8. Quantum field theory on the lattice 8.1. Fundamentals • Quantum field theory (QFT) can be defined using Feynman’s path integral [R. Feynman, Rev. Mod. Phys. 20, 1948]: Z = dφ(x) exp[iS(φ)] (1) x Z Y 4 S = d xL(φ, @tφ) Z Here x = (x0; x1; x2; x3), and gµν = diag(+,-,-,-) • Physical observables can be evaluated if we add a source S ! S + Jxφx. For example, 1- and connected 2-point functions are: @ 1 hφxi = log Z = [dφ]φx exp[iS] i@Jx J=0 Z Z @ @ 1 2 hφxφyi = log Z = [dφ]φxφy exp[iS] − hφxi i@Jx i@Jy Z Z 1 • These can be computed using perturbation theory, if the coupling constants in S are small (see any of the oodles of QFT textbooks). • However: if – 4-volume is finite, and – 4-coordinate x is discrete (x 2 aZ4, a lattice spacing), the integral in (1) has finite dimensionality and can, in principle, be evaluated by brute force ( = computers) 7! gives fully non-perturbative results. • Need to extrapolate: V ! 1; a ! 0 in order to recover continuum physics. 2 • But: - the dimensionality of the integral is (typically) huge - exp iS is complex+unimodular: every configuration fφg contributes with equal magnitude 7! extremely slow convergence in numerical computations (useless in practice). • This is (largely) solved by using imaginary time formalism (Eu- clidean spacetime). • Imaginary time also admits non-zero temperature T . 3 Units: Standard HEP units, where c = h¯ = kB = 1; and thus [length]−1 = [time]−1 = [mass] = [temperature] = [energy] = GeV: Mass m = rest energy mc2 = (Compton wavelength)−1 mc=h¯. (1 GeV)−1 ≈ 0:2 fm 4 8.2. Path integral in imaginary time Consider scalar field theory in Minkowski spacetime: 3 3 1 µ S = d x dt L(φ, @tφ) = d x dt 2@µφ @ φ − V (φ) Z Z h i We obtain the (classical) Hamiltonian by Legendre transformation: H = d3x dt πφ_ − L Z h i 3 1 2 1 2 = d x dt 2π + 2(@iφ) + V (φ) Z h i Here π is canonical momentum for φ: π = δL/δφ_. Quantum field theory: consider Hilbert space of states jφi, jπi, and field operators (φ ! φ^, π ! π^, H ! H^ ), with the usual properties: • φ^(x¯)jφi = φ(x¯)jφi • [ dφ(x¯)]jφihφj = 1 [ dπ(x¯)=(2π)]jπihπj = 1 x x Z Y Z Y • [φ^(x¯); φ^(x¯0)] = −iδ3(x¯ − x¯0) • hφjπi = exp [i d3xπ(x¯)φ(x¯)] Z 5 Time evolution operator is U(t) = e−iH^ t Feynman showed that the quantum theory defined by H^ and the Hilbert space is equivalent to the path integral (1). We shall now do this in imaginary time. Let us now consider the quantum system in imaginary time: t ! τ = it ; exp[−iH^ t] ! exp[−τH^ ] This makes the spacetime Euclidean: g = (+ − − −) ! (+ + + +). We shall keep the Hamiltonian H^ as-is, and also the Hilbert space. Let us also discretize space (convenient for us but not necessary at this stage), and use finite volume: x¯ = an;¯ ni = 0 : : : Ns d3x ! a3 Z x 1 X @ φ ! [φ(x¯ + e¯ a) − φ(x¯)] ≡ ∆ φ(x¯) i a i i 6 Thus, the Hamiltonian is: ^ 3 1 2 1 ^ 2 ^ H = a 2[π^(x)] + 2[∆iφ(x¯)] + V [φ(x¯)] x X −(τB −τA)H^ Let us now consider the amplitude hφBje jφAi: 1) divide time in small intervals: τB − τA = Nτ aτ . Obviously −(τB −τA)H^ −aτ H^ Nτ hφBje jφAi = hφBj(e ) jφAi: 2) insert 1-operators [ dφi]jφiihφij, [ dπi]jπiihπij x x Z Y Z Y −τH^ hφBje jφAi = Nτ Nτ −aτ H^ dφi(x) dπi(x) hφBjπNτ ihπNτ je jφNτ i 2 x 3 2 x 3 Z iY=2 Y iY=1 Y 4 5 4 5 −aτ H^ −aτ H^ ×hφNτ jπNτ −1ihπNτ −1je jφNτ −1i : : : × hφ2jπ1ihπ1je jφAi 7 Thus, we need to calculate −aτ H^ 3 1 2 1 2 hπije jφii = exp −a aτ 2πi + 2(∆iφi) + V [φi] " x # X 2 × hπijφii + O(aτ ) and 3 3 ia πi(x¯)φi+1(x¯) −ia πi(x¯)φi(x¯) hφi+1jπiihπijφii = e x e x P 3 P = exp[ia aτ πi(x¯)∆0φi(x¯)] (2) x X 1 where we have defined ∆0φi = (φi+1 − φi). aτ We can now integrate over πi(x¯): 3 1 2 [ πi(x¯)] exp[a aτ −2πi + (i∆0φi)πi] Z x x Y 3 X NS=2 2π 3 1 2 = × exp[−a aτ 2(∆0φi(x¯)) ] "a3a # x τ X 8 Repeating this for i = 1 : : : Nτ , and identifying the time coordinate x0 = τ = aτ i, we finally obtain the path integral −τH^ −SE hφBje jφAi / [ dφ]e x Z Y where SE is the Euclidean (imaginary time) action 3 1 2 SE = a aτ (2(∆µφ) + V [φ]) x X 4 1 2 ! d x(2(@µφ) + V [φ]) Z and we have the boundary conditions φ(τA) = φA, φ(τB) = φB. The path integral is precisely in the form of a partition function for a 4-dimensional classical statistical system, with the identification SE $ H=(kBT ). For convenience, we make the system periodic in time by identifying φA = φB and integrating over φA. 9 We can now make a connection between the correlation functions of the “statistical” theory and the Green’s functions of the quantum field theory. −aτ H^ First, note that we can interpret Tφi+1,φi = hφi+1je jφii as a transfer matrix. In terms of T the partition function is Z = [dφ]e−SE = Tr (T Nτ ) Z Let us label the eigenvalues of T by λ0; λ1 : : :, so that λ0 > λ1 ≥ : : :. Note that λi = exp −Ei, where Ei are eigenvalues of H^ . Thus, λ0 corresponds to the state of lowest energy, vacuum j0i. If we now take Nτ to be very large (while keeping aτ constant; i.e. take ∆τ to be large), Nτ Nτ Nτ Z = λi = λ0 [1 + O((λ1/λ0) )] Xi For example, a 2-point function can be written as (let i − j > 0) 1 −SE 1 Nτ −i+j i−j hφiφji = [dφ]φiφje = Tr (T φT^ φ^): Z Z Z Taking now Nτ ! 1, and recalling aτ (i − j) = τi − τj, i−j hφiφji = h0jφ^(T/λ0) φ^j0i = h0jφ^(τi)φ^(τj)j0i; 10 where we have introduced time-dependent operators ^ ^ φ^(τ) = eτH φ^e−τH : Allowing for both positive and negative time separations of τi and τj, we can identify hφ(τi)φ(τj)i = h0jT [φ^(τi)φ^(τj)]j0i; where T is the time ordering operator. Mass spectrum: • Green’s functions in time τ: 1 h0jΓ(τ)Γy(0)j0i = [dφ]Γ(τ)Γy(0)e−S Z Z ^ ^ = h0jeHτ Γ(0)e−Hτ Γy(0)j0i −H^ τ y = h0jΓ(0) jEnihEnje Γ (0)j0i n X 11 −Enτ 2 = e jh0jΓ(0)jEnij n X −E0τ 2 ! e jh0jΓ(0)jE0ij as τ ! 1 where jE0i is the lowest energy state with non-zero matrix element h0jΓ(0)jE0i. ! measure masses (E0) from the exponential fall-off of correlation func- tions. 12 8.3. Finite temperature Connection Euclidean QFT $ classical statistical mechanics was de- rived for zero-temperature quantum system. However, this can be read- ily generalized to finite temperature: Quantum thermodynamics w. the Gibbs ensemble: ^ ^ Z = Tr e−H=T = [dφ] hφje−H=T jφi Z Expression is of the same form as the one which gave us the Euclidean path integral for T = 0 theory! The difference here is 1) Finite + fixed “imaginary time” interval 1=T 2) Periodic boundary condition: φ(1=T ) = φ(0). Repeating the previous derivation, the partition function becomes 1=T −SE 3 Z(T ) = [dφ] e = [dφ] exp[− dτ d xLE ] Z Z Z0 Z Thus, a connection between: – Quantum statistics in 3d: Z = Tr e−H^ =T 13 – Classical statistics in 4d: Z = [dφ]e−S Euclidean P.I. is a very commonR tool for finite T field theory analysis [J. Kapusta, Finite Temperature Field Theory, Cambridge University Press] 14 8.4. Some terminology: In numerical work, lattice is a finite box with finite lattice spacing a. In order to obtain continuum results, we should take 2 limits: • V ! 1 thermodynamic limit • a ! 0 continuum limit Both have to be controlled – expensive! 1) Perform simulations with fixed a, various V . Extrapolate V ! 1. 2) Repeat 1) using different a’s. Extrapolate a ! 0. 3) [ In QCD, one often has to extrapolate mq ! mq;phys:. ] 15 T = 0: 1) V ! 1: Nτ ; Ns ! 1, a constant. 2) continuum: a ! 0. T > 0: 1) V ! 1: Ns ! 1, Nτ , a constant. 2) continuum: 1 a ! 0, T = aNτ constant. 16 8.5. Scalar field Free scalar field on a finite d-dimensional lattice with periodic boundary conditions: xµ = anµ; nµ 2 Z Action: d 1 1 2 1 2 2 d 1 1 2 2 S = a 2 (φx+µ − φx) + m φ = a 2φx x;yφy + 2m φx x 2 µ a2 2 3 X X h i 4 5 (implicit sum over x; y), and the lattice d’Alembert operator is 2 2φx − φx+µ^ − φx−µ^ x;yφy = −∆ φ = µ a2 X 1 The action is of form S = 2φxMx;yφy, and Z = [dφ]e−S = (Det M=2π)−1=2 Z 17 Fourier transforms: f~(k) = ade−ikxf(x) x X Since x = an, f~(k + 2πn) = f~(k), and we restrict k to Brillouin zone: −π=2 < kµ ≤ π=2 Inverse transform: d π=a d k ikx ~ f(x) = d e f(k) Z−π=a (2π) Note: often it is convenient to use dimensionless natural lattice units xµ 2 Z, −π < kµ ≤ π.