Appendix A Notational Conventions

Transcription

1 Appendix A Notational Conventions Throughout the book we use Einstein s implicit summation convention: repeated indices in an expression are automatically summed over. We work in natural units where the Planck constant is = 1 and the speed of light is c = 1. In these units, energy and mass have the same mass dimension equal to +1 and time and length have the same mass dimension equal to 1. A useful relation is 1 = 197 MeVfm, which allows the conversion between energy in units of MeV 1 and length in units of fm. 2 In Lattice Quantum Chromodynamics the fields representing the quarks and the gluons are defined on a Euclidean lattice in four dimensions with lattice spacing a. The directions on the lattice are labelled by μ = 0, 1, 2, 3. The lattice points have coordinates x = (x 0, x 1, x 2, x 3 ) which are integer multiples of a: x μ = n μ a, n μ N for μ = 0, 1, 2, 3. The coordinate x 0 is referred to as the Euclidean time and the spatial coordinates are collectively denoted by x = (x 1, x 2, x 3 ). The nearest neighbor of a lattice point x in positive μ direction is denoted by x + a ˆμ, where ˆμ is the unit vector in direction μ. If the lattice has a finite volume V = T L 3 then the ranges of the coordinates are x 0 /a = 0, 1,...,T/a 1 and x k = 0, 1,...,L/a 1 for k = 1, 2, 3. Unless otherwise specified, in finite volume we impose periodic boundary conditions, which mean that the points x + T ˆ0 and x + L ˆk for k = 1, 2, 3 are identified with x. 1 MeV means Mega electronvolt; 1 ev is the potential energy that an electron aquires when a tension of 1 V is applied. 2 fm means fermi; 1 fm = m. The Author(s) 2017 F. Knechtli et al., Lattice Quantum Chromodynamics, SpringerBriefs in Physics, DOI /

2 136 Appendix A: Notational Conventions A.1 SU(N) Notation An SU(N) gauge field on the lattice assign an element U μ (x) of SU(N) to the link connecting the point x + a ˆμ with x. SU(N) is the special unitary group. In the fundamental representation, the elements U are N N complex matrices which satisfy U SU(N) U 1 = U (U T ), det(u) = 1. (A.1) The Lie algebra su(n) of SU(N) may be identified with the linear space of all anti-hermitian traceless N N matrices X: X = X, and tr X = 0. (A.2) We denote the generators (basis) of su(n) by T i, i = 1,...,N 2 1. Any element X of su(n) can written as X = X i T i, X i = 2tr { XT i}, in terms of real components X i. The normalisation of the generators is given by tr { T i T j} = 1 2 δij. (A.3) The natural scalar product in su(n) is X, Y =X i Y i = 2tr {XY}. (A.4) The structure constants f ijk defined by the commutation relation [T i, T j ]= f ijk T k (A.5) are real and totally anti-symmetric in the indices i, j, k. They satisfy f ikl f jkl = Nδ ij. The exponential map e X X k = I + k!, maps an element X su(n) onto the group SU(N). Differentiation of a function of the lattice gauge field f (U) with respect to a link U μ (x) is defined by the link differential operators [1] k=1 x,μ f (U) = T i i x,μ f (U) with i x,μ f (U) = d f (U s) ds. (A.6)

3 Appendix A: Notational Conventions 137 In the derivative with respect to s the gauge field U s is given by { e st i U (U s ) ν (y) = μ (x) if (y,ν)= (x,μ) U ν (y) otherwise. (A.7) Useful formulae are d(u s ) μ (x) ds = T i U μ (x), and where the second expression follows from 0 = d(u s) 1 d(u s ) 1 μ (x) = Uμ 1 ds (x)t i, μ (x)(u s) μ (x) ds. A.2 Fermions A fermion field assigns Dirac spinors ψ α j (x) and ψ α j (x) to each lattice point x. The spinor index is α = 1, 2, 3, 4 and the colour index in the fundamental representation of SU(N) is j = 1, 2,...,N. The components of the fermion field are Grassmann numbers. The Grassmann algebra of q Grassmann numbers {η 1,η 2,...,η q } is defined as {η i,η j }=0, i, j = 1, 2,...,q. (A.8) For a single Grassmann variable, the Berezin integral is dη = 0, and dη η= 1. These are not integrals in our usual intuitive sense of the area under a curve but should be considered as abstract and simple rules that define the fermion path integral in a physically sensible way in analogy to the bosonic path integral. Since η 2 = 0fora Grassmannian, any function of this single variable can be written f (η) = f 0 + f 1 η, with f 0 and f 1 two constants and so the integral of any function is then dη f (η) = f 1 and we see the first peculiar property of a Grassmann variable; integration is equivalent to differentiation. Now consider integrating q Grassmannians. The rules above

4 138 Appendix A: Notational Conventions apply straightforwardly, but care is needed to take the order of integration appropriately since Grassmann variables anti-commute. We have D[η] η 1 η 2...η q = 1 (A.9) with D[η] = q k=1 dη k. All other integrals, which have at least one of the variables missing, vanish. A function of q Grassmannians will have 2 N coefficients in its most general form and can be written in a binary notation. For example, with q = 3 f (η 1,η 2,η 3 ) = f f 100 η 1 + f 010 η 2 + f 001 η 3 + f 110 η 1 η 2 + f 101 η 1 η 3 + f 011 η 2 η 3 + f 111 η 1 η 2 η 3, and the integral result follows easily; D[η] f (η 1,η 2,...,η q ) = f An important role is played by Gaussian integrals. The Matthews Salam formula is D[η]D[ η] e η i M ij η j = det(m), (A.10) where { η 1, η 2,..., η q } is a second set of independent Grassmann numbers. Consider a linear transformation of the Grassmann variables η = Aη, where A is a complex matrix. Using the chain of equalities det(m) = D[Aη ]D[ η]e ηm(aη ) = J(A) D[η ]D[ η]e η(ma)η = J(A) det(ma) (A.11) we conclude that the Jacobian of the transformation is J(A) = det(a) 1. Similarly, the transformation η = η B leads to a Jacobian D[ η B]=J(B)D[ η ] with J(B) = det(b) 1. On an infinite lattice we can represent the fermion fields as integrals ψ(x) = ψ(x) = π/a π/a π/a π/a d 4 p (2π) 4 eipx ψ(p), d 4 p e ipx ψ(p). (A.13) (2π) 4 (A.12) The momenta p = (p μ ) are defined over the Brillouin zone p μ [ π/a,π/a].the Fourier components can be computed by Fourier transformation as

5 Appendix A: Notational Conventions 139 ψ(p) = a 4 x ψ(p) = a 4 x e ipx ψ(x), e ipx ψ(x). (A.14) (A.15) The Wilson Dirac operator acts on fermion fields according to Eq. (1.83). There, the 4 4 Dirac matrices γ μ, μ = 0, 1, 2, 3 act in spinor space. They are Hermitian (γ μ ) = γ μ and satisfy the anti-commutation relation {γ μ,γ ν }=2δ μν. (A.16) Since γ μ is Hermitian, it follows (γμ )αβ = γμ βα μ. An explicit choice is given by the chiral representation of the Dirac matrices, where γ μ = A possible choice for the 2 2 matrices e μ is where σ k are the Pauli matrices σ 1 = ( ) 0 eμ (e μ ). (A.17) 0 e 0 = I, e k = iσ k, k = 1, 2, 3, (A.18) ( ) 0 1, σ = We define γ 5 = γ 0 γ 1 γ 2 γ 3 with the properties ( ) ( ) 0 i 1 0, σ i 0 3 =. 0 1 (γ 5 ) = γ 5, (γ 5 ) 2 = I, {γ μ,γ 5 }=0 μ. (A.19) In the chiral representation Eq. (A.17) of the Dirac matrices we have γ 5 = and for the matrices σ μν = i 2 [γ μ,γ ν ] σ 0k = ( ) I 0 0 I ( ) ( ) σk 0 σk 0, σ 0 σ ij = ε ijk, k 0 σ k where ε ijk is the totally anti-symmetric tensor with ε 123 = 1. A pseudofermion field is defined like the fermion field to have space x, spinα and color i indices but it takes complex values instead of being Grassmann-valued. The scalar product of two pseudofermion fields φ and ψ is defined as

6 140 Appendix A: Notational Conventions φ,ψ = x,α, j φ α j (x)ψ α j (x). (A.20) We remind the useful properties φ,ψ = ψ, φ and φ, A ψ = A φ,ψ for any matrix A acting on the pseudofermion fields. The scalar product in Eq. (A.20) can be rewritten as φ,ψ = [ tr σ,c ψ(x)φ (x) ]. (A.21) x Here, M = ψ(x)φ (x) is a matrix in colour and spinor space with elements M αβ jk = ψ α j (x)φ β k (x) and tr c means the trace over the colour indices and tr σ the trace over the spinor indices. A.3 Probability Spaces A probability space (Ω, F, P) consists of a non-empty set Ω,aσ-algebra F and a probability measure P mapping F onto [0, 1]. Hereby a set of subsets of Ω is called σ -algebra, if the following three conditions hold: 1. Ω F. 2. A F Ω \ A F. 3. A i F for i = 1, 2,... i=1 A i F. For P being a probability measure, P(Ω) = 1 has to hold, as well as P( i=1 A i) = i=1 P(A i) for A i being pairwise disjoint. A random variable X : Ω R n is a measurable function mapping Ω onto R n, i.e., f ( 1) (B) F for all open sets B R n. Any measurable function f with the property P(A Ω) = dp = fdx X 1 A for all open set A R n is called a probability density function. A Reference 1. M. Lüscher, Commun. Math. Phys. 293, 899 (2010). doi: /s