirror decoupling. Since a lattice on-site internal symmetry can be gauged, we thus can further define a nonperturbative regularization of any anomaly-free U(1) chiral gauge theory in
1+1D. As an additional remark, we show a topological nonperturbative proof of the equivalence relation between the
G-symmetric 't Hooft anomaly cancellation conditions and the
G-symmetry-preserving gapping rules (e.g. Haldane's stability conditions for Luttinger liquid) for multiple U(1) symmetries. We expect that our result holds universally regardless of spatial Hamiltonian or spacetime Lagrangian/path integral formulation of quantum theory. Numerical tests on our proposal are demanding tasks but highly desirable for future work.
Quantum field theory is a powerful language for the description of the subatomic constituents of the physical world and the laws and principles that govern them. This book contains up-to-date in-depth analyses, by a group of eminent physicists and philosophers of science, of our present understanding of its conceptual foundations, of the reasons why this understanding has to be revised so that the theory can go further, and of possible directions in which revisions may be promising and productive. These analyses will be of interest to graduate students and research workers in physics who want to know about the foundational problems of their subject. The book will also be of interest to professional philosophers, historians and sociologists of science, because it contains much material for metaphysical and methodological reflections, for historical and cultural analyses, and for sociological analyses of the way in which various factors contribute to the way the foundations are revised.
The formulation of Quantum Field Theory (QFT) in phase space offers a unique alternative to operator and path-integral paradigms, providing distinct conceptual advantages for semiclassical expansions. In this work, we present a systematic and self-consistent functional framework that maps stationary Schrodinger functional equations directly onto phase-space star-eigenvalue equations across different spin statistics. Operating within a non-manifestly covariant equal-time formalism, we derive explicit vacuum Wigner functionals for scalar, gauge, and fermionic fields, establishing the rigorous theoretical consistency of the formalism from first principles prior to phenomenological applications. We analyze how ordering prescriptions and continuous symmetries manifest under the functional star-product, including an explicit phase-space formulation of Noether’s theorem and field regularization. Finally, the framework is extended to interacting systems via a functional Rayleigh–Schrodinger perturbative scheme, illustrated explicitly through the non-trivial first-order Wigner functional correction W(1) and the vacuum energy correction for a ϕ4 self-interacting theory, establishing a solid foundation for evaluating real-time quantum field dynamics.