10 Field Theory on a Lattice 10.1 Scalar Fields To represent a hermitian field φ(x), we put a real number φ(i, j, k, `) at each vertex of the lattice and set x = a(i, j, k, `) (10.1) where a is the lattice spacing. The derivative @iφ(x) is approximated as φ(x + ˆi) φ(x) @ φ(x) − (10.2) i ⇡ a in which x is the discrete position 4-vector and ˆi is a unit 4-vector pointing in the i direction. So the euclidian action is the sum over all lattice sites of 1 1 S = (@ φ(x))2 a4 + m2 φ2(x) a4 e 2 i 2 x X 1 2 1 = φ(x + ˆi) φ(x) a2 + m2 φ2(x) a4 2 − 2 0 (10.3) x ⇣ ⌘ X1 1 = φ(x + ˆi)φ(x) a2 + 8+m2 φ2(x) a4 − 2 2 0 xi X if the self interaction happens to be quartic. 1 1 λ S = (@ φ(x))2 a4 + m2 φ2(x) a4 + φ4(x) a4 (10.4) e 2 i 2 0 4 if the self interaction happens to be quartic. 10.2 Pure Gauge Theory 249 10.2 Pure Gauge Theory The gauge-covariant derivative is defined in terms of the generators ta of a compact Lie algebra [ta,tb]=ifabctc (10.5) b and a gauge-field matrix Ai = igAi tb as D = @ A = @ igAb t (10.6) i i − i i − i b summed over all the generators, and g is a coupling constant. Since the group is compact, we may raise and lower group indexes without worrying about factors or minus signs. The Faraday matrix is F =[D ,D ]=[I@ A (x),I@ A (x)] = @ A +@ A +[A ,A ] (10.7) ij i j i − i j − j − i j j i i j in matrix notation. With more indices exposed, it is (F ) =( @ A + @ A +[A ,A ]) ij cd − i j j i i j cd b b b b b e e (10.8) = igtcd @iAj @jAi + [igt Ai ,igt Aj] . − − cd ⇣ ⌘ ⇣ ⌘ Summing over repeated indices, we get b b b 2 b e b e (Fij)cd = igtcd @iAj @jAi g Ai Aj [t ,t ] − − − cd ⇣ ⌘ ⇣ ⌘ = igtb @ Ab @ Ab g2AbAeif tf − cd i j − j i − i j bef cd b ⇣ b b⌘ 2 b e f = igtcd @iAj @jAi ig Ai Ajfbef tcd − − − (10.9) ⇣ ⌘ = igtb @ Ab @ Ab ig2Af Aef tb − cd i j − j i − i j feb cd ⇣ ⌘ = igtb @ Ab @ Ab + gAf Aef − cd i j − j i i j feb ⇣ ⌘ = igtb @ Ab @ Ab + gf Af Ae = igtb F b − cd i j − j i bfe i j − cd ij ⇣ ⌘ where F b = @ Ab @ Ab + gf Af Ae (10.10) ij i j − j i bfe i j is the Faraday tensor. The action density of this tensor is 1 L = F b F ij. (10.11) F − 4 ij b 250 Field Theory on a Lattice The trace of the square of the Faraday matrix is ij b b c ij Tr FijF =Tr igt Fij igt Fc ⇥ ⇤ = hg2F b F ij Tr(tbtci)= g2F b F ij kδ (10.12) − ij c − ij c bc = kg2 F b F ij. − ij b So the Faraday action density is 1 1 1 L = F b F ij = Tr F F ij = Tr F F ij . (10.13) F − 4 ij b 4kg2 ij 2g2 ij ⇥ ⇤ ⇥ ⇤ The theory described by this action density, without scalar or spinor fields, is called pure gauge theory. The purpose of a gauge field is to make the gauge-invariant theories. So if a field a is transformed by the group element g(x) a0 (x)=g(x)ab b(x), (10.14) then we want the covariant derivative of the field to go as (Di (x))0 = @i A0 (x) g(x) (x) − i (10.15) = g(x)D (x)=g(x)[(@ A (x)) (x)] . i i − i So the gauge field must go as @ g A0 g = gA (10.16) i − i − i or as 1 1 Ai0 (x)=g(x)Ai(x)g− (x)+(@ig(x))g− (x). (10.17) So if g(x)= exp( i✓a(x)ta), then a gauge transforms as − i✓a(x)ta i✓a(x)ta i✓a(x)ta i✓a(x)ta Ai0 (x)=e− Ai(x) e +(@ie− ) e . (10.18) How does an exponential of a path-ordered, very short line integral of gauge fields go? We will evaluate how the path-ordered exponential in which g0 is a coupling constant i 1 1 i P exp g0Ai(x)dx 0 = P exp g(x)g0Ai(x)g− (x)+(@ig(x))g− (x) dx ⇣h 1 i i ⌘ = P exp g(x)g0Ai(x)g− (x)+@i log g(x) dx ⇣h i ⌘(10.19) changes under the gauge transformation (10.18) in the limitt dxi 0. We ! 10.3 Pure Gauge Theory on a Lattice 251 find i i g0Ai(x)dx 1 P exp g0Ai(x)dx 0 = P g(x)e g− (x) logh g(x+dxi/2) log g(x)+log g(x) log g(x dxi/2) e − − − ⇥ i g0A (x)dx 1 i = P g(x)e i g− (x) (10.20) h i 1 1 i g(x + dx /2)g− (x)g(x)g− (x dx /2) ⇥ − i i g0A (x)dx 1 i i = g(x + dx /2)e i g− (x dx /2). − Putting together a chain of such infinitesimal links, we get x x i 0 i 1 P exp g0Ai(x0)dx0 = g(x)P exp g0Ai(x0)dx0 g− (y). (10.21) ✓Zy ◆ ✓Zy ◆ In particular, this means that the trace of a closed loop is gauge invariant i 0 i 1 TrP exp g0Ai(x)dx =Trg(x)P exp g0Ai(x)dx g− (x) ✓I ◆ ✓I ◆ i =TrP exp g0Ai(x)dx . ✓I ◆ (10.22) 10.3 Pure Gauge Theory on a Lattice Wilson’s lattice gauge theory is inspired by these last two equations. Another source of inspiration is the approximation for a loop of tiny area dx dy ^ which in the joint limit dx 0 and dy 0is ! ! i W = P exp g0Ai(x)dx =exp g0 Ay,x Ax,y + g0[Ax,Ay] dxdy − ✓I ◆ h ⇣ ⌘ i =exp g F dxdy . − 0 xy ⇣ ⌘ (10.23) To derive this formula, we will ignore the bare coupling constant g0 for the moment and apply the Baker-Campbell-Hausdorf identity 1 1 1 eA eB =exp A + B + [A, B]+ [A, [A, B]] + [B,[B,A]] + ... 2 12 12 ✓ (10.24)◆ 252 Field Theory on a Lattice to the product W =exp Ax dx exp Ay dy + Ay,x dxdy (10.25) ⇣ ⌘ ⇣ ⌘ exp Ax dx Ax,y dxdy exp Ay dy . ⇥ − − − We get ⇣ ⌘ ⇣ ⌘ 1 W =exp Ax dx + Ay dy + Ay,x dxdy + [Ax dx, (Ay + Ay,x dx)dy] 2 ⇣ 1 ⌘ exp Ax dx Ay dy Ax,y dxdy + [(Ax + Ax,y dy)dx, Ay dy] . ⇥ − − − 2 ⇣ (10.26)⌘ Applying again the BCH identity, we get W =exp Ay,x Ax,y +[Ax,Ay] dxdy =exp Fxy dxdy , (10.27) − an identity thath⇣ is the basis of lattice gauge⌘ i theory. ⇣ ⌘ Restoring g0, we divide the trace of W by the dimension n of the matrices ta and subtract from unity 1 1 1 Tr P exp g A (x)dxi =1 Tr exp g F dxdy − n 0 i − n 0 xy ✓I ◆ (10.28) 1 1 h2 ⇣ ⌘i = Tr g F dxdy + g F dxdy . − n 0 xy 2 0 xy h ⇣ ⌘ i The generators of SU(n), SO(n), and Sp(2n) are traceless, so the first term vanishes, and we get 1 1 2 1 Tr P exp g A (x)dxi = Tr g F dxdy . (10.29) − n 0 i − 2n 0 xy ✓I ◆ ⇣ ⌘ Recalling the more explicit form (10.9) of the Faraday matrix, we have 1 g2 kg2 1 Tr(W )= 0 Tr t F a t F b dxdy 2 = 0 F a dxdy 2 (10.30) − n 2n a xy b xy 2n xy h i in which k is the constant of the normalization Tr(tatb)=kδab. For SU(2) with ta = σa/2 and for SU(3) with ta = λa/2, this constant is k =1/2. The Wilson action is a sum over all the smallest squares of the lattice, called the plaquettes, of the quantity n 1 1 a 2 4 S2 = 1 Tr(W ) = F a (10.31) 2kg2 − n 4 ij 0 in which a is the lattice spacing. The full Wilson action is the sum of this quantity over all the elementary squares of the lattice and over i, j = 1, 2, 3, 4..