8.323 Relativistic Quantum Field Theory I

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1 MIT OpenCourseWare Relatvstc Quantum Feld Theory I Sprng 2008 For nformaton about ctng these materals or our Terms of Use, vst:

2 1 The Lagrangan: 8323 Lecture Notes 1: Quantzaton of the Free Scalar Feld, p1 MASSACHUSETTS INSTITUTE OF TECHNOLOGY Physcs Department 8323: Relatvstc Quantum Feld Theory I Quantzaton of the F r e e Scalar Feld February 14, , February 14, 2008 QUANTIZATION OF THE FREE SCALAR FIELD L = d 3 x L where L = 1 2 µφ µ φ 1 2 m 2 φ 2 = 1 2 φ φ φ 1 2 m 2 φ , February 14,

3 8323 Lecture Notes 1: Quantzaton of the Free Scalar Feld, p2 2 Canoncal Quantzaton: L = L(q, q,t) L p q H = p q L Schrödnger pcture: If H s ndependent of tme, q,p j ] = hδ j h ψ(t) = H ψ(t) t ψ(t) = e Ht/h ψ(0) Guth Alan Massachusetts Insttute Technology of 8323, February , 2 Hesenberg pcture: O(t) = e Ht/ h O e Ht/ h, so that ψ O(t) ψ = ψ(t) O ψ(t) Relatvstc quantum feld theory s usually done n the Hesenberg pcture: puts space and tme on equal footng 8323, February 14,

4 8323 Lecture Notes 1: Quantzaton of the Free Scalar Feld, p3 3 Feld Quantzaton by Lattce Approxmaton: Motvaton: 1) Can use canoncal formulaton exactly, nstead of as a startng pont for a natural generalzaton Note that δ j mght naturally generalze to δ 3 ( x x), but the two expressons do not even have the same unts δ j s dmensonless, whle δ 3 ( x x) has unts of 1/L 3 2) For nteractng theores, the lattce formulaton s the easest way to understand renormalzaton When strong couplng s essental, as n QCD (Quantum Chromodynamcs) and the strong nteractons, the lattce s even the best way to calculate 8323, February 14, Start wth dscrete lattce, spacng = a, and fnte volume Later take lmts a 0and V For the free theory, these lmts are trval L = L k V, k where V = a 3 and 1 L k = 2 φ 2 k 1 2 φ k φ k 1 2 m 2 φ 2 k, and the lattce dervatve s defned by φ k φ k (k,) φ k a, where k (k, ) denotes the lattce ste that s a dstance a nthe th drecton from the lattce ste k 8323, February 14,

5 8323 Lecture Notes 1: Quantzaton of the Free Scalar Feld, p4 Canoncal momenta: L p k = = φ k V φ k Defne a canoncal momentum densty: π k p k V = φ k Hamltonan: ] ] H = p k φ k L = π k φ k L k V Canoncal commutaton relatons: k φ k,φ k ] =0, p k,p k ] =0, and φ k,p k ] = hδ k k In terms of the canoncal momentum denstes, hδ k k φ k,φ k ] =0, π k,π k ] =0, and φ k,π k ] = V k 8323, February 14, Contnuum lmt: k = x π k 1 L L k L = = π( x,t)= V φ k φ k φ ( x,t) = φ ( x,t) Hamltonan becomes ] H = π k φ k L k V = k ] H = d 3 x πφ L 8323, February 14,

6 4 Revew of Smple Harmonc Oscllator: 8323 Lecture Notes 1: Quantzaton of the Free Scalar Feld, p5 Canoncal commutaton relatons: φ( x,t), φ( x,t) ] =0 and π( x,t), π( x,t) ] =0, φ k,π k ] = hδ k k V snce = φ( x,t), π( x,t) ] = hδ( x x ), k R { δ k k V V = 1 f k R 0 otherwse = x R d 3 xδ 3 ( x x) Note: Drac delta functon s defned by { d 3 xf( x) δ( x x ) f( x ) f x R x R 0 otherwse 8323, February 14, Choose m 1, so L = 1 q 2 1 ω 2 q 2, 2 2 L p = = q, q H = pq L = 1 p ω 2 q Canoncal commutaton relaton: q, p] = h Defne creaton and annhlaton operators ω a = q + ω p, a = q p, 2 h 2 hω 2 h 2 hω so that ] a, a =1 8323, February 14,

7 8323 Lecture Notes 1: Quantzaton of the Free Scalar Feld, p6 The Hamltonan can be rewrtten as wth egenstates n wth egenvalues ( ) H = hω a a + 1, 2 ( ) 1 E n = n + hω 2 Then ] H,a = hω a, H,a] = hω a, whch mples that a lowers, and a rases, the egenvalues of H by hω a 0 = 0, 8323, February 14, Normalzed nth excted state: Solve for q and p: Inthe Hesenberg pcture, 1 ( ) n n = a n, n! h ( ) hω ( q = a + a, p = a a ) 2ω 2 h ( ) ωt ωt q(t) = ae + a e 2ω and hω ( ) p(t) = q(t) = ae ωt a e ωt , February 14,

8 8323 Lecture Notes 1: Quantzaton of the Free Scalar Feld, p7 5 Quantzaton of the Scalar Feld: Contnuum lmt of the lattce: and L π( x,t)= = φ ( x,t), φ ( x,t) ] H = d 3 x πφ L, ] ] φ( x,t), φ( x,t) =0 and π( x,t), π( x,t) =0, ] φ( x,t), π( x,t) = hδ ( x x ) Use these relatons to buld the quantum theory Fourer transform: φ ( k,t) d 3 xe k x φ( x,t), and π ( k,t) d 3 xe k x π( x,t), Guth Alan Massachusetts Insttute Technology of 8323, February , 12 Fourer nverson: and φ( x,t)= π( x,t)= d 3 k (2π) 3 e k x φ( k,t) d 3 k (2π) 3 e k x π( k,t) φ( x,t)real = Hermtanquantum operator = φ( k,t)= φ ( k,t), Hesenberg equatons of moton: π( k,t)= π ( k,t) 2 φ t 2 2 φ + m 2 φ = , February 14,

9 8323 Lecture Notes 1: Quantzaton of the Free Scalar Feld, p8 Snce h = 1, obeys m has the unts of an nverse length, not a mass Fourer transform General soluton: where 2 φ ( k,t)+( k 2 + m 2 ) φ ( k,t)=0 t 2 φ ( k,t)= φ 1 ( k)e ω pt + φ 2 ( k)e ω pt, ω p = k 2 + m 2 Realty φ ( k,t)= φ ( k,t) mples φ 2 ( k)= φ 1 ( k), so Snce π = φ, φ ( k,t)= φ 1 ( k)e ω pt + φ 1 ( k)e ω pt π ( k,t)= ω p φ 1 ( k)e ω pt + ω p φ 1 ( k)e ω pt 8323, February 14, These two equatons can be solved smultaneously to gve: φ 1 ( k)= 1 φ( 2 k,t)+ ] π( k,t) e ω pt ω p = 1 ] d 3 xe ( k x ω p t) φ( x,t)+ π( x,t) 2 ω p Usng canoncal commutaton relatons, ] 1 φ 1 ( d 3 d 3 ( k x ω k t) ( q y ω q t) k), φ 1 ( q ) = x ye e 4 ] φ( x,t)+ π( x,t), φ( y,t)+ π( y,t) ω k ω q 1 d 3 d 3 ( k x ω k t) ( q y ω q t) = x ye e 4 { } hδ 3 ( x y ) hδ 3 ( x y ) ω q ω k, = 15

10 8323 Lecture Notes 1: Quantzaton of the Free Scalar Feld, p9 Usng canoncal commutaton relatons, φ 1 ( ] k ), φ 1 ( q ) = 1 4 = 1 4 = h 4 d 3 x φ( x,t)+ d 3 x d 3 ye ( k x ω k t) e ( q y ω q t) ω k π( x,t), φ( y,t)+ d 3 ye ( k x ω k t) e ( q y ω q t) { ω q hδ 3 ( x y ) ] π( y,t) ω q } hδ 3 ( x y ) ω k d 3 xe ( ( k + q ) x (ω k +ω q )t ) { 1 ω k 1 ω q } 8323, February 14, But d 3 xe ( k + q ) x =(2π) 3 δ 3 ( k + q ), whch requres q = k, but thenthe factor ncurly brackets vanshes Thus, φ 1 ( ] k), φ 1 ( q ) =0, as expected f φ 1 ( k) s proportonal to an annhlaton operator 8323, February 14,

11 8323 Lecture Notes 1: Quantzaton of the Free Scalar Feld, p10 Smlarly, φ 1 ( ] k), φ 1 ( q ) = h 4 ( ) { d 3 xe ( k q ) x (ω k ω q )t } ω k ω q = h 4 (2π)3 δ 3 ( k q ) 2 ω k = h 2ω k (2π) 3 δ 3 ( k q ) Contnuum normalzaton conventon: Defne a( k)= so that 2ωk h φ 1 ( k) a( ] k), a( q ) =0, a( ] k ), a ( q ) =(2π) 3 δ 3 ( k q ), 8323, February 14, Then φ( x,t)= d 3 k h { a( (2π) k ) e ( k x ω k t) + a ( } k )e ( k x ω k t) 3 2ω k Note: n 2nd term, I changed varables of ntegraton k k, Canoncal momentum densty: π( x,t)= φ( x,t) d 3 k hωk = (2π) 3 2 { a( k) e ( k x ω k t) a ( } k)e ( k x ω k t) 8323, February 14,

12 8323 Lecture Notes 1: Quantzaton of the Free Scalar Feld, p11 Creaton and annhlaton operators: a( k)= ωk 2 h d 3 xe ( k x ω p t) φ( x,t)+ ] π( x,t) ω p a ( k)= ωk 2 h d 3 xe ( k x ω p t) φ( x,t) ] π( x,t) ω p 8323, February 14,