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Quantization of Scalar Field Quantization of Scalar Field Wei Wang 2017.10.12 Wei Wang(SJTU) Lectures on QFT 2017.10.12 1 / 41 Contents 1 From classical theory to quantum theory 2 Quantization of real scalar field 3 Quantization of complex scalar field 4 Propagator of Klein-Gordon field 5 Homework Wei Wang(SJTU) Lectures on QFT 2017.10.12 2 / 41 Free classical field Klein-Gordon Spin-0, scalar Klein-Gordon equation µ 2 @µ@ φ + m φ = 0 Dirac 1 Spin- 2 , spinor Dirac equation i@= − m = 0 Maxwell Spin-1, vector Maxwell equation µν µν @µF = 0;@µF~ = 0: Gravitational field Wei Wang(SJTU) Lectures on QFT 2017.10.12 3 / 41 Klein-Gordon field scalar field, satisfies Klein-Gordon equation µ 2 (@ @µ + m )φ(x) = 0: Lagrangian 1 L = @ φ∂µφ − m2φ2 2 µ Euler-Lagrange equation @L @L @µ − = 0: @(@µφ) @φ gives Klein-Gordon equation. Wei Wang(SJTU) Lectures on QFT 2017.10.12 4 / 41 From classical mechanics to quantum mechanics Mechanics: Newtonian, Lagrangian and Hamiltonian Newtonian: differential equations in Cartesian coordinate system. Lagrangian: Principle of stationary action δS = δ R dtL = 0. Lagrangian L = T − V . Euler-Lagrangian equation d @L @L − = 0: dt @q_ @q Wei Wang(SJTU) Lectures on QFT 2017.10.12 5 / 41 Hamiltonian mechanics Generalized coordinates: q ; conjugate momentum: p = @L i j @qj Hamiltonian: X H = q_i pi − L i Hamilton equations @H p_ = − ; @q @H q_ = : @p Time evolution df @f = + ff; Hg: dt @t where f:::g is the Poisson bracket. Wei Wang(SJTU) Lectures on QFT 2017.10.12 6 / 41 Quantum Mechanics Quantum mechanics Hamiltonian: Canonical quantization Lagrangian: Path integral Canonical quantization Observables: operators commutation relations [qi; qj] = [pi; pj] = 0; [qi; pj] = i~δij Poisson bracket ! commutation bracket: f:::g ! 1 [:::] i~ Time evolution (Heisenberg equation) q_i = i[H; qi]; p_i = i[H; pi]: For any observable F (q; p), F_ (q; p) = i[H; F ]: Wei Wang(SJTU) Lectures on QFT 2017.10.12 7 / 41 1D harmonic oscillator (classical) Lagrangian r ! 1 m!2 K L = mq_2 − q2; ! = 2 2 m Canonical momentum @L p = = mq:_ @q_ Hamiltonian 1 H = pq_ − L = (p2 + m2!2q2) 2m Hamilton equation @H p_ = − = −m!2q; @q @H p q_ = = : @p m Wei Wang(SJTU) Lectures on QFT 2017.10.12 8 / 41 1D harmonic oscillator (quantum) equal-time commutation relation [q; p] = i; [q; q] = [p; p] = 0 equation of motion p_ = i[H; p] = −Kq; p q_ = i[H; q] = : m raising and lowering operators r 1 r 1 a = (p − im!q); ay = (p + im!q): 2m! 2m! [a; ay] = 1; [a; a] = 0; [ay; ay] = 0: Wei Wang(SJTU) Lectures on QFT 2017.10.12 9 / 41 particle-number representation Hamiltonian: 1 H = !(aay + ) 2 particle-number operator 1 N = aay;H = !(N + ); 2 1 Njni = njni;Hjni = (n + )!jni 2 Vacuum state: j0i ! Nj0i = 0;Hj0i = j0i: 2 Creation and annihilation p ayjni = n + 1jn + 1i; p ajni = njn − 1i: Wei Wang(SJTU) Lectures on QFT 2017.10.12 10 / 41 From mechanics to field theory Mechanics: finite degree of freedom. field: infinite (continuum) degree of freedom. canonical coordinates: x ! φ(~x;t). canonical momentum: p ! π(~x;t) = @L(φ,∂µφ) @φ_(~x;t) Hamiltonian: H = R d3xH(π(~x;t); φ(~x;t)) = R d3x(πφ_ − L), H: Hamiltonian density. Wei Wang(SJTU) Lectures on QFT 2017.10.12 11 / 41 Analogy between mechanics and field theory Discretization: Z 1 3 φi(t) = d xφ(~x;t) ∆Vi (∆Vi) φi(t) is the average value in ∆Vi. Continuum ! denumerable. Lagrangian: Z 3 X L = d xL(φ(x);@µφ(x)) ! ∆ViL¯i(φi(t); φ_i(t); φi±s(t); ··· ): i Z 1 3 @ φ_i(t) = d x φ(x; t): ∆Vi (∆Vi) @t Wei Wang(SJTU) Lectures on QFT 2017.10.12 12 / 41 Analogy between mechanics and field theory canonical momentum: @L @L¯i pi(t) = = ∆Vi ≡ ∆Viπi(t): @φ_i(t) @φ_i(t) Hamiltonian X X H = piφ_i − L = ∆Vi(πiφ_i − L¯i): i i Canonical quantization [φi(t); pi(t)] = iδij; [φi(t); φj(t)] = [pi(t); pj(t)] = 0: Heisenberg equation φ_i(t) = i[H; φi(t)]; p_i(t) = i[H; pi(t)]: Wei Wang(SJTU) Lectures on QFT 2017.10.12 13 / 41 Continuum limit δ When ∆V ! 0, ij ! δ3(~x − ~x0) i ∆Vi commutation relation 0 3 0 [φ(t; ~x); π(t; ~x )] = i~δ (~x − ~x ); (~ = 1) [φ(t; ~x); φ(t; ~x0)] = [π(t; ~x); π(t; ~x0)] = 0; Heisenberg equation (equation of motion) φ_(~x;t) = i[H; φ(~x;t)]; π_ (~x;t) = i[H; π(~x;t)]: For any physical quantity F , F_ = i[H; F ]: Wei Wang(SJTU) Lectures on QFT 2017.10.12 14 / 41 Quantization of real scalar field Lagrangian density 1 1 L = @µφ∂ φ − m2φ2: 2 µ 2 Euler-Lagrange equation (Klein-Gordon equation) µ 2 (@ @µ + m )φ(x) = 0 canonical momentum π(x) = @L = φ_(x) @φ_(x) Hamiltonian density 1 h i 1 H = π@ φ − L = (@ φ)2 + (r~ φ)2 + m2φ2: 0 2 0 2 Wei Wang(SJTU) Lectures on QFT 2017.10.12 15 / 41 quantization of real scalar field Introducing commutation relation for φ and π [φ(~x;t); π(~x0; t)] = iδ3(~x − ~x0); [φ(~x;t); φ(~x0; t)] = [π(~x;t); π(~x0; t)] = 0 Heisenberg equation φ_(~x;t) = i[H; φ(~x;t)]; π_ (~x;t) = i[H; π(~x;t)]: Wei Wang(SJTU) Lectures on QFT 2017.10.12 16 / 41 Mode expansion Plane-wave expansion Z 3 d k ~ −ik·x y ~ ik·x φ(x) = 3 [a(k)e + a (k)e ]; (2π) 2!k p 2 2 with !k = ~k + m . For π(x; t), we have Z 3 _ d k ~ −ik·x y ~ ik·x π(x) = φ(x) = 3 (−i!k)[a(k)e − a (k)e ]; (2π) 2!k a and ay can be expressed by the field operator Z Z 3 ik·x ! y 3 −ik·x ! a(~k) = i d xe @0 φ(x; t); a (~k) = −i d xe @ 0φ(x; t) Wei Wang(SJTU) Lectures on QFT 2017.10.12 17 / 41 Commutation relation for a and ay y 0 3 3 0 [a(~k); a (k~ )] = (2π) 2!kδ (~k − ~k ); [a(~k); a(k~0)] = [ay(~k); ay(k~0)] = 0: Hamiltonian 1 Z H = d3x[π(x; t)2 + jrφ(x; t)j2 + m2φ(x; t)2] 2 Z 3 1 d k ~ y ~ y ~ ~ = 3 !k[a(k)a (k) + a (k)a(k)]: 2 (2π) 2!k Momentum Z P = − d3xπ(x; t)rφ(x; t) Z 3 1 d k ~ ~ y ~ y ~ ~ = 3 k[a(k)a (k) + a (k)a(k)] 2 (2π) 2!k Wei Wang(SJTU) Lectures on QFT 2017.10.12 18 / 41 vacuum zero point energy Z 3 1 d k ~ y ~ y ~ ~ H = 3 !k[a(k)a (k) + a (k)a(k)] 2 (2π) 2!k Z 3 d k y ~ ~ !k 3 = 3 [!ka (k)a(k) + δ (0)] (2π) 2!k 2 Z 3 d k y ~ ~ V = 3 !k a (k)a(k) + 3 : (2π) 2!k 2(2π) Z d3k V h0jHj0i = 3 !k 3 : (2π) 2!k 2(2π) The vacuum is not empty! Infinity! The infinity can be dropped (no worry). Wei Wang(SJTU) Lectures on QFT 2017.10.12 19 / 41 Normal ordering Operator ordering in quantum theory: normal (Wick), anti-normal, Weyl-Wigner,... Normal ordering : O(a; ay): ,move all ay(k) to the left of a(k). e.g., : a(~k)ay(k) :=: ay(~k)a(k) := ay(~k)a(~k): zero point energy is dropped h0j : O(a; ay): j0i = 0: Wei Wang(SJTU) Lectures on QFT 2017.10.12 20 / 41 Hamiltonian and particle number operator Define Hamiltonian by normal ordering 1 Z d3k H = ! :[a(~k)ay(~k) + ay(~k)a(~k)] : 3 k 2 (2π) 2!k Z d3k = ! ay(~k)a(~k) 3 k (2π) 2!k Momentum Z d3k P~ = ~kay(~k)a(~k): 3 (2π) 2!k four-momentum Z d3k P µ = kµay(~k)a(~k): 3 (2π) 2!k Particle number operator Z d3k N = ay(~k)a(~k): [N; P µ] = 0 for free field. 2 (2π) 2!k Wei Wang(SJTU) Lectures on QFT 2017.10.12 21 / 41 Fock space and particle interpretation Basis: all the eigenstate of N. j~ki = ay(~k)j0i; y y j~k1;~k2i = a (~k1)a (~k2)j0i; . vacuum state a(~k)j0i = 0; h0j0i = 1: y One-particle state: Pµj~ki = Pµa (~k)j0i = kµj~ki: With energy ~ 2 2 2 momentum relation jkj + m = !k. normalization 0 3 3 0 h~k j~ki = (2π) 2!kδ (~k − ~k ) Wei Wang(SJTU) Lectures on QFT 2017.10.12 22 / 41 Quantization for many real scalar fields n scalar fields φr(x; t); (r = 1; ··· ; n) @L π(x; t)r = @φ_r(x; t) Hamiltonian n X H(πr; ··· ; φr; ··· ) = πrφ_r − L; r=1 Z H = d3xH: Commutation relation 0 3 0 [φr(x ; t); πs(x; t)] = iδrsδ (x − x ); 0 0 [φr(x; t); φs(x ; t)] = [πr(x; t); πs(x ; t)] = 0: Wei Wang(SJTU) Lectures on QFT 2017.10.12 23 / 41 Quantization for many real scalar fields Heisenberg equation φ_r(x; t) = i[H; φr(x; t)]; π_ r(x; t) = i[H; πr(x; t)]: or @ φr(x) = i[Pµ; φr(x)]: @xµ solution: iP ·b −iP ·b φr(x + b) = e φr(x)e : Wei Wang(SJTU) Lectures on QFT 2017.10.12 24 / 41 Complex scalar field real scalar field: Hermitian, particle=anti-particle.
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