Section 4: The Quantum Scalar Field

Transcription

1 Physics Section 4: The Quantum Scalar Field February 2012 c 2012 W. Taylor Section 4: Quantum scalar field 1 / 19

2 4.1 Canonical Quantization Free scalar field equation (Klein-Gordon) ( µ µ + m 2 ) φ(x) = 0 Why scalar fields? Simpler than spin 1 2 (e,...) or spin 1 (γ, W,,...) (no indices) May observe scalar at LHC in next few years (Higgs) Scalar field useful in inflation models Scalar fields may arise from fundamental theory Klein-Gordon equation in momentum space ( ( p 2 + m 2 )) φ(t, p) = 0 Equation describes many decoupled SHO s c 2012 W. Taylor Section 4: Quantum scalar field 2 / 19

3 Classical free scalar field L = 1 2 µφ µ φ 1 2 m2 φ 2 π( x) = L = φ( x) {φ( x), φ( x) π( y)} = δ 3 ( x y) Quantum scalar field: need operators [φ( x), π( y)] = iδ 3 ( x y) Write in terms of Fourier modes φ(p) = φ ( p) Notation: so we write /d k p = dk p (2π) k φ( x) = d 3 p (2π) 3 ei p x φ( p), /δ k ( p q) = (2π) k δ (k) ( p q) φ( x) = /d 3 p e i p x φ( p) c 2012 W. Taylor Section 4: Quantum scalar field 3 / 19

4 Expansion in mode operators φ( x) = /d 3 p e i p x φ( p) Each mode φ( p) acts as an independent SHO. Recall that for the standard SHO x = 1 2ω (a + a ), so for each mode we write where a ±p = a (p) r φ( p) = 1 p 2ωp (a p + a p), ω p = p p 2 + m 2 ± ia (p) i. [notation: drop vector on ω p, a p, a p,...] φ( x) = /d 3 1 p p (a p + a p) e i p x 2ωp For the standard SHO p = i p ω 2 (a a ) π( x) = /d 3 p i r ωp «(a p a 2 p) e i p x Commutators [φ( x), π( y)] = iδ (3) ( x y) [a p, a p ] = /δ 3 ( p p ) Similarly, [φ( x), φ( y)] = [π( x), π( y)] = 0 c 2012 W. Taylor Section 4: Quantum scalar field 4 / 19

5 Hamiltonian for free scalar field H = d 3 x 1 2 [π2 ( x) + ( φ( x)) 2 + m 2 φ 2 ( x)] = = /d 3 p ω p ( 1 2 apa p a pa p) /d 3 p ω p (a pa p [ap, a p]) system of many SHO s. R Drop ( ) ground state energy 1 2 [ap, a p] irrelevant without GR (cosm. constant) Normal ordering convention: : aa : = a a (a s on right.) H = /d 3 p N p ω p, N p = a pa p P i = π iφ = /d 3 p N p p i a p creates a particle excitation with momentum p energy E p = ω p = p p 2 + m 2 [a p, a q] = 0 Bose-Einstein statistics (bosons) c 2012 W. Taylor Section 4: Quantum scalar field 5 / 19

6 Hilbert Space Hilbert space H = ph p, H p = SHO Hilbert space. H = Fock Space Vacuum state is 0, with a p 0 = 0 General state: a p 1 a p 2 a p k 0 is a k-particle state ( p i: not necessarily distinct) We can decompose H into H = H 0 H 1 H 2 H 3 H 0 = { 0 }, vacuum H 1 = {a p 0 }, one-particle states H 2 = {a pa q 0 }, two-particle states... Normalization of ground state: 0 0 = 1 c 2012 W. Taylor Section 4: Quantum scalar field 6 / 19

7 One-particle states Lorentz invariant measure: d 3 p (2π) 3 2ω p Proof: use invariant δ function on hyperboloid d 4 p (2π) 4 2πδ(p2 m 2 ) θ(p 0) = /d 3 p dp 0 δ((p 0 + ω p)(p 0 ω p)) θ(p 0) = /d 3 p dp0 2p 0 δ(p 0 ω p) = /d 3 p( 1 2ω p ) is Lorentz invariant 2E p /δ 3 ( p q) is also Lorentz invariant Normalization of one-particle states p = p 2E pa p 0 p p = 2E p /δ 3 ( p p ) so φ( x) 0 = p : state with 1 particle, /d 3 p 1 2E p e i p x p p µ = (E p, p) represents a state with 1 particle at position x c 2012 W. Taylor Section 4: Quantum scalar field 7 / 19

8 Heisenberg picture From e iht a p e iht = a p e iept φ(t, x) = e iht φ( x) e iht e iht a p e iht = a p e iept we have (x a four-vector) φ(x) = /d 3 1 p p (a p e ip x + a p e ip x ) 2Ep φ is a field operator. Note association: creation operators a : e iωt, ω > 0 annihilation operators a : e iωt, ω > 0 (explains E states: associated with a operators ( holes )) [HW : complex KG field] Time derivatives: φ(t, x) = π(t, x), φ(t, x) = ( 2 m 2 ) φ(t, x) [ d2 brings down ( E 2 ); can also see from i dt 2 t φ = [ φ, H])] c 2012 W. Taylor Section 4: Quantum scalar field 8 / 19

9 4.2 Causality and Propagators QM: Time-ordered propagator D QM(t, t ) = D F(t, t ) = T{x(t)x(t )} Time-ordering not well-defined in SR Given y 0 > x 0, (x y) 2 < 0, Λ : x = Λx, y = Λy, with (x ) 0 > (y ) 0. x x y y Scalar field propagator: Compute D(x y) = 0 φ(x)φ(y) 0 = = /d 3 p 1 2E p e ip(x y) /d 3 p /d 3 q e ip x+iq y p 2Ep 2E q 0 a pa q 0 D(x y) 0 for (x y) 2 < 0 [HW] Problem with causality? c 2012 W. Taylor Section 4: Quantum scalar field 9 / 19

10 Feynman propagator (scalar field) Recall, [φ( x), φ( y)] = 0 for x y. For (x y) 2 < 0, [φ(x), φ(y)] = 0 [φ(x), φ(y)] 0 = D(x y) D(y x) = 0 follows by Lorentz invariance of D since (x y) (y x) causality OK! Define Feynman Propagator j D(x y), x 0 > y 0 D F(x y) = 0 T{φ(x)φ(y)} 0 = D(y x), y 0 > x 0 Well-defined since D(x y) = D(y x) when (x y) 2 < 0 Most useful propagator for perturbation theory Green s function: From we have D F(x y) = θ(x 0 y 0 ) 0 φ(x)φ(y) 0 + θ(y 0 x 0 ) 0 φ(y)φ(x) 0 ( 2 x + m 2 )D F(x y) = ( 0δ(x 0 y 0 )) 0 [φ(x), φ(y)] δ(x 0 y 0 ) 0 [ φ(x), φ(y)] 0 = iδ 4 (x y) So D F(x y) is Green s function for Klein-Gordon equation c 2012 W. Taylor Section 4: Quantum scalar field 10 / 19

11 Feynman propagator in momentum space Just like the SHO D F(x y) = /d 4 i p e ip (x y) p 2 m 2 + iɛ Poles at ±p 0 = ± p p 2 + m 2 iɛ /d 3 p e i p ( x y) dp 0 2πi 1 0 (x 0 y 0 ) (p 0 p 0 )(p 0 + p 0 ) e ip p0 x 0 > y 0 : /d 3 p 1 e iep(x 0 y 0 )+i p ( x y) = D(x y) 2E p p0 y 0 > x 0 : /d 3 p 1 e iep(x 0 y 0 )±i p ( x y) = D(y x) 2E p p0 c 2012 W. Taylor Section 4: Quantum scalar field 11 / 19

12 Other Propagators Retarded propagator p0 D R(x y) = θ(x 0 y 0 ) 0 [φ(x), φ(y)] 0 Advantages: = 0 for (x y) 2 < 0. causality clear. But: not so useful for perturbation theory doesn t match as well with PI s. Advanced propagator p0 D A(x y) = θ(y 0 x 0 ) 0 [φ(y), φ(x)] 0 similar properties to retarded propagator c 2012 W. Taylor Section 4: Quantum scalar field 12 / 19

13 Wick s theorem Same as for Quantum Mechanics T{φ(x 1) φ(x 2n)} = X T{φ(x i1 )φ(x j1 )} T{φ(x in )φ(x jn )} sum over i k < i k+1, i k < jk General Proof: (operators: HW, P & S, ; PI: HW, Brown, Ryder, below) Write Q(x 1,... x 2n) = T{φ(x 1) φ(x 2n)} Assume without loss of generality t i > t i+1 φ(x) = R dα (f x(α)a α + g x(α)a α) Q(x 1,... x 2n) = dα 1 dα 2n 0 (f x1 (α 1)a α1 + g x1 (α 1)a α 1 ) (f x2n (α 2n)a α2n + g x2n (α 2n)a α 2n ) 0 Compute by moving each a α to right nonzero terms from commutators. Q(x 1,..., x 2n) = P 2n k=2 Q(x1, x k) Q(x 2,..., x k 1, x k+1,..., x 2n) results by induction. c 2012 W. Taylor Section 4: Quantum scalar field 13 / 19

14 4.3 Path Integrals Gaussian integrals dx e a 2 x2 = r 2π a dx x 2n e a 2 x2 = dx x 2 e a 2 x2 = 1 a r (2n 1)!! 2π a n a r 2π a Wick s theorem (2n 1) (2n 3) 1 Generating function (j) = r dx e 2 a x2 +xj 2π = a e j 2 2a x 2n = R x 2n e a 2 x2 R e a 2 x2 = 2n j (j) j=0 = j j... j e j2 2a j=0 (0) = # of pairings 1 a n = (2n 1)!! 1 a n c 2012 W. Taylor Section 4: Quantum scalar field 14 / 19

15 Matrices x = (x 1,..., x N), Dx = dx 1dx 2... dx N Dx e 1 2 x ia ij x j = (2π) N 2 DetA, A symmetric, positive definite Generating function: use J = (J 1, J 2,..., J N) [J] = Dx e 1 2 x A x+j x = (2π) N 2 e 1 2 J A 1 J DetA so Wick: x ix j = Ji Jj e 1 2 J A 1 J J=0 = (A 1 ) ij x k1 x k2n = Jk1 Jk2n [J] J=0 / [0] = Complex variables dz d z e αz z = 2i (z z) n = dx R (z z) n e αz z R e αz z X dy e α(x2 +y 2) = 2πi α i m<i m+1,i m<j m (A 1 ) ki1 k j1 (A 1 ) kin k jn = n! αn Wick again: n n! z n z c 2012 W. Taylor Section 4: Quantum scalar field 15 / 19

16 Scalar field in 4D S[φ] = 1 2 = D[φ(x)] e is[φ] d 4 x ( µφ µ φ m 2 φ 2 ) = 1 2 d 4 x φ(x) ( + m 2 ) φ(x) where = µ µ. Formally: φ(x)φ(y) = i ( + m 2 ) 1 (x,y) fixes propagator up to BC s. (pole prescription in p space) ( + m 2 ) φ(x)φ(y) = iδ(x y) Want 0 e iht 0 Pick out ground state: t t(1 iɛ) Equivalent to (assuming mass gap): p 2 + m 2 p 2 + m 2 iɛ [H H(1 iɛ)] So i D F(x y) = [ ( + m 2 iɛ) ] (x,y) = /d 4 p i p 2 m 2 + iɛ e ip (x y) c 2012 W. Taylor Section 4: Quantum scalar field 16 / 19

17 Less formally, discretize p. periodic coordinates L 2 t, x, y, z L 2 : pµ p µ (n) = 2πnµ L, n0,1,2,3 φ(p) φ n, /d k p X n 1 L k, φ n = φ n /δk (p + q) L k δ k n+m,0 S = 1 d 4 x φ(x) ( + m 2 iɛ) φ(x) = X [( 2πn 2 L )2 m iɛ] L 4 φn φ n n µ 4 /d 4 p [p 2 m 2 + iɛ] φ(p) φ( p) R e is dimensional complex Gaussian (iɛ convergent) il 4 i φ nφ m = p 2 (n) m2 + iɛ δ4 n+m cts φ(p)φ(q) = p 2 m 2 + iɛ /δ4 (p + q) D F(x y) = φ(x)φ(y) = /d 4 i p e ip (x y) p 2 m 2 + iɛ c 2012 W. Taylor Section 4: Quantum scalar field 17 / 19

18 Wick again: φ(x 1)... φ(x 2n) = R D[φ] φ(x1)... φ(x 2n) e is R D[φ] e is = X φ(x i1 )φ(x j1 )... φ(x in )φ(x jn ) Generating functional for correlators [J(x)] = Dφ e i R d 4 x[l[φ, φ]+j(x)φ(x)] = 0 e 2 1 R d 4 x d 4 y J(x)DF (x y)j(y) Two-point function Four-point function δ φ(x)φ(y) = ( i δj(x) )( i δ )[J] J=0/0 = DF(x y) δj(y) φ(x)φ(y)φ(z)φ(w) = δ δ δ δ δj(x) δj(y) δj(z) δj(w) [J] J=0/0 = D F(x y) D F(z w) + D F(x z) D F(y w) + D F(x w) D F(y z) x y x y x y etc.... z w z w z w c 2012 W. Taylor Section 4: Quantum scalar field 18 / 19

19 Comments on path integral for free scalar field Connection to statistical mechanics: t ix 0 dt 2 d x 2 (dx 0 ) 2 d x 2 Statistical mechanics partition function J External magnetic field Further discussion: Peskin & Schroeder 9.3, KB lectures J acts as a source L L + Jφ Classical equation of motion ( + m 2 ) φ(x) = J(x) e is e S Euclidean H H Jφ ( + m 2 ) φ(x) = ( + m 2 ) i δ[j] «1 0 δj(x) = ( + m 2 ) d 4 y i D F(x y) J(y) = J(x) Note though: BC at t = +, t = nontrivial For trivial as t, use retarded propagator c 2012 W. Taylor Section 4: Quantum scalar field 19 / 19