Quantization of Scalar Field

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1 Quantization of Scalar Field Wei Wang Wei Wang(SJTU) Lectures on QFT / 41

2 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 / 41

3 Free classical field Klein-Gordon Spin-0, scalar Klein-Gordon equation µ µ φ + m 2 φ = 0 Dirac Spin- 1 2, spinor Dirac equation i /ψ mψ = 0 Maxwell Spin-1, vector Maxwell equation µ F µν = 0, µ F µν = 0. Gravitational field Wei Wang(SJTU) Lectures on QFT / 41

4 Klein-Gordon field scalar field, satisfies Klein-Gordon equation Lagrangian ( µ µ + m 2 )φ(x) = 0. L = 1 2 µφ µ φ m 2 φ 2 Euler-Lagrange equation ( ) L µ L ( µ φ) φ = 0. gives Klein-Gordon equation. Wei Wang(SJTU) Lectures on QFT / 41

5 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 dtl = 0. Lagrangian L = T V. Euler-Lagrangian equation d L dt q L q = 0. Wei Wang(SJTU) Lectures on QFT / 41

6 Hamiltonian mechanics Generalized coordinates: q i ; conjugate momentum: p j = L q j Hamiltonian: H = i q i p i L Hamilton equations Time evolution ṗ = H q, df dt = f t q = H p. where {...} is the Poisson bracket. + {f, H}. Wei Wang(SJTU) Lectures on QFT / 41

7 Quantum Mechanics Quantum mechanics Hamiltonian: Canonical quantization Lagrangian: Path integral Canonical quantization Observables: operators commutation relations [q i, q j ] = [p i, p j ] = 0, [q i, p j ] = i δ ij Poisson bracket commutation bracket: {...} 1 i [...] Time evolution (Heisenberg equation) For any observable F (q, p), q i = i[h, q i ], p i = i[h, p i ]. F (q, p) = i[h, F ]. Wei Wang(SJTU) Lectures on QFT / 41

8 1D harmonic oscillator (classical) Lagrangian L = 1 2 m q2 mω2 2 q2, ( ω = ) K m Canonical momentum p = L q = m q. Hamiltonian Hamilton equation H = p q L = 1 2m (p2 + m 2 ω 2 q 2 ) ṗ = H q = mω2 q, q = H p = p m. Wei Wang(SJTU) Lectures on QFT / 41

9 1D harmonic oscillator (quantum) equal-time commutation relation equation of motion [q, p] = i, [q, q] = [p, p] = 0 ṗ = i[h, p] = Kq, q = i[h, q] = p m. raising and lowering operators 1 1 a = 2mω (p imωq), a = (p + imωq). 2mω [a, a ] = 1, [a, a] = 0, [a, a ] = 0. Wei Wang(SJTU) Lectures on QFT / 41

10 particle-number representation Hamiltonian: particle-number operator H = ω(aa ) N = aa, H = ω(n ), N n = n n, H n = (n )ω n Vacuum state: 0 N 0 = 0, H 0 = ω 2 0. Creation and annihilation a n = n + 1 n + 1, a n = n n 1. Wei Wang(SJTU) Lectures on QFT / 41

11 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 = d 3 xh(π( x, t), φ( x, t)) = d 3 x(π φ L), H: Hamiltonian density. Wei Wang(SJTU) Lectures on QFT / 41

12 Analogy between mechanics and field theory Discretization: φ i (t) = 1 d 3 xφ( x, t) V i ( V i ) φ i (t) is the average value in V i. Continuum denumerable. Lagrangian: L = d 3 xl(φ(x), µ φ(x)) i φ i (t) = 1 d 3 x φ(x, t). V i ( V i ) t V i Li (φ i (t), φ i (t), φ i±s (t), ). Wei Wang(SJTU) Lectures on QFT / 41

13 Analogy between mechanics and field theory canonical momentum: p i (t) = L φ i (t) = V L i i φ i (t) V iπ i (t). Hamiltonian H = i p i φi L = i V i (π i φi L i ). Canonical quantization [φ i (t), p i (t)] = iδ ij, [φ i (t), φ j (t)] = [p i (t), p j (t)] = 0. Heisenberg equation φ i (t) = i[h, φ i (t)], ṗ i (t) = i[h, p i (t)]. Wei Wang(SJTU) Lectures on QFT / 41

14 Continuum limit δ When V i 0, ij V i δ 3 ( x x ) commutation relation [φ(t, x), π(t, x )] = i δ 3 ( x x ), ( = 1) [φ(t, x), φ(t, x )] = [π(t, x), π(t, x )] = 0, Heisenberg equation (equation of motion) For any physical quantity F, φ( x, t) = i[h, φ( x, t)], π( x, t) = i[h, π( x, t)]. F = i[h, F ]. Wei Wang(SJTU) Lectures on QFT / 41

15 Quantization of real scalar field Lagrangian density L = 1 2 µ φ µ φ 1 2 m2 φ 2. Euler-Lagrange equation (Klein-Gordon equation) canonical momentum π(x) = Hamiltonian density H = π 0 φ L = 1 2 ( µ µ + m 2 )φ(x) = 0 L = φ(x) φ(x) [ ( 0 φ) 2 + ( φ) 2] m2 φ 2. Wei Wang(SJTU) Lectures on QFT / 41

16 quantization of real scalar field Introducing commutation relation for φ and π [φ( x, t), π( x, t)] = iδ 3 ( x x ), [φ( x, t), φ( x, t)] = [π( x, t), π( x, 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 / 41

17 Mode expansion Plane-wave expansion φ(x) = (2π) 3 [a( 2ω k)e ik x + a ( k)e ik x ], k with ω k = k 2 + m 2. For π(x, t), we have π(x) = φ(x) = (2π) 3 2ω k ( iω k )[a( k)e ik x a ( k)e ik x ], a and a can be expressed by the field operator a( k) = i d 3 xe ik x 0 φ(x, t), a ( k) = i d 3 xe ik x 0 φ(x, t) Wei Wang(SJTU) Lectures on QFT / 41

18 Commutation relation for a and a [a( k), a ( k )] = (2π) 3 2ω k δ 3 ( k k ), [a( k), a( k )] = [a ( k), a ( k )] = 0. Hamiltonian H = 1 d 3 x[π(x, t) 2 + φ(x, t) 2 + m 2 φ(x, t) 2 ] 2 = 1 2 (2π) 3 ω k [a( 2ω k)a ( k) + a ( k)a( k)]. k Momentum P = = 1 2 d 3 xπ(x, t) φ(x, t) (2π) 3 2ω k k[a( k)a ( k) + a ( k)a( k)] Wei Wang(SJTU) Lectures on QFT / 41

19 vacuum zero point energy H = 1 2 (2π) 3 ω k [a( 2ω k)a ( k) + a ( k)a( k)] k = (2π) 3 [ω k a ( 2ω k)a( k) + ω k k 2 δ3 (0)] d 3 [ k = (2π) 3 ω k a ( 2ω k)a( k) + V ] k 2(2π) 3. 0 H 0 = V (2π) 3 ω k 2ω k 2(2π) 3. The vacuum is not empty! Infinity! The infinity can be dropped (no worry). Wei Wang(SJTU) Lectures on QFT / 41

20 Normal ordering Operator ordering in quantum theory: normal (Wick), anti-normal, Weyl-Wigner,... Normal ordering : O(a, a ) :,move all a (k) to the left of a(k). e.g., : a( k)a (k) :=: a ( k)a(k) := a ( k)a( k). zero point energy is dropped 0 : O(a, a ) : 0 = 0. Wei Wang(SJTU) Lectures on QFT / 41

21 Hamiltonian and particle number operator Define Hamiltonian by normal ordering Momentum four-momentum H = 1 2 = (2π) 3 2ω k ω k : [a( k)a ( k) + a ( k)a( k)] : (2π) 3 2ω k ω k a ( k)a( k) P = P µ = Particle number operator N = (2π) 3 2ω k ka ( k)a( k). (2π) 3 2ω k k µ a ( k)a( k). (2π) 2 2ω k a ( k)a( k). [N, P µ ] = 0 for free field. Wei Wang(SJTU) Lectures on QFT / 41

22 Fock space and particle interpretation Basis: all the eigenstate of N. vacuum state k = a ( k) 0, k 1, k 2 = a ( k 1 )a ( k 2 ) 0,. a( k) 0 = 0, 0 0 = 1. One-particle state: P µ k = P µ a ( k) 0 = k µ k. With energy momentum relation k 2 + m 2 = ω 2 k. normalization k k = (2π) 3 2ω k δ 3 ( k k ) Wei Wang(SJTU) Lectures on QFT / 41

23 Quantization for many real scalar fields n scalar fields φ r (x, t), (r = 1,, n) π(x, t) r = L φ r (x, t) Hamiltonian H(π r,, φ r, ) = H = d 3 xh. Commutation relation n π r φr L, r=1 [φ r (x, t), π s (x, t)] = iδ rs δ 3 (x x ), [φ r (x, t), φ s (x, t)] = [π r (x, t), π s (x, t)] = 0. Wei Wang(SJTU) Lectures on QFT / 41

24 Quantization for many real scalar fields Heisenberg equation or solution: φ r (x, t) = i[h, φ r (x, t)], π r (x, t) = i[h, π r (x, t)]. x µ φ r(x) = i[p µ, φ r (x)]. φ r (x + b) = e ip b φ r (x)e ip b. Wei Wang(SJTU) Lectures on QFT / 41

25 Complex scalar field real scalar field: Hermitian, particle=anti-particle. complex scalar field: particle anti-particle, e.g. π ±, K ±, etc. Lagrangian for free complex scalar field L = ( µ φ ) µ φ m 2 φ φ expressed with two real scalar field φ = φ 1 + iφ 2 2, φ = φ 1 iφ 2 2 Lagrangian in terms of φ 1 and φ 2 L = 1 2 ( µφ 1 µ φ 1 + µ φ 2 µ φ 2 ) 1 2 m2 (φ φ 2 2). Wei Wang(SJTU) Lectures on QFT / 41

26 complex scalar field Euler-Lagrange equation: Klein-Gordon equation conjugate momentum ( + m 2 )φ(x) = 0, ( + m 2 )φ (x) = 0. π = L φ = φ, π = L = φ. φ Hamiltonian H = d 3 x(π φ + π φ L) = d 3 x(π π + φ φ + m 2 φ φ). Wei Wang(SJTU) Lectures on QFT / 41

27 Quantization of complex scalar field commutation relation (and 0 for others) [φ(x, t), π(x, t)] = [φ (x, t), π (x, t)] = iδ 3 (x y), Mode expansions φ(x) = (2π) 3 [a( k)e ik x + b ( k)e ik x ], 2ω k φ (x) = (2π) 3 [a ( k)e ik x + b( k)e ik x ] 2ω k φ annihilates a and creates b. Plane wave expansion for the two real field φ i (x) = a,b can be expressed by a i as (2π) 3 2ω k [a i ( k)e ik x + a i ( k)e ik x ]. a( k) = 1 2 [a 1 ( k) + ia 2 ( k)], a ( k) = 1 2 [a 1 ( k) ia 2 ( k)], b( k) = 1 2 [a 1 ( k) ia 2 ( k)], b ( k) = 1 2 [a 1 ( k) + ia 2 ( k)]. Wei Wang(SJTU) Lectures on QFT / 41

28 Quantization of complex scalar field commutation relation for a and b (and 0 for others) [a( k), a ( k )] = [b( k), b ( k )] = (2π) 3 2ω k δ 3 ( k k ). commutation relation for φ and φ [φ(x), φ (y)] = i (x y), [φ i (x), φ j (y)] = iδ ij (x y). Number operator for a and b N a = N b = (2π) 3 2ω k a ( k)a( k), (2π) 3 2ω k b ( k)b( k). Wei Wang(SJTU) Lectures on QFT / 41

29 Quantization for complex scalar field four-momentum P µ = (2π) 3 2ω k k µ [a ( k)a( k) + b ( k)b( k)] vacuum stsate a( k) 0 = b( k) 0 = 0. U(1) symmetry of complex scalar field: invariant under U(1) transformation or φ e iα φ, φ e iα φ δφ = iδαφ, δφ = iδαφ. Wei Wang(SJTU) Lectures on QFT / 41

30 Quantization of complex scalar field Noether current conserved charge j µ = iφ µ φ, Q = d 3 xj 0 (x) = i d 3 xφ 0 φ conserved charge as an operator Q = = (2π) 2 [a ( k)a( k) b( k)b ( k)] 2ω k (2π) 3 [a ( k)a( k) b ( k)b( k)] = N a N b. 2ω k In QM, φ is interpreted as wave function, j µ is probability density. Probability can be negative! In QFT, φ is interpreted as field operator, j µ is the charge current. Can be negative. Wei Wang(SJTU) Lectures on QFT / 41

31 Propagator of Klein-Gordon field propagator of real scalar field d 3 k [φ(x), φ(y)] = (2π) 3 2ω k (2π) 3 2ω k [[a(k), ] a (k )]e ik x+ik y + [a (k), a(k )]e ik x ik y d 3 k = (2π) 3 2ω k (2π) 3 2ω k (2π) 3 2ω k δ 3 ( k k ) [e ik x+ik y e ik x ik y] = (2π) 3 (e ik (x y) e ik (x y) ) 2ω k i (x y), with ω k = k 2 + m 2. Wei Wang(SJTU) Lectures on QFT / 41

32 Propagator of Klein-Gordon field propagator of real scalar field i (x y) = (2π) 3 (e ik (x y) e ik (x y) ) 2ω k = (2π) 3 e i k ( x y) (e iω k(x 0 y 0) e iω k(x 0 y 0) ), 2ω k with ω k = then k 2 + m 2. Introducing on-shell condition δ(k 2 m 2 ), (x) = 1 i where ε(k 0 ) = k0 k 0. d 4 k (2π) 4 2πδ(k2 m 2 )ε(k 0 )e ik x, Wei Wang(SJTU) Lectures on QFT / 41

33 Propagator of Klein-Gordon field The field φ is the sum of the positive and negative frequency parts: φ(x) = φ (+) (x) + φ ( ) (x), with φ (+) (x) = φ ( ) (x) = (2π) 3 2ω k a( k)f k (x), (2π) 3 2ω k a ( k)f k (x), where f k (x) = e iω kt+i k x The Green s function (+) (x x ) = ( ) (x x ) = d 3 k (2π) 3 f k 2ω (x )f k (x) = k (2π) 3 2ω k f k (x )f k (x) = d 4 k (2π) 3 θ(k 0)δ(k 2 m 2 )e ik (x x ), d 4 k (2π) 3 θ(k 0)δ(k 2 m 2 )e ik (x x ), (x x ) = i( (+) (x x ) ( ) (x x )) = d3 k (2π) 3 sin ω k (t t ω k ) e i k ( x x ) Wei Wang(SJTU) Lectures on QFT / 41

34 Properties of Green s function (+) (x) = ( ) ( x) ( + m 2 ) (x) = 0, ( + m 2 ) (±) (x) = 0 (x) t=0 = 0, and 0 for x 2 < 0. [ ] x 0 (x) = x 0 =0 δ3 (x) [ x i (x)] x=0 = 0 ( x, x 0 ) = ( x, x 0 ) ( x, x 0 ) = ( x, x 0 ) ( x, x 0 ) = ( x, x 0 ) Wei Wang(SJTU) Lectures on QFT / 41

35 Retarded and advanced Green s functions Define retarded and advanced Green s functions R (x) = 1 2 (1 + ε(x 0)) (x) = θ(x 0 ) (x), A (x) = 1 2 (1 ε(x 0)) (x) = θ( x 0 ) (x), (x) = A (x) R (x). Represented by contour integral: (x) = c dk 0 2π e ik x (2π) 3 m 2 k 2. Wei Wang(SJTU) Lectures on QFT / 41

36 Green s function C Wei Wang(SJTU) Lectures on QFT / 41

37 Green s function For retarded and advanced Green s functions: R/A (x) = dk0 e ik x 2π (2π) 3 m 2 k 2. C R/A C R C A Wei Wang(SJTU) Lectures on QFT / 41

38 time-ordered product Dyson s time-ordered product T φ(x )φ (x) = θ(t t)φ(x )φ (x) + θ(t t )φ (x)φ(x ) satisfying ( x + m 2 )it φ(x )φ (x) = δ 4 (x x). Feynman s propagation function i F (x x) = 0 T φ(x )φ (x) 0, ( x + m 2 ) F (x x) = δ 4 (x x). Wei Wang(SJTU) Lectures on QFT / 41

39 Feynman s propagator From the definition, i F (x x) = θ(t t) 0 φ(x )φ (x) 0 + θ(t t ) 0 φ (x)φ(x ) 0 = (2π) 3 [θ(t t)e ik (x x) + θ(t t )e ik (x x) ] 2ω k d 4 k = e ik (x x) i C F (2π) 4 k 2 m 2 d 4 k = e ik (x x) i (2π) 4 k 2 m 2 + iɛ Wei Wang(SJTU) Lectures on QFT / 41

40 C F Wei Wang(SJTU) Lectures on QFT / 41

41 Homework Derive the propagator for a scalar field: d 4 p i 0 T φ(x)φ(y) 0 = e ip (x y) (2π) 4 p 2 m 2. + iɛ Peskin and Schroeder s book: Exercise 2.2 Peskin and Schroeder s book: Exercise 2.3 Prove the following identity: [P µ, a(k)] = k µ a(k), [P µ, a (k)] = k µ a (k) with P µ = (2π) 3 2ω k k µ a ( k)a( k). Wei Wang(SJTU) Lectures on QFT / 41

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