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Domain Wall Fermion
In lattice field theory, domain wall (DW) fermions are a fermion discretization avoiding the fermion doubling problem. They are a realisation of Ginsparg–Wilson fermions in the infinite separation limit L_s\rightarrow\infty where they become equivalent to overlap fermions. DW fermions have undergone numerous improvements since Kaplan's original formulation such as the reinterpretation by Shamir and the generalisation to Möbius DW fermions by Brower, Neff and Orginos. The original d-dimensional Euclidean spacetime is lifted into d+1 dimensions. The additional dimension of length L_s has open boundary conditions and the so-called domain walls form its boundaries. The physics is now found to ″live″ on the domain walls and the doublers are located on opposite walls, that is at L_s\rightarrow\infty they completely decouple from the system. Kaplan's (and equivalently Shamir's) DW Dirac operator is defined by two addends : D_\text(x,s;y,r) = D(x;y)\delta_ + \delta_D_(s;r)\, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Lattice Field Theory
In physics, lattice field theory is the study of lattice models of quantum field theory, that is, of field theory on a space or spacetime that has been discretised onto a lattice. Details Although most lattice field theories are not exactly solvable, they are of tremendous appeal because they can be studied by simulation on a computer, often using Markov chain Monte Carlo methods. One hopes that, by performing simulations on larger and larger lattices, while making the lattice spacing smaller and smaller, one will be able to recover the behavior of the continuum theory as the continuum limit is approached. Just as in all lattice models, numerical simulation gives access to field configurations that are not accessible to perturbation theory, such as solitons. Likewise, non-trivial vacuum states can be discovered and probed. The method is particularly appealing for the quantization of a gauge theory. Most quantization methods keep Poincaré invariance manifest but sacrifice m ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Fermion
In particle physics, a fermion is a particle that follows Fermi–Dirac statistics. Generally, it has a half-odd-integer spin: spin , spin , etc. In addition, these particles obey the Pauli exclusion principle. Fermions include all quarks and leptons and all composite particles made of an odd number of these, such as all baryons and many atoms and nuclei. Fermions differ from bosons, which obey Bose–Einstein statistics. Some fermions are elementary particles (such as electrons), and some are composite particles (such as protons). For example, according to the spin-statistics theorem in relativistic quantum field theory, particles with integer spin are bosons. In contrast, particles with half-integer spin are fermions. In addition to the spin characteristic, fermions have another specific property: they possess conserved baryon or lepton quantum numbers. Therefore, what is usually referred to as the spin-statistics relation is, in fact, a spin statistics-quantum ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Fermion Doubling
In lattice field theory, fermion doubling occurs when naively putting fermionic fields on a lattice, resulting in more fermionic states than expected. For the naively discretized Dirac fermions in d Euclidean dimensions, each fermionic field results in 2^d identical fermion species, referred to as different tastes of the fermion. The fermion doubling problem is intractably linked to chiral invariance by the Nielsen–Ninomiya theorem. Most strategies used to solve the problem require using modified fermions which reduce to the Dirac fermion only in the continuum limit. Naive fermion discretization For simplicity we will consider a four-dimensional theory of a free fermion, although the fermion doubling problem remains in arbitrary dimensions and even if interactions are included. Lattice field theory is usually carried out in Euclidean spacetime arrived at from Minkowski spacetime after a Wick rotation, where the continuum Dirac action takes the form : S_F psi, \bar \psi= \in ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Ginsparg–Wilson Equation
In lattice field theory, the Ginsparg–Wilson equation generalizes chiral symmetry on the lattice in a way that approaches the continuum formulation in the continuum limit. The class of fermions whose Dirac operators satisfy this equation are known as Ginsparg–Wilson fermions, with notable examples being overlap, domain wall and fixed point fermions. They are a means to avoid the fermion doubling problem, widely used for instance in lattice QCD calculations. The equation was discovered by Paul Ginsparg and Kenneth Wilson in 1982, however it was quickly forgotten about since there were no known solutions. It was only in 1997 and 1998 that the first solutions were found in the form of the overlap and fixed point fermions, at which point the equation entered prominence. Ginsparg–Wilson fermions do not contradict the Nielsen–Ninomiya theorem because they explicitly violate chiral symmetry. More precisely, the continuum chiral symmetry relation D\gamma_5+\gamma_5 D=0 (where D ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Overlap Fermion
In lattice field theory, overlap fermions are a fermion discretization that allows to avoid the fermion doubling problem. They are a realisation of Ginsparg–Wilson fermions. Initially introduced by Neuberger in 1998, they were quickly taken up for a variety of numerical simulations. By now overlap fermions are well established and regularly used in non-perturbative fermion simulations, for instance in lattice QCD. Overlap fermions with mass m are defined on a Euclidean spacetime lattice with spacing a by the overlap Dirac operator : D_ = \frac1a \left(\left(1+am\right) \mathbf + \left(1-am\right)\gamma_5 \mathrm gamma_5 Aright)\, where A is the ″kernel″ Dirac operator obeying \gamma_5 A = A^\dagger\gamma_5, i.e. A is \gamma_5-hermitian. The sign-function usually has to be calculated numerically, e.g. by rational approximations. A common choice for the kernel is : A = aD - \mathbf 1(1+s)\, where D is the massless Dirac operator and s\in\left(-1,1\right) is a free pa ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Euclidean Space
Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, that is, in Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics there are Euclidean spaces of any positive integer dimension, including the three-dimensional space and the ''Euclidean plane'' (dimension two). The qualifier "Euclidean" is used to distinguish Euclidean spaces from other spaces that were later considered in physics and modern mathematics. Ancient Greek geometers introduced Euclidean space for modeling the physical space. Their work was collected by the ancient Greek mathematician Euclid in his ''Elements'', with the great innovation of '' proving'' all properties of the space as theorems, by starting from a few fundamental properties, called ''postulates'', which either were considered as evident (for example, there is exactly one straight line passing through two points), or seemed impossible to prove (paral ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Dirac Equation
In particle physics, the Dirac equation is a relativistic wave equation derived by British physicist Paul Dirac in 1928. In its free form, or including electromagnetic interactions, it describes all spin- massive particles, called "Dirac particles", such as electrons and quarks for which parity is a symmetry. It is consistent with both the principles of quantum mechanics and the theory of special relativity, and was the first theory to account fully for special relativity in the context of quantum mechanics. It was validated by accounting for the fine structure of the hydrogen spectrum in a completely rigorous way. The equation also implied the existence of a new form of matter, ''antimatter'', previously unsuspected and unobserved and which was experimentally confirmed several years later. It also provided a ''theoretical'' justification for the introduction of several component wave functions in Pauli's phenomenological theory of spin. The wave functions in the Dirac t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Chirality (physics)
A chiral phenomenon is one that is not identical to its mirror image (see the article on mathematical chirality). The spin of a particle may be used to define a handedness, or helicity, for that particle, which, in the case of a massless particle, is the same as chirality. A symmetry transformation between the two is called parity transformation. Invariance under parity transformation by a Dirac fermion is called chiral symmetry. Chirality and helicity The helicity of a particle is positive (“right-handed”) if the direction of its spin is the same as the direction of its motion. It is negative (“left-handed”) if the directions of spin and motion are opposite. So a standard clock, with its spin vector defined by the rotation of its hands, has left-handed helicity if tossed with its face directed forwards. Mathematically, ''helicity'' is the sign of the projection of the spin vector onto the momentum vector: “left” is negative, “right” is positive. The chirali ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Multi-index Notation
Multi-index notation is a mathematical notation that simplifies formulas used in multivariable calculus, partial differential equations and the theory of distributions, by generalising the concept of an integer index to an ordered tuple of indices. Definition and basic properties An ''n''-dimensional multi-index is an ''n''-tuple :\alpha = (\alpha_1, \alpha_2,\ldots,\alpha_n) of non-negative integers (i.e. an element of the ''n''-dimensional set of natural numbers, denoted \mathbb^n_0). For multi-indices \alpha, \beta \in \mathbb^n_0 and x = (x_1, x_2, \ldots, x_n) \in \mathbb^n one defines: ;Componentwise sum and difference :\alpha \pm \beta= (\alpha_1 \pm \beta_1,\,\alpha_2 \pm \beta_2, \ldots, \,\alpha_n \pm \beta_n) ;Partial order :\alpha \le \beta \quad \Leftrightarrow \quad \alpha_i \le \beta_i \quad \forall\,i\in\ ;Sum of components (absolute value) :, \alpha , = \alpha_1 + \alpha_2 + \cdots + \alpha_n ;Factorial :\alpha ! = \alpha_1! \cdot \alpha_2! \cdots \alpha_n ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Nielsen–Ninomiya Theorem
In lattice field theory, the Nielsen–Ninomiya theorem is a no-go theorem about placing chiral fermions on the lattice. In particular, under very general assumptions such as locality, hermiticity, and translational symmetry, any lattice formulation of chiral fermions necessarily leads to fermion doubling, where there are the same number of left-handed and right-handed fermions. It was originally proved by Holger Bech Nielsen and Masao Ninomiya in 1981 using two methods, one that relied on homotopy theory and another that relied on differential topology. Another proof provided by Daniel Friedan uses differential geometry. The theorem was also generalized to any regularization scheme of chiral theories. One consequence of the theorem is that the Standard Model cannot be put on the lattice. Common methods for overcoming the fermion doubling problem is to use modified fermion formulations such as staggered fermions, Wilson fermions, or Ginsparg–Wilson fermions. Lattice regula ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Lattice Field Theory
In physics, lattice field theory is the study of lattice models of quantum field theory, that is, of field theory on a space or spacetime that has been discretised onto a lattice. Details Although most lattice field theories are not exactly solvable, they are of tremendous appeal because they can be studied by simulation on a computer, often using Markov chain Monte Carlo methods. One hopes that, by performing simulations on larger and larger lattices, while making the lattice spacing smaller and smaller, one will be able to recover the behavior of the continuum theory as the continuum limit is approached. Just as in all lattice models, numerical simulation gives access to field configurations that are not accessible to perturbation theory, such as solitons. Likewise, non-trivial vacuum states can be discovered and probed. The method is particularly appealing for the quantization of a gauge theory. Most quantization methods keep Poincaré invariance manifest but sacrifice m ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
