In conventional quantum field theory (QFT) the local observables most frequently considered are energy density, charge density, and correlation functions. These quantities encode the dynamical and kinematic properties of fields but do not directly capture the structural intricacy of the quantum state at each point in space–time. In this work we propose a new local observable, the Local Complexity Density of Quantum Fields (LCDQF), denoted ρ_C(x), which quantifies the minimal computational resources required to prepare the quantum state in an infinitesimal neighbourhood of the space–time point x from a fixed reference state. The definition of ρ_C(x) is grounded in the circuit complexity framework of quantum information theory and is constructed to be a local scalar density, invariant under spatial diffeomorphisms and Lorentz transformations. We derive a set of general properties for ρ_C(x), including positivity, monotonicity under coarse‑graining, and covariance under field redefinitions. In addition, we establish a formal relationship between ρ_C(x) and the energy density ρ_E(x) by showing that, for a wide class of free and weakly interacting fields, the two densities are linked through a differential operator that captures the local scaling of the field modes. Explicit calculations for the free scalar, the φ^4 theory, and Maxwell electrodynamics illustrate how ρ_C(x) behaves in different regimes and highlight its sensitivity to ultraviolet cutoffs, renormalisation group flows, and topological excitations. We discuss the implications of LCDQF for the AdS/CFT correspondence, arguing that the local complexity density may serve as a bridge between bulk geometric quantities such as the volume of extremal slices and boundary field-theoretic complexity. Finally, we outline prospective avenues for experimental probing of ρ_C(x) via quantum simulators that can implement local state preparation protocols, suggesting that LCDQF may become a measurable characteristic in near‑term quantum devices. This new local observable opens a novel window into the internal structure of quantum fields and promises to enrich both theoretical and experimental studies of quantum many‑body systems.
In conventional quantum field theory (QFT) the local observables most frequently considered are energy density, charge density, and correlation functions. These quantities encode the dynamical and kinematic properties of fields but do not directly capture the structural intricacy of the quantum state at each point in space–time. In this work we propose a new local observable, the Local Complexity Density of Quantum Fields (LCDQF), denoted ρ_C(x), which quantifies the minimal computational resources required to prepare the quantum state in an infinitesimal neighbourhood of the space–time point x from a fixed reference state. The definition of ρ_C(x) is grounded in the circuit complexity framework of quantum information theory and is constructed to be a local scalar density, invariant under spatial diffeomorphisms and Lorentz transformations. We derive a set of general properties for ρ_C(x), including positivity, monotonicity under coarse‑graining, and covariance under field redefinitions. In addition, we establish a formal relationship between ρ_C(x) and the energy density ρ_E(x) by showing that, for a wide class of free and weakly interacting fields, the two densities are linked through a differential operator that captures the local scaling of the field modes. Explicit calculations for the free scalar, the φ^4 theory, and Maxwell electrodynamics illustrate how ρ_C(x) behaves in different regimes and highlight its sensitivity to ultraviolet cutoffs, renormalisation group flows, and topological excitations. We discuss the implications of LCDQF for the AdS/CFT correspondence, arguing that the local complexity density may serve as a bridge between bulk geometric quantities such as the volume of extremal slices and boundary field-theoretic complexity. Finally, we outline prospective avenues for experimental probing of ρ_C(x) via quantum simulators that can implement local state preparation protocols, suggesting that LCDQF may become a measurable characteristic in near‑term quantum devices. This new local observable opens a novel window into the internal structure of quantum fields and promises to enrich both theoretical and experimental studies of quantum many‑body systems.
This thesis aims to establish the study of a somewhat novel form of entanglement in field-theoretical quantum systems: that between degrees of freedom in momentum space. Although scattered results can be found in the literature, with some even going back decades, this work, based on the papers published during the PhD, provides a systematic investigation of the matter and arrive at non-perturbative results, shedding light on unexplored structures present in Quantum Field Theories (QFTs) and their ground states. This line of research is motivated by the inherent connection between the notion of momentum modes of fields and the Wilsonian formulation of the Renormalization Group (RG), a cornerstone of the modern understanding of field theories and phases of matter which is naturally defined by the partition of the degrees of freedom into “fast' and “slow' modes, according to their momenta. Furthermore, by glancing at the achievements of techniques such as the Density Matrix Renormalization Group and Tensor Networks, which are based on ideas from renormalization in real space, there is reason to believe that similarly efficient methods may be derived by connecting the fundaments of the RG with quantum information in momentum-space. With this in mind, the thesis begins by introducing the main notions of information theory and renormalization used throughout, with the inclusion of a brief discussion on the use of von Neumann algebras in quantum physics, a formalism which is not directly applied here, but nevertheless provides a solid basis for many intuitions on the nature of entanglement. Afterwards, the idea of degrees of freedom in momentum-space is defined and its different subtleties clarified. A key insight, that any field theory possesses multiple definitions of partitioning its Hilbert space into momentum modes, based on a choice of a complete set of local and commuting observables or field variables in a path integral, is then obtained. Given this solid grasp on the fundaments of quantum information in momentum-space, two directions of further investigation are pursued. First, a path-integral method is developed for the calculation of the associated entanglement entropies in ground states and applied perturbatively to several models, which also allows for the quantitative confirmation of certain predictions. Secondly, the Wilsonian renormalization procedure is translated to this formalism and is shown to be equivalent to a so-called quantum channel which acts on momentum-space. While an interesting result on its own right, this also leads directly to a powerful constraint on the structure of ground states of renormalization group fixed points: they must be separable, and thus uncorrelated, when the degrees of freedom are partitioned based on their momentum scale. This opens the way for further analytical and numerical studies of these special QFTs beyond the scope of this thesis. Finally, the concept of fidelity of quantum states is invoked in order to better understand the information aspects of the renormalization group flow for theories in the critical manifold of a given fixed point. It is used to quantify the idea that such QFTs are indistinguishable from the fixed point itself when long-distance observables are measured and new “critical exponents' are introduced in this context, providing tight bounds on the differences of expectation values of operators in the critical and fixed-point theories. The results obtained in this work mark only the beginning of the in-depth research on different structures present in the momentum-space of Quantum Field Theories and may, hopefully, serve as a basis for further developments.:Contents Acknowledgements iii 1 Introduction 1 1.1 Notions of quantum information 5 1.2 Comments on von Neumann algebras 10 1.3 Basics of Wilsonian renormalization 14 2 Defining momentum-space entanglement 18 2.1 Momentum modes from the path integral 19 2.2 Non-uniqueness of partitions 24 3 The replica method for non-local actions 33 3.1 Construction of the replica trick 34 3.2 Entanglement between momentum scales in perturbation theory 38 3.2.1 Perturbation theory and the replica trick 39 3.2.2 Coupled Oscillators 40 3.2.3 Perturbative interacting QFT 42 3.2.4 Sine-Gordon and Thirring models 48 4 Wilsonian RG as a quantum channel 53 4.1 Quantum operations from the RG steps 54 4.2 Separability of ground states at fixed points 59 4.3 Further implications 63 4.3.1 The RG is a quantum dynamical semigroup 64 4.3.2 First-order phase transitions 67 5 Critical theories and operator estimations 74 5.1 Quantum state distinguishability 75 5.2 Fidelity in translation-invariant systems 77 5.3 Hyperscaling in the critical manifold 80 6 Conclusions and further directions 86 Bibliography 88
In the successful framework of the Renormalization Group (RG) approach to Quantum Field Theories (QFT), establishing general constraints on the RG flow has become a central problem. This is formally addressed by the monotonicity theorems, such as Zamolodchikov’s c-theorem in d=2, the a-theorem in d=4, and the F-theorem in d=3. Essentially, these theorems state that if a RG trajectory leads from a UV fixed point to a IR one, each described by a Conformal Field Theory (CFT), then a suitable positive quantity decreases. In both even and odd dimensions, this quantity can be extracted, at fixed points, from the free energy of the CFT on a sphere, leading to the conjecture of a generalized F-theorem valid in continuous dimension d. In this thesis we investigate this framework in a multicritical multiscalar theory with a generic sextic interaction for N massless scalar fields below its upper critical dimension d=3. This model captures, among many others, the critical features of the Tricritical Ising universality class. A first step is to establish whether the RG flow admits a gradient structure, i.e. that there exists a monotonic function A whose gradient yields the beta function up to a positive-definite metric in the space of couplings. This is one of the strongest realizations of monotonicity, as it guarantees the irreversibility of the RG flow and gives an intrinsic geometric structure to the theory space. Given the known beta function up to next-to-leading order, we solve the gradient flow equation perturbatively, finding a parametric family of solutions for the monotonic A-function together with the metric at and below the upper critical dimension, so that the RG flow admits a gradient structure at this order. As this procedure leaves some parameters undetermined, to turn it into a physical statement these functions must be matched to suitable observables of the CFTs describing the fixed points. We therefore compute the CFT candidates suggested by the generalized F-theorem, namely the sphere free energy of the CFT and the Zamolodchikov metric. Specializing to the O(N)-symmetric fixed point, we consistently match the RG and CFT results, fixing all the free parameters in a unique way. This matching is the central result of this work: it establishes that the RG flow admits a gradient structure perturbatively, with F as the monotonic function and Zamolodchikov's metric as the metric on the theory space, recovered from the geometric construction for a specific and unique choice of the free parameters. Importantly, this result provides a nontrivial, perturbative confirmation of the F-theorem in its strongest form, at and below the upper critical dimension, for renormalizable three-dimensional multiscalar theories.
We demonstrate a precise relation between the rate of complexity of quantum states excited by local operators in two-dimensional conformal field theories and the radial momentum of particles in three-dimensional anti–de Sitter spacetimes. Similar relations have been anticipated based on qualitative models for operator growth. Here, we make this correspondence sharp with two key ingredients: the precise definition of quantum complexity given by the spread complexity of states, and the match of its growth rate to the bulk momentum measured in the proper radial distance coordinate.
This research investigates computational complexity problems through the lens of quantum field theory (QFT). Traditional approaches to complexity theory often rely on algorithmic and logical arguments, while QFT provides a powerful framework for analyzing systems with high dimensionality and entanglement. This paper explores how QFT concepts – such as renormalization, path integrals, and the representation of states as quantum fields – can be applied to understand and potentially mitigate the challenges inherent in certain computational problems. Specifically, we examine how the inherent probabilistic nature of QFT, combined with its ability to handle complex, high-dimensional spaces, might offer new perspectives on problems like NP-completeness and the limits of computation. We propose a framework for translating complexity problems into QFT-inspired models and discuss potential avenues for leveraging these models to explore the landscape of computational complexity. The core contribution of this work lies in establishing a novel methodology for tackling complexity questions by drawing upon the established tools and techniques of QFT. We investigate the potential for utilizing QFT's inherent parallelism and its ability to represent exponentially large state spaces to overcome limitations faced by classical algorithms.
A Modular Time-Dynamics Framework Combining Relativistic Physics with the Proposed VFE1 and QCC0 Formalisms Abstract We propose a modular computational framework for investigating time evolution across relativistic, quantum, and geometric regimes. The framework separates established physical models from two proposed components: Vibrational Field Equation One (VFE1), intended to describe an additional field-dependent contribution to temporal evolution, and Quantum Computerization Consciousness Zero (QCC0), introduced as a quantitative measure of quantum-state correlation and computational coherence within the model. The framework does not replace established relativity or quantum mechanics. Instead, special relativity, general relativity, and conventional quantum-mechanical descriptions provide the reference layer against which the proposed VFE1 and QCC0 formulations can be evaluated. The Kapnack Solver provides the numerical engine, while the 369 vortex formulation is treated as a proposed geometric/kinematic transformation layer within the broader SDKP architecture. The central objective is falsifiable comparison: for a specified physical system, the framework calculates the prediction obtained from established physics, calculates the corresponding prediction obtained after inclusion of the proposed VFE1/QCC0 terms, and compares both predictions with independently measured data. Any improvement must be demonstrated quantitatively and must survive independent reproduction. 1. Introduction Time is a fundamental parameter in modern physics, but its operational behavior depends upon the physical state and reference frame of the system under consideration. In special relativity, proper time depends upon relative motion. For an inertial observer, the differential relation between proper time and coordinate time is [d\tau = dt\sqrt{1-\frac{v^2}{c^2}},] where (v) is relative velocity and (c) is the invariant speed of light. In general relativity, spacetime geometry is determined by the gravitational field, and the proper time measured by a clock depends upon its worldline through spacetime. In the weak-field limit, gravitational time dilation can be approximated by the corresponding metric or gravitational-potential correction. Quantum mechanics introduces a separate description of physical evolution in which states evolve according to a dynamical equation such as [ i\hbar\frac{\partial}{\partial t}\left|\psi(t)\right\rangle \hat{H}\left|\psi(t)\right\rangle .] These established theories provide the reference framework for the present proposal. The purpose of the present work is to investigate whether additional geometric, density, kinetic, positional, or field-state variables can provide useful predictive information beyond the standard reference calculations. Within the proposed SDKP architecture, these additional variables are organized through the Scale-Density-Kinetic/kinematic framework and associated computational methods. Two components are particularly relevant to temporal modeling: Vibrational Field Equation One (VFE1) — a proposed field-based temporal correction or extension. Quantum Computerization Consciousness Zero (QCC0) — a proposed quantitative descriptor of quantum correlation/coherence within the computational representation. Both are treated here as hypotheses requiring empirical validation rather than as established physical laws. 2. Established Physical Baseline 2.1 Special Relativity The first reference model is special-relativistic proper time. For motion at constant velocity, [d\tau =dt\sqrt{1-\frac{v^2}{c^2}}.] Integration gives [\Delta\tau =\Delta t\sqrt{1-\frac{v^2}{c^2}}] for constant (v). This relationship provides an experimentally established baseline for evaluating any proposed modification to velocity-dependent temporal evolution. The framework therefore requires the relativistic calculation to remain available independently of VFE1 and QCC0. 2.2 General Relativity For gravitational systems, the reference calculation is obtained from the spacetime metric [ds^2 = g_{\mu\nu}dx^\mu dx^\nu .] For a timelike trajectory, [d\tau^2 =-\frac{ds^2}{c^2}] under the appropriate metric-signature convention. Thus, gravitational time dilation is not introduced as an independent assumption of the proposed framework. It is calculated from the established spacetime geometry. In weak gravitational fields, the resulting prediction may be expressed approximately in terms of gravitational potential and velocity. These established predictions constitute the baseline against which the proposed VFE1 contribution is tested. 3. Proposed SDKP Temporal Representation The proposed SDKP architecture introduces a state representation containing geometric scale, density, kinetic state, and position: [\Psi =(S,\rho,K,P).] Here, (S) represents structural or geometric scale, (\rho) represents a density or density distribution, (K) represents kinetic state, (P) represents spatial position. The important distinction is that these quantities are treated as model variables rather than being assumed automatically to constitute new physical laws. The computational objective is therefore to determine whether an appropriate transformation [\Psi\longrightarrowT] can reproduce known temporal behavior and, more importantly, whether it produces independently testable predictions that improve upon established models. 4. Vibrational Field Equation One (VFE1) VFE1 is proposed as a field-dependent extension of temporal dynamics. The general form may be represented as [ \mathcal{T}_{\mathrm{VFE1}} \mathcal{F}\left(S,\rho,K,P,\Phi_v\right),] where (\Phi_v) represents the proposed vibrational-field state. The precise functional form of (\mathcal{F}) must be specified for each physical application and must not be selected solely to reproduce an existing observation. A useful formulation is therefore to decompose the prediction into a standard component and an additional VFE1 contribution: [ T_{\mathrm{model}} T_{\mathrm{standard}}+\Delta T_{\mathrm{VFE1}}.] The term [\Delta T_{\mathrm{VFE1}}] is the quantity that must be independently constrained. This construction has an important scientific advantage: the VFE1 hypothesis can be falsified. If [\Delta T_{\mathrm{VFE1}} \rightarrow 0] is consistently required by experimental data, then VFE1 provides no experimentally distinguishable temporal correction in that regime. Conversely, if a nonzero correction is predicted before measurement and is subsequently observed, the result provides evidence requiring further investigation. 5. Quantum Computerization Consciousness Zero (QCC0) QCC0 is proposed as a computational quantity associated with the representation of quantum-state correlation, coherence, or information consistency. For a quantum state described by density operator (\rho_q), established quantum mechanics provides quantities such as purity, [ \mathcal{P} \mathrm{Tr}(\rho_q^2),] and entanglement or correlation measures may be defined according to the physical system and experimental protocol. QCC0 should therefore be defined in relation to measurable quantum quantities rather than assumed to represent a new physical substance or consciousness mechanism. A general proposed mapping can be written as [QCC0 =\mathcal{Q}\left(\rho_q,\mathcal{C},\mathcal{E},\mathcal{I}\right),] where the arguments represent appropriately defined quantum-state and information-theoretic quantities. The exact definition of (\mathcal{Q}) is an empirical question. If QCC0 is intended to represent quantum coherence, then its numerical value should be compared with independently measurable coherence metrics. If it is intended to represent a broader computational invariant, that invariant must be explicitly defined and tested for reproducibility. Accordingly, QCC0 is treated in this work as a proposed computational construct rather than an established physical constant. 6. Relationship Between VFE1 and QCC0 The two proposed components serve different conceptual purposes. VFE1 is associated primarily with physical field evolution and temporal dynamics, whereas QCC0 is associated primarily with quantum-state correlation and computational representation. A combined model may therefore be expressed schematically as [ \mathcal{M} \mathcal{M}{\mathrm{GR/SR}}+\Delta\mathcal{M}{\mathrm{VFE1}}+\Delta\mathcal{M}_{\mathrm{QCC0}}.] The terms should not be interpreted as automatically additive physical forces or energies. They represent model contributions whose physical interpretation must be established through dimensional analysis and experiment. The framework consequently imposes three requirements: Every new variable must have a defined mathematical meaning. Every predicted observable must have physical units or be explicitly dimensionless. Every proposed correction must generate an experimentally distinguishable prediction. 7. Kapnack Solver The Kapnack Solver is the computational engine responsible for evaluating the model. Its role is not to establish physical validity by itself. Instead, it provides a reproducible numerical implementation of the equations being tested. A typical calculation follows: [\text{Initial Conditions}\rightarrow\text{Established Physics}\rightarrow\text{VFE1}\rightarrow\text{QCC0}\rightarrow\text{Observable Prediction}.] The solver should preserve the intermediate results so that an investigator can determine whether an observed difference originates from: the relativistic baseline, the VFE1 contribution, the QCC0 contribution, numerical approximation, parameter selection, or measurement uncertainty. This separation is essential for scientific reproducibility. 8. 369 Vortex Representation The 369 vortex formulation is incorporated as a proposed geometric or kinematic transformation within the computational framework. Rather than ass
Quantum circuit complexity is a fundamental concept whose importance permeates quantum information, computation, many-body physics and high-energy physics. While extensively studied in closed systems, its characterization and behaviors in the widely important setting where the system is embedded within a larger one -- encompassing measurement-assisted state preparation -- lack systematic understanding. We introduce the notion of embedded complexity that characterizes the complexity of projected states and measurement operators in this general setting incorporating auxiliary systems and measurements. For random circuits and certain strongly interacting time-independent Hamiltonian dynamics, we show that the embedded complexity is lower-bounded by the circuit volume -- the total number of gates acting on both the subsystem and its complement. This strengthens the complexity linear growth theorems, enriches the understanding of deep thermalization, and indicates that measurement-assisted methods generically cannot yield significant advantages in state preparation cost, contrary to expectations. We further demonstrate a spacetime conversion of certain circuit models that concentrates circuit volume onto a subsystem, and showcase applications for random circuit sampling and shadow tomography. Our theory establishes a unified framework for space and time aspects of quantum circuit complexity, yielding profound new insights and applications across quantum information and physics.