typically, divergent integrals appear.'* This is the problem of renormaliza- tion in quantum field theory. Thus … renormalization scheme one employs to handle divergent integrals that can arise when one uses the EFT … regularization of divergent integrals is performed by restricting the range of momentum variables in integrals over

We reconsider the thermal scalar Casimir effect for p -dimensional hypercubic cavity inside D +1-dimensional Minkowski space-time.The thermal Casimir free energy can be divided into the divergent zero-temperature part and the automatically finite temperature-dependent part through standard quantum field theory treatments.Due to the finiteness,the regularization of the temperature-dependent part,which is also required for the convergency of the Casimir energy and the vanishing of the Casimir force with the separation increasing to infinity,is neglected in some literatures.We derive rigorously the regularization of the zero temperature part as well as the temperature-dependent part of the free energy by making use of the zeta function technique and the Abel-Plana formula.In the cases of D =3, p =1 and D =3, p =3,we precisely recover the results of parallel plates and three-dimensional box in the literature.And explicit expressions of the Casimir free energy in both low temperature (small separations) and high temperature (large separations) regimes are given,through which we find that after the regularization of both parts,with the side length going to infinity the force always tends to zero for different boundary conditions.Our study may be helpful in providing a comprehensive and complete understanding of this old problem.

The extreme divergence of vacuum energy constitutes the most severe predictive failure of standard quantum field theory. We present a novel, non-perturbative regularization of the cosmological vacuum by explicitly mapping the universe's apparent horizon to the thermodynamic event horizon of an inverted black hole. Leveraging the Hardy-Ramanujan partition formula—traditionally utilized in string theory and mock modular forms to count black hole microstates—we compute the exact quantum degeneracy of the cosmological interval field. We demonstrate that evaluating the canonical partition trace over the horizon-bounded field via the steepest-descent method naturally circumvents the catastrophic zero-point ultraviolet divergences. This strictly statistical approach dynamically yields the exact exponential background scaling required for the Ramanujan-Bose Interval Cosmology (RBIC), completely resolving the cosmological constant problem without supersymmetric cancellations or anthropic fine-tuning.

Abstract Most primitively divergent Feynman diagrams are well defined in x -space but too singular at short distances for transformation to p -space. A new method of regularization is developed in which singular functions are written as derivatives of less singular functions which contain a logarithmic mass scale. The Fourier transform is then defined by formal integration by parts. The procedure is extended to graphs with divergent subgraphs. No explicit cutoff or counter-terms are required, and the method automatically delivers renormalized amplitudes which satisfy Callan-Symanzik equations. These features are thoroughly explored in massless φ 4 theory through 3-loop order, and the method yields explicit functional forms for all amplitudes with less difficulty than conventional methods which use dimensional regularization in p -space. The procedure also appears to be compatible with gauge invariance and the chiral structure of the standard model. This aspect is tested in extensive 1-loop calculations which include the Ward identity in quantum electrodynamics, the chiral anomaly, and the background field algorithm in non-abelian gauge theories.

In particle physics, the history of quantum field theory starts with its creation by Paul Dirac, when he attempted to quantize the electromagnetic field In particle physics, the history of quantum field theory starts with its creation by Paul Dirac, when he attempted to quantize the electromagnetic field in the late 1920s. Major advances in the theory were made in the 1940s and 1950s, leading to the introduction of renormalized quantum electrodynamics (QED). The field theory behind QED was so accurate and successful in predictions that efforts wer Despite its early successes quantum field theory was plagued by several serious theoretical difficulties. Basic physical quantities, such as the self-energy of the electron, the energy shift of electron states due to the presence of the electromagnetic field, gave infinite, divergent contributions—a nonsensical result—when computed using the perturbative techniques available in the 1930s and most of the 1940s. The electron self-energy problem was already a serious issue in the classical electromagnetic field theory, where the attempt to attribute to the electron a finite size or extent (the classical electron-radius) led immediately to the question of what non-electromagnetic stresses would need to be invoked, which would presumably hold the electron together against the Coulomb repulsion of its finite-sized "parts". The situation was dire, and had certain features that reminded many of the "Rayleigh–Jeans catastrophe". What made the situation in the 1940s so desperate and gloomy, however, was the fact that the correct ingredients (the second-quantized Maxwell–Dirac field equations) for the theoretical description of interacting photons and electrons were well in place, and no major conceptual change was needed analogous to that which was necessitated by a finite and physically sensible account of the radiative behavior of hot objects, as provided by the Planck radiation law. Moreover,…

Scheming in Dimensional Regularization We consider the most general loop integral that appears in non-relativistic effective field theories with no light particles. The divergences of this integral are in correspondence with simple poles in the space of complex space-time dimensions. Integrals related to the original integral by subtraction of one or more poles in dimensions other than D=4 lead to nonminimal subtraction schemes. Subtraction of all poles in correspondence with ultraviolet divergences of the loop integral leads naturally to a regularization scheme which is precisely equivalent to cutoff regularization. We therefore recover cutoff regularization from dimensional regularization with a nonminimal subtraction scheme. We then discuss the power-counting for non-relativistic effective field theories which arises in these alternative schemes. Published as: J.Phys.A32:3397-3407,1999 DOI: 10.1088/0305-4470/32/18/313 arXiv categories: hep-th hep-ph nucl-th

Path Integral Treatment of Singular Problems and Bound States: Quantum Mechanics A path-integral approach for the computation of quantum-mechanical propagators and energy Green's functions is presented. Its effectiveness is demonstrated through its application to singular interactions, with particular emphasis on the inverse square potential--possibly combined with a delta-function interaction. The emergence of these singular potentials as low-energy nonrelativistic limits of quantum field theory is highlighted. Not surprisingly, the analogue of ultraviolet regularization is required for the interpretation of these singular problems. Published as: Int.J.Mod.Phys. A19 (2004) 1413-1440 DOI: 10.1142/S0217751X04017926 arXiv categories: hep-th hep-ph math-ph math.MP nucl-th quant-ph

A unifying theme of this thesis is the study of Archimedean zeta functions defined by complex-geometric data. Classically, Archimedean zeta functions over C are parameter-dependent integrals over a domain in Cn, where the parameter is a complex power of the modulus of a holomorphic function. We consider a global complex-geometric generalization in which the integration is over a complex manifold or reduced analytic space, and where the modulus of a holomorphic function is replaced by the norm of a holomorphic section of a vector bundle.The first two papers concern finite parts of divergent integrals on reduced complex analytic spaces. Given a singular differential form whose singularities are determined a holomorphic section of a vector bundle, a Hermitian metric induces a natural regularization of the divergent integral, giving rise to an Archimedean zeta function. A finite part of the divergent integral is defined as the constant term in the Laurent expansion of this zeta function about 0. Paper I establishes an explicit formula describing the dependence of the resulting finite part on the choice of Hermitian metric. Paper II develops a current calculus adapted to this setting and derives decomposition formulas that permit explicit computations of certain finite parts. We illustrate these formulas with a family of examples on projective space, where the resulting finite parts turn out to be multiple zeta values.The second pair of papers studies Archimedean zeta functions arising as partition functions of Gibbs ensembles on compact Kähler manifolds. In Paper III, we consider systems of particles on the two-dimensional sphere, interacting through logarithmic pair potentials. Depending on the numerical values of the coupling constants, the resulting partition functions are either examples of Archimedean zeta functions or slight generalizations thereof. Using techniques from complex algebraic geometry, in particular the Fulton—MacPherson compactification of configuration space, we establish the meromorphic continuation of these partition functions and relate the location of their critical inverse temperatures to a discrete optimization problem governing both integrability and particle clustering.Paper IV concerns Berman's probabilistic approach to Kähler—Einstein metrics on log Fano manifolds X. In this framework, the Kähler--Einstein geometry of X is encoded by a canonical random point process admitting a statistical-mechanical interpretation in terms of a family of Gibbs measures on the products XN, whose associated partition functions define Archimedean zeta functions. The main contribution of the paper is an extension of this framework to log Fano manifolds with non-discrete automorphism groups. To this end, we propose a symmetry-breaking procedure based on a moment-map constraint for the Gibbs measures, and introduce an algebraic notion of Gibbs polystability, conjecturally equivalent to the existence of a Kähler—Einstein metric on X. Moreover, we conjecture that if X is Gibbs polystable, then the unique Kähler—Einstein metric with vanishing moment emerges when sampling N points on X subject to the moment-map constraint as N tends to infinity. Inter alia, we verify several of our conjectures for log Fano curves.

In this article, we present a new implementation of the Laporta algorithm to reduce scalar multi-loop integrals---appearing in quantum field theoretic calculations---to a set of master integrals. We extend existing approaches by using an additional algorithm based on modular arithmetic to remove linearly dependent equations from the system of equations arising from integration-by-parts and Lorentz identities. Furthermore, the algebraic manipulations required in the back substitution are optimized. We describe in detail the implementation as well as the usage of the program. In addition, we show benchmarks for concrete examples and compare the performance to Reduze 2 and FIRE 5. In our benchmarks we find that Kira is highly competitive with these existing tools.

To study quantum field theories on a quantum computer, we must begin with Hamiltonians defined on a finite-dimensional Hilbert space and then take appropriate limits. This approach can be seen as a new type of regularization for quantum field theories, which we refer to as qubit regularization. A related finite-dimensional regularization, known as the D-theory approach, was proposed long ago as a general framework for all quantum field theories. In this framework, the dimensionality of the local Hilbert space at each spatial point can increase as needed through an additional flavor index. To reproduce asymptotically free QFTs, most studies assume that qubit-regularized theories require extending the local Hilbert space to infinity. However, contrary to this common belief, recent discoveries in (1+1) dimensions have revealed two examples where asymptotic freedom appears to emerge within a strictly finite-dimensional local Hilbert space through a novel renormalization group (RG) flow. These findings motivate further investigation into whether asymptotically free gauge theories could also emerge within a strictly finite-dimensional local Hilbert space. To support these explorations, we propose an orthonormal basis called the monomer-dimer-tensor-network (MDTN) basis and use it to construct new types of qubit-regularized lattice gauge theories.

We present an algorithm for calculating the metric perturbations and gravitational self-force for extreme-mass-ratio inspirals (EMRIs) with eccentric orbits. The massive black hole is taken to be Schwarzschild and metric perturbations are computed in Lorenz gauge. The perturbation equations are solved as coupled systems of ordinary differential equations in the frequency domain. Accurate local behavior of the metric is attained through use of the method of extended homogeneous solutions and mode-sum regularization is used to find the self-force. We focus on calculating the self-force with sufficient accuracy to ensure its error contributions to the phase in a long term orbital evolution will be $\delta\Phi \lesssim 10^{-2}$ radians. This requires the orbit-averaged force to have fractional errors $\lesssim 10^{-8}$ and the oscillatory part of the self-force to have errors $\lesssim 10^{-3}$ (a level frequently easily exceeded). Our code meets this error requirement in the oscillatory part, extending the reach to EMRIs with eccentricities of $e \lesssim 0.8$, if augmented by use of fluxes for the orbit-averaged force, or to eccentricities of $e \lesssim 0.5$ when used as a stand-alone code. Further, we demonstrate accurate calculations up to orbital separations of $a \simeq 100 M$, beyond that required for EMRI models and useful for comparison with post-Newtonian theory. Our principal developments include (1) use of fully constrained field equations, (2) discovery of analytic