The Monte Carlo Method in Quantum Field Theory Colin Morningstar Carnegie Mellon University HUGS June 2006 1 / 169 Outline Path integrals in quantum mechanics Monte Carlo integration and Markov chains Monte Carlo evaluation of path integral in quantum mechanics Free Klein-Gordon scalar field theory in 2 + 1 dimensions Interacting φ4 scalar field theory in 2 + 1 dimensions Applications in quantum chromodynamics Topics for future study 2 / 169 Part I Path integrals in quantum mechanics 3 / 169 Transition amplitudes in quantum mechanics key quantity in quantum mechanics: transition amplitude Z(b, a) ≡ hxb(tb) | xa(ta)i Z(b, a) is probability amplitude for particle to go from point xa at time ta to point xb at time tb in this talk, will work in Heisenberg picture state vectors |Ψi are stationary operators and their eigenvectors evolve with time x(t) = eiHt/~ x(0) e−iHt/~ |x(t)i = eiHt/~ |x(0)i often will shift Hamiltonian so ground state energy is zero H |φn(t)i = En |φn(t)i, E0 = 0 |φ0(t)i = |φ0(0)i ≡ |0i 4 / 169 Spectral representation of transition amplitude insert complete (discrete) set of Heisenberg-picture eigenstates |φn(t)i of Hamiltonian H into transition amplitude X Z(b, a) ≡ hxb(tb) | xa(ta)i = hxb(tb) |φn(tb)ihφn(tb)| xa(ta)i n iHt/ iEnt/ now use |φn(t)i = e ~|φn(0)i = e ~|φn(0)i to obtain X −iEn(tb−ta)/ Z(b, a) = e ~hxb(tb) |φn(tb)ihφn(ta)| xa(ta)i n finally, hx(t)|φn(t)i ≡ ϕn(x) is the wavefunction in coordinate space, so X ∗ −iEn(tb−ta)/~ Z(b, a) = ϕn(xb)ϕn (xa) e n transition amplitude contains information about all energy levels and all wavefunctions → spectral representation 5 / 169 Vacuum saturation take ta = −T and tb = T in the limit T → (1 − i)∞ −iHT/ iH(−T)/ hxb(T)|xa(−T)i = hxb(0)|e ~ e ~|xa(0)i ∞ X −2iEnT/ = hxb(0)|φn(0)ihφn(0)|xa(0)i e ~ n=0 → hxb(0)|0ih0|xa(0)i insert complete set of energy eigenstates, use En+1 ≥ En, E0 = 0, assume nondegenerate vacuum possibility of probing ground state (vacuum) properties 6 / 169 Vacuum expectation values now apply limit T → (1 − i)∞ to more complicated amplitude hxb(T)|x(t2)x(t1)|xa(−T)i −iHT/~ −iHT/~ = hxb(0)|e x(t2)x(t1) e |xa(0)i X = hxb(0)|φn(0)ihφn(0)|x(t2)x(t1)|φm(0)ihφm(0)|xa(0)i n,m ×e−i(En+Em)T/~ → hxb(0)|0ih0|x(t2)x(t1)|0ih0|xa(0)i hence, vacuum expectation values from hxb(T)|x(t2)x(t1)|xa(−T)i h0|x(t2)x(t1)|0i = lim T→(1−i)∞ hxb(T)|xa(−T)i result generalizes to higher products of position operator 7 / 169 Observables from correlation functions all observables can be extracted from the correlation functions (vacuum expectation values) example: energies of the stationary states h0|x(t)x(0)|0i = h0|eiHt/~x(0)e−iHt/~x(0)|0i X −iHt/ = h0|x(0)e ~|φn(0)ihφn(0)|x(0)|0i n X 2 −iEnt/ = |h0|x(0)|φn(0)i| e ~ n similarly for more complicated correlation functions h0|x2(t)x2(0)|0i = h0|eiHt/~x2(0)e−iHt/~x2(0)|0i X 2 2 −iEnt/ = |h0|x (0)|φn(0)i| e ~ n but difficult to extract energies En from above oscillatory functions → much easier if we had decaying exponentials 8 / 169 The imaginary time formalism can get decaying exponentials if we rotate from the real to the imaginary axis in time (Wick rotation) t → −iτ X 2 −Enτ/ h0|x(t)x(0)|0i = |h0|x(0)|φn(0)i| e ~ n τ→∞ 2 2 −E1τ/~ −→ |h0|x(0)|0i| + |h0|x(0)|φ1(0)i| e later, will see this imaginary time formalism provides another important advantage for Monte Carlo applications 9 / 169 Quantum mechanics and path integrals in the 1940s, Feynman developed an alternative formulation of quantum mechanics (his Ph.D. thesis) Richard Feynman, Rev Mod Phys 20, 367 (1948) quantum mechanical law of motion: probability amplitude from sum over histories X Z(b, a) ∼ exp (iS[x(t)]/~) all paths x(t) from a to b all paths contribute to probability amplitude, but with different phases determined by the action S[x(t)] classical limit: when small changes in path yield changes in action large compared to ~, phases cancel out and path of least action δS = 0 dominates sum over histories 10 / 169 Defining the path integral action = time integral of Lagrangian (kinetic minus potential energy) Z Z S = dt L(x, x˙) = dt K − U divide time into steps of width ε where Nε = tb − ta path integral is defined as 1 Z ∞ dx dx dx Z(b, a) = lim 1 2 ··· N−1 eiS[x(t)]/~ ε→0 A −∞ A A A where A is a normalization factor depending on ε chosen so path integral well-defined in nonrelativistic theory, paths cannot double-back in time 11 / 169 Schrödinger equation probability amplitude ψ(xb, tb) at time tb given amplitude ψ(xa, ta) at earlier time ta given by Z ψ(xb, tb) = Z(b, a) ψ(xa, ta) dxa take ta = t and tb = t + ε one time slice away Z ∞ 1 iε xb + xa xb − xa ψ(xb, t + ε) = exp L , ψ(xa, t) dxa A −∞ ~ 2 ε in L, take x˙ = (xb − xa)/ε and mid-point prescription x → (xb + xa)/2 1 2 particle in potential: L = 2 mx˙ −V(x, t), write xb = x, xa = x + η ∞ 1 Z 2 ψ(x, t + ε) = eimη /(2~ε)e−iεV(x+η/2,t)/~ψ(x + η, t) dη A −∞ 12 / 169 Schrödinger equation (continued) ∞ 1 Z 2 ψ(x, t + ε) = eimη /(2~ε)e−iεV(x+η/2,t)/~ψ(x + η, t) dη A −∞ 2 √ rapid oscillation of eimη /(2~ε) except when η ∼ O( ε) → integral dominated by contributions from η having values of this order 2 expand to O(ε) and O(η2), except eimη /(2~ε) (ψ refers to ψ(x, t)) ∞ 2 2 ∂ψ 1 Z 2 h iε ih ∂ψ η ∂ ψ i ψ + ε = eimη /(2~ε) − V(x, t) ψ+η + dη 1 2 ∂t A −∞ ~ ∂x 2 ∂x ∞ 2 2 1 Z 2 h iε ∂ψ η ∂ ψ i = eimη /(2~ε) ψ − V(x, t)ψ + η + dη 2 A −∞ ~ ∂x 2 ∂x 13 / 169 Schrödinger equation (continued) ∞ 2 2 ∂ψ 1 Z 2 h iε ∂ψ η ∂ ψ i ψ + ε = eimη /(2~ε) ψ − V(x, t)ψ + η + dη 2 ∂t A −∞ ~ ∂x 2 ∂x matching leading terms on both sides determines A (analytic continuation to evaluate integral) ∞ 1/2 1/2 1 Z 2 1 2πi ε 2πi ε 1 = eimη /(2~ε)dη = ~ ⇒ A = ~ A −∞ A m m two more integrals: ∞ ∞ 1 Z 2 1 Z 2 i ε eimη /(2~ε) η dη = 0, eimη /(2~ε) η2dη = ~ A −∞ A −∞ m O(ε) part of equation at top yields ∂ψ 2 ∂2ψ −~ = − ~ + V(x, t)ψ i ∂t 2m ∂x2 the Schrödinger equation! 14 / 169 Free particle in one dimension 1 2 Lagrangian of free particle in one dimension L = 2 mx˙ amplitude for particle to travel from xa at time ta to location xb at later time tb is Z b hxb(tb)|xa(ta)i = Dx(t) exp(iS[b, a]/~) a summing over all allowed paths with x(ta) = xa and x(tb) = xb. classical path xcl(t) from δS = 0 and boundary conditions: (t − ta) ¨xcl(t) = 0, xcl(t) = xa + (xb − xa) (tb − ta) classical action is Z tb 2 1 2 m(xb − xa) Scl[b, a] = dt 2 mx˙cl = ta 2(tb − ta) write x(t) = xcl(t) + χ(t) where χ(ta) = χ(tb) = 0 then Z tb 1 2 S[b, a] = Scl[b, a] + dt 2 mχ˙ ta where Scl[b, a] is classical action; no terms linear in χ(t) since Scl is extremum 15 / 169 Path integral for free particle amplitude becomes (T = tb − ta) Z(b, a) = F(T) exp(iScl/~) Z 0 im Z T F(T) = Dχ exp dt χ˙ 2 0 2~ 0 partition time into discrete steps of length ε, use midpoint prescription, and note that χ0 = χN = 0 Z 0 Z ∞ N−1 ! 1/2 1 Y dχl 2πi ε Dχ = A = ~ A A m 0 −∞ l=1 N−1 Z T 1 X dt χ˙ 2 = (χ −χ )2 ε j+1 j 0 j=0 N−1 ! m N/2Z ∞ Y im F(T) = dχ exp χ M χ 2πi ε l 2 ε j jk k ~ −∞ l=1 ~ 16 / 169 Gaussian integration a multivariate Gaussian integral remains N−1 ! m N/2 Z ∞ Y im F(T) = dχ exp χ M χ 2πi ε l 2 ε j jk k ~ −∞ l=1 ~ where M is a symmetric (N − 1) × (N − 1) matrix 2 −1 0 0 ··· −1 2 −1 0 ··· M = 0 −1 2 −1 ··· . . .. Gaussian integrals of symmetric matrix A easily evaluated n ! 1/2 Z ∞ Y πn dχ exp −χ A χ = i j jk k det A −∞ i=1 result: m 1/2 F(T) = 2πi~ε det M 17 / 169 Determinant evaluation now need to compute det(M) consider n × n matrix Bn of form 2b −b 0 0 ··· −b 2b −b 0 ··· Bn = 0 −b 2b −b ··· .