Quantum field theory is defined by its lattice regularization
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Quantum field theory is defined as a theoretical framework combining field theory, special relativity, and quantum mechanics, while lattice regularization is merely one of several non-perturbative or computational approaches (along with continuum and dimensional regularizations).
In this chapter, the notions of dimensional continuation and dimensional regularization are introduced, by defining a continuation of Feynman diagrams to analytic functions of the space dimension. Dimensional continuation, which is essential for generating Wilson–Fisher's famous ϵexpansion in the theory of critical phenomena, and dimensional regularization seem to have no meaning outside the perturbative expansion of quantum field theory (QFT). Dimensional regularization is a powerful regularization technique, which is often used, when applicable because it leads to much simpler perturbative calculations. Dimensional regularization performs a partial renormalization, cancelling what would show up as power-law divergences in momentum or lattice regularization. In particular it cancels the commutator of quantum operators in local QFTs. These cancellations may be convenient but may also, occasionally, remove divergences that have an important physical meaning. It is not applicable when some essential property of the field theory is specific to the initial dimension. For example, in even space dimensions, the relation between γS (identical to γ5 in four dimensions) and the other γ matrices involving the completely antisymmetric tensor ϵμ1···μd, may be needed in theories violating parity symmetry. Its use requires some care in massless theories because its rules may lead to unwanted cancellations between ultraviolet and infrared logarithmic divergences. Explicit calculations at two-loop order in a scalar QFT with a general four-field interaction are performed.
We present a new regularization method, for d dim (Euclidean) quantum field theories in the continuum formalism, based on the domain wall configuration in (1+d) dim space-time. It is inspired by the recent progress in the chiral fermions on the lattice. The wall "height" is given by 1/M, where M is a regularization mass parameter and appears as a 1+d dim Dirac fermion mass. The present approach gives a thermodynamic view to the domain wall or the overlap formalism in the lattice field theory. We will show qualitative correspondence between the present continuum results and those of the lattice. The extra dimension is regarded as the (inverse) temperature t. The domains are defined by the directions of the "system movement", not by the sign of M as in the original overlap formalism. Physically the parameter M controls both the chirality selection and the dimensional reduction to d dimension. From the point of regularization, the limit $Mt\ra 0$ regularize the infra-red behaviour whereas the condition on the momentum ($k^\m$) integral, $|k^\m|\leq M$, regularize the ultra-violet behaviour. To check the new regularization works correctly, we take the 4 dim QED and 2 dim chiral gauge theory as examples. Especially the consistent and covariant anomalies are correctly obtained. The choice of solutions of the higher dim Dirac equation characterize the two anomalies. The projective properties of the positive and negative energy free solutions are exploited in calculation. Some integr
We present a new regularization method, for d dim (Euclidean) quantum field theories in the continuum formalism, based on the domain wall configuration in (1+d) dim space-time. It is inspired by the recent progress in the chiral fermions on lattice. The wall "height" is given by 1/M, where M is a regularization mass parameter and appears as a (1+d) dim Dirac fermion mass. The present approach gives a {\it thermodynamic view} to the domain wall or the overlap formalism in the lattice field theory. We will show qualitative correspondence between the present continuum results and those of lattice. The extra dimension is regarded as the (inverse) {\it temperature} t. The domains are defined by the {\it directions} of the "system evolvement", not by the sign of M as in the original overlap formalism. We take the 4 dim QED and 2 dim chiral gauge theory as examples. Especially the consistent and covariant anomalies are correctly obtained.
physics, quantum field theory (QFT) is a theoretical framework that combines field theory, special relativity and quantum mechanics. QFT is used in particle
In theoretical physics, quantum field theory (QFT) is a theoretical framework that combines field theory, special relativity and quantum mechanics. QFT is used in particle physics to construct physical models of subatomic particles and in condensed matter physics to construct models of quasiparticles. The current Standard Model of particle physics is based on QFT.
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Lattice QCD is a well-established non-perturbative approach to solving the quantum chromodynamics (QCD) theory of quarks and gluons. It is a lattice gauge
Lattice QCD is a well-established non-perturbative approach to solving the quantum chromodynamics (QCD) theory of quarks and gluons. It is a lattice gauge theory formulated on a grid or lattice of points in space and time. When the size of the lattice is taken infinitely large and its sites infinitesimally close to each other, the continuum QCD is recovered. Lattice QCD was developed in the 1970s
Lattice QCD is a well-established non-perturbative approach to solving the quantum chromodynamics (QCD) theory of quarks and gluons. It is a lattice gauge theory formulated on a grid or lattice of points in space and time. When the size of the lattice is taken infinitely large and its sites infinitesimally close to each other, the continuum QCD is recovered. Lattice QCD was developed in the 1970s by Nobel laureate Kenneth Wilson. It was developed within a short interval of time after the theory of quantum chromodynamics had been discovered.
Analytic or perturbative solutions in low-energy QCD are hard or impossible to obtain due to the highly nonlinear nature of the strong force and the large coupling constant at low energies. The formulation of QCD in discrete rather than continuous spacetime naturally introduces a momentum cut-off at the order 1/a, where a is the lattice spacing, which regularizes the theory. As a result, lattice QCD is mathematically well-defined. Most importantly, lattice QCD provides a framework for investigation of non-perturbative phenomena such as confinement and quark–gluon plasma formation.
In lattice QCD, fields representing quarks are defined at lattice sites (which leads to fermion doubling), while the gluon fields are defined on the links connecting neighboring sites. This approximation approaches continuum QCD as the spacing between lattice sites is reduced to zero. Because the computational cost of numerical simulations increases as the lattice spacing decreases, results must be extrapolated to a = 0 (the continuum limit) by repeated calculations at different lattice spacings a.
Numerical lattice QCD calculations using Monte Carlo methods can be extremely computationally intensive, requiring the use of the largest available supercomputers. To reduce the computational burden, the so-called quenched approximation can be used, in which the quark fields are treated as non-dynamic "frozen" variables. While this was common in early lattice QCD calculations, "dynamical" fermions are now standard.
At present, lattice QCD is primarily applicable at low baryon densities where the numerical sign problem does not interfere with…
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