Mandl and Shaw reading assignments

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1 Mandl and Shaw reading assignments Chapter 2 Lagrangian Field Theory 2.1 Relativistic notation 2.2 Classical Lagrangian field theory 2.3 Quantized Lagrangian field theory 2.4 Symmetries and conservation laws Problems; Chapter 3 The Klein-Gordon Field 3.1 The real Klein-Gordon field 3.2 The complex Klein-Gordon field 3.3 Covariant commutation relations 3.4 The meson propagator Problems;

2 Chapter 3 The Klein-Gordon Field 3.1 The Real Klein-Gordon Field 3.2 The Complex Klein-Gordon Field 3.3 Covariant Commutation Relations 3.4 The Meson Propagator 3.2 THE COMPLEX KLEIN-GORDON FIELD Notations ħ = 1 and c = 1 (usually) See Section 6.1. x = ( t, x) or x α = (x 0, x 1, x 2, x 3 ) Relativistic notations x μ = {x 0, x 1, x 2, x 3 } = {ct, x, y, z} x μ = {x 0, x 1, x 2, x 3 } = {ct, x, y, z} = g μν x ν g μν = g μν = DIAG(1, 1, 1, 1) A B = g μν A μ B ν = g μν A μ B ν = A 0 B 0 A B (4D vec. product) (3D vec. product) φ,α = φ / x α and φ,α = φ / x α = g αβ φ, β Lagrangian field theory = 1/2 ( φ,α φ,α μ 2 φ φ ) μ = mc/ħ = m ( + μ 2 ) φ = 0 and ( + μ 2 ) φ = 0 π(x) = φ / t and π (x) = φ / t Equal-time commutation relations [ φ(t,x), φ(t,x ) ] = 0 [ φ(t,x), π(t,x )] = iħc 2 δ 3 ( x x ) etc. 2

3 Fourier expansion φ(x) = N ( a k e ik.x + b k e+ik.x ) k where N = ( 1 /2Ωω) 1 /2 and k.x = k μ x μ = ω t k.x = k 0 x 0 k i x i (very flexible but involved notations! ) Creation and annihilation operators [ a k, a k ] = 0 and [ a k, a k ] = δ k,k [ b k, b k ] = 0 and [ b k, b k ] = δ k,k [ a k, b k ] = 0 and [ a k, b k ] = 0 Important operators P α = (H/c,P) = the 4-momentum operator P α = See 3.12 and 3.13; H = H (π,φ) d 3 x P α = k ħ k α ( N a (k) + N b (k) ) Q = the charge operator Q = See 3.31 ; NEEDS ELECTROMAGNETISM TO INTERPRET Q = q k ( N a (k) N b (k) ) opposite charges Real (or, hermitian) field operators We could define φ 1 (x) and φ 2 (x) by φ 1 = 1/ 2 ( φ + φ ) and φ 2 = i / 2 ( φ φ ) ; these imply φ = 1/ 2 ( φ 1 + i φ 2 ) and φ = 1/ 2 ( φ 1 i φ 2 ). i.e., there are two different mesons The complex scalar field is equivalent to a particle ( a ) and antiparticle ( k b ) k theory of two real fields 3

4 3.3 COVARIANT COMMUTATION RELATIONS CALCULATION (Set ħ = 1 and c = 1.) Mandl and Shaw: define then That is, For a real scalar field, [ φ(x), φ(y) ] = i ħc Δ( x y ); ( x means x μ ) Δ(ξ) = c d 3 k/(2π) 3 sin(k.ξ). Δ(ξ μ ) = c d 3 k/(2π) 3 sin(ωξ 0 k.ξ). where ω=c ( k 2 +μ 2 ). ω k Here let s consider the complex scalar field. Define [ φ(x), φ (y) ] = i ħc Δ( x y ). ω k [ φ(x), φ (y) ] = N N [ (a k e ik.x + b k e+ik.x ), (a k e+ ik.y + b k e ik.y ) ] = N N { δ(k,k ) e ik.(x y) δ(k,k ) e +ik.(x y) } = N 2 ( 2i) sin( k.(x y) ) = d 3 k/(2π) 3 1/(2ω) ( 2i) sin( k.(x y) ) = i Δ( x y ) where again sin(k.ξ) Δ(ξ) = d 3 k/(2π) 3. k.ξ = ωξ 0 k.ξ φαμħ π δ Ωω 4 ω k

5 COVARIANCE OF THE COMMUTATION RELATION (ħ=1 and c=1) THEOREM. Δ(ξ) = i/(2π) 3 d4k δ(k.k μ 2 ) ε(k 0 ) e ik.ξ where ε(k 0 ) = ± 1 for k 0 0. PROOF. δ(k 2 μ 2 ) = δ[ (k 0 ) 2 ω 2 ] So, = δ[ (k 0 ω )(k 0 + ω ) ] = δ[ k 0 ω ] /2ω + δ[ k 0 + ω ] /2ω [sign fun.] = 1 /(2ω) { δ[ k 0 ω ] + δ[ k 0 + ω ] } Δ(ξ) = i/(2π) 3 d 3 k 1/(2ω) e i ( ωt k.ξ) = i/(2π) 3 d 3 k /(2ω) e i ( ωt k.ξ) + i/(2π) 3 d 3 k /(2ω) e i ( ωt+k.ξ) = i/(2π) 3 d 3 k /(2ω) e i ( ωt k.ξ) + i/(2π) 3 d 3 + i ( ωt k.ξ) k /(2ω) e = + i d 3 k/(2π) 3 /(2ω) 2i sin( ωt k.ξ ) = d 3 k/(2π) 3 sin(k.ξ). Q.E.D. The theorem shows that [ φ(x), φ(y) ] is covariant : because the integral on the right-hand side is manifestly covariant. Microcausality [ φ(x μ ), φ(y μ ) ] = 0 if ( x y ) 2 < 0, i.e., if x and y have spacelike separation. i/(2π) 3 d 3 k ( 1)/(2ω) e i ( ωt k.ξ) Proof: covariance implies we can set x 0 = y 0 ; and the ETCR is 0. 5 ω k

6 Calculate the commutator function 6

7 τ 2 = m 2 (x 1 x 2 ) 2 7

8 3.4 THE MESON PROPAGATOR Mandl and Shaw: for a real scalar field, define <0 T { φ(x) φ(y) } 0> = i ħc Δ F (x y); then Δ F (ξ) = CF d 4 k/(2π) 4, e ik.x (k 2 μ 2 ) where the contour CF is appropriately chosen. Let s consider instead the complex field. Define the propagator by <0 T { φ(x) φ (y) } 0> = i ħc Δ F (x y). Now derive the function Δ F (x y). There are several ways to derive the result. Define Note that i ħc Δ + (x y) = <0 φ(x) φ (y) 0> i ħc Δ (x y) = <0 φ (y) φ(x) 0> i ħc ( Δ + (x y) + Δ (x y) ) = <0 [ φ(x), φ (y) ] 0> = <0 i ħc Δ(x y) 0> = i ħc Δ(x y) Δ + (x y) + Δ (x y) = Δ(x y) 8

9 Now the propagator... <0 T { φ(x) φ (y) } 0> = i ħc Δ F (x y) = <0 θ(x 0 y 0 ) φ(x) φ (y) + θ(y 0 x 0 ) φ (y) φ(x) 0> FOUR FUNCTIONS (check these eqs. in Section 3.4) Δ(ξ) = d 3 k/(2π) 3 sin(k.ξ) ω k = θ(x 0 y 0 ) i ħc Δ + (x y) + θ(y 0 x 0 ) ( i ħc) Δ (x y) Δ + (ξ) = d 3 k/(2π) 3 Δ (ξ) = d 3 k/(2π) 3 i exp( i k.ξ) 2 ω k ( i) exp( ik.ξ) 2 ω k Δ F ( x y ) = θ(x 0 y 0 ) Δ + ( x y ) θ(y 0 x 0 ) Δ ( x y ) Δ F (ξ) = d 3 k/(2π) 3 i 2 ω k { θ(ξ 0 ) exp( i k.ξ) + θ( ξ 0 ) exp( i k.ξ) } 9

10 THEOREM Δ F (ξ) = d 4 k (2π) 4 i k.ξ e k 2 μ 2 + i ε This is the crucial equation for understanding Feynman diagrams. Prove the theorem next time. Homework Problems due Friday Feb 10 Problem 16. Equal time commutation relations. We have, in the Schroedinger picture, [ ψ(x), ψ (x ) ] = δ 3 (x x ), [ ψ(x), ψ(x ) ] = 0, etc. (a ) Show that in the Heisenberg picture, this commutation relation holds at all equal times. (b ) What is the commutation relation for different times? Problem 17. (a ) Do problem 2.1 in Mandl and Shaw. (b ) Do problem 2.2 in Mandl and Shaw. (c ) Do problem 2.3 in Mandl and Shaw. Problem 18. For the free real scalar field, (A ) Write H in terms of π(x) and φ(x). (B ) Write H in terms of a k and a k. Problem 19. (A ) Mandl and Shaw problem 3.3. (B ) Mandl and Shaw problem 3.4. Problem 20. The Yukawa theory problem. 10