1/89J urgen¨ R. Reuter Theoretical Particle Physics DESY, 07/2015 Introduction to Theoretical Particle Physics Jurgen¨ R. Reuter DESY Theory Group Hamburg, 07/2015 2/89J urgen¨ R. Reuter Theoretical Particle Physics DESY, 07/2015 Literature I Georgi: Weak Interactions and Modern Particle Physics, Dover, 2009 I Quigg: Gauge Theories of the Strong, Weak, and Electromagnetic Interactions, Perseus 1997 I Peskin: An Introduction to Quantum Field Theory Addison Wesly, 1994 I Weinberg, The Quantum Theory of Fields, Vol. I/II/(III) Cambridge Univ. Press, 1995-98 I Itzykson/Zuber, Quantum Field Theory, McGraw-Hill, 1980 I Bohm/Denner/Joos,¨ Gauge Theories of Strong and Electroweak Interactions, Springer, 2000 I Kugo, Eichtheorie, Springer, 2000 (in German) I ...and many more 3/89J urgen¨ R. Reuter Theoretical Particle Physics DESY, 07/2015 ”I have nothing to offer but blood, toil, tears and sweat.” 4/89J urgen¨ R. Reuter Theoretical Particle Physics DESY, 07/2015 ...but it is fun, though.... 5/89J urgen¨ R. Reuter Theoretical Particle Physics DESY, 07/2015 Part I (Vorabend) From Lagrangians to Feynman Rules: Quantum Field Theory with Hammer and Anvil 6/89J urgen¨ R. Reuter Theoretical Particle Physics DESY, 07/2015 Why Quantum Field Theory? Subatomic realm: typical energies and length scales are of order • (~c) 200MeV fm use of both special relativity and quantum mechanics∼ mandatory· ) Particles (quantum states) are created and destroyed, hence particle • number not constant: beyond unitary time evolution of a single QM system Schrodinger¨ propagator (time-evolution operator) violates • microcausality Scattering on a potential well for relativistic wave equation leads to • unitarity violation Use quantized fields: can be viewed as continuos limit of QM • many-body system with many (discrete) degrees of freedom Least Action Principle leads to classical equations of motion • (Euler-Lagrange equations) @ @ S = dtL = dtd3x = d4x (φ, @ φ) @ L = L L L µ ) µ @(@ φ) @φ Z Z Z µ 7/89J urgen¨ R. Reuter Theoretical Particle Physics DESY, 07/2015 What kind of fields? Classical wave equations must be Lorentz-covariant • Action and Lagrangian (density) are Lorentz scalars • Fields classified according to irreps of Lorentz group • Simplest case: Lorentz scalars (real/complex), Klein-Gordon equation • = (@ φ)∗(@µφ) m2 φ 2 ( + m2)φ (@2 ~ 2 + m2)φ = 0 L µ − j j ) ≡ t − r Spin 1/2 particles: Dirac equation • = iΨ(i=@ + m)Ψ (i=@ m)Ψ = 0 where =a a γµ L ) − ≡ µ 0 σµ γµ = σµ = (1; ~σ)σ ¯µ = (1; ~σ) γµ; γν = 2ηµν 1 σ¯µ 0 − f g γ5 = iγ0γ1γ2γ3 is the chiral projector: = 1 (1 γ5) PL=R 2 ∓ 8/89J urgen¨ R. Reuter Theoretical Particle Physics DESY, 07/2015 The Dawn of gauge theories Spin 1 particles: Maxwell’s equations • 1 = F F µν with F = @ A @ A +g [A ;A ] L −4 µν µν µ ν − ν µ µ ν ) Invariant under (local) gauge transformation, G: 1 A A0 = GA G−1 (@ G)G−1 µ −! µ µ − g µ Electric and magnetic fields are defined via: _ E~ = A~ ~ A g A ; A~ 0 0 µν µν − − r − @µF = 0 i [Aµ;F ] B~ = ~ A~ g A~ h; A~ i − ! r × − 2 × h i µ0 0 = DµF = ~ E~ + g A~ ; E~ r · i 8 0 = D F µi = E_ i + (~h B~ )ii g A ;Ei + g A~ ; B~ < µ − r × − 0 × h i : 8/89J urgen¨ R. Reuter Theoretical Particle Physics DESY, 07/2015 The Dawn of gauge theories Spin 1 particles: Maxwell’s equations • 1 = F F µν with F = @ A @ A +g [A ;A ] L −4 µν µν µ ν − ν µ µ ν ) Invariant under (local) gauge transformation, G: 1 A A0 = GA G−1 (@ G)G−1 µ −! µ µ − g µ Electric and magnetic fields are defined via: _ E~ = A~ ~ A g A ; A~ 0 0 µν µν − − r − @µF = ig [Aµ;F ] B~ = ~ A~ g A~ h; A~ i − ! r × − 2 × h i µ0 0 = DµF = ~ E~ + g A~ ; E~ r · i 8 0 = D F µi = E_ i + (~h B~ )ii g A ;Ei + g A~ ; B~ < µ − r × − 0 × h i : 9/89J urgen¨ R. Reuter Theoretical Particle Physics DESY, 07/2015 ...for the curious... Spin 3/2 particles: Rarita-Schwinger equation • 1 1 = µνρσΨ γ γ @ Ψ mΨ [γµ; γν ]Ψ L 2 µ 5 ν ρ ν − 4 µ ν ) m (=@γν Ψ γν =@Ψ ) = 0 ν − ν γµΨ = 0 @µΨ = 0 (i=@ m)Ψ = 0 µ µ − µ Leads only to sensible theory in supergravity Spin 2 particles: de Donder equation • 4πG = N det g R with R = Rµ ν and L − c4 j j µ ν p Rρ = @ Γρ @ Γρ + Γρ Γλ Γρ Γλ σµν µ νσ − ν µσ µλ νσ − νλ µσ Γρ = 1 gρσ (@ g @ g @ g ) define g = η + 16πG h µν 2 µ νσ − ν µσ − ρ µν µν µν N µν h 1 η hρ = @ @ρ h 1 η hσ + (µ pν) µν − 2 µν ρ µ ρν − 2 ρν σ $ 10/89J urgen¨ R. Reuter Theoretical Particle Physics DESY, 07/2015 How to quantize a field (Analogous to QM) Field and its canonically conjugate momentum: π = @ =@φ_ • L Canonically transform to the Hamiltonian of the system: • @ 2 = π L 1 π2 + 1 ~ φ + 1 m2φ2 H _ − L ! 2 2 r 2 @φ φ_=φ_(φ,π) Impose equal-time commutation relations: • [φ(~x); π(~y)] = iδ3(~x ~y)[φ(~x); φ(~y)] = [π(~x); π(~y)] = 0 − Creation and annihilation operators to diagonalize the Hamiltonian: • y 3 3 0 [a ; a 0 ] = 2E (2π) δ (~p ~p )[a ; a 0 ] = 0 p p p − ~p ~p Heisenberg/interaction picture: (field) operators time dependent • Creation/annihilation operators Fourier coefficients of field operators: • 3 −ik·x y +ik·x d k φ(x) = dk ake + ake dk 3 ≡ (2π) (2Ek) Z Z Z f f 11/89J urgen¨ R. Reuter Theoretical Particle Physics DESY, 07/2015 Hamiltonian is a sum of harmonic oscillators: • d3p = E ay a + const: H (2π)3 p p p Z Infinite constant is abandoned by normal ordering renormalization • d3p : := E : ay a : Note: 0 : : 0 = 0 H (2π)3 p p p h O i Z Creation operator creates a 1-particle state with well-defined • momentum (plane wave): ay 0 = p p j i j i Field operator creates a 1-particle state at x: φ(x) 0 = dp eip·x p • j i j i Consider a complex field: R • f −ik·x y +ik·x y −ik·x y +ik·x φ(x) = dk ake + bke ; φ (x) = dk bke + ake Z Z Field operator:f positve frequency modes (e−ik·x) havef annihilation operator for particles, negative frequency modes (e+ik·x) have creation operator for anti-particles a, b are independent, commute (independent Fourier components!) Got rid of negative energies as in classical wave equations • 12/89J urgen¨ R. Reuter Theoretical Particle Physics DESY, 07/2015 Microcausality and the Feynman propagator Amplitude for a particle to propagate from y to x: • 4 d p j~x−~yj!1 D(x y) 0 φ(x)φ(y) 0 = e−ip(x−y) e−mj~x−~yj − ≡ h i (2π)4 ∼ Z falls off exponentially outside light cone, but is non-zero Measurement is determined through the commutator: • [φ(x); φ(y)] = D(x y) D(y x) = 0 (for (x y)2 < 0) − − − − Cancellation of causality-violating effects by Feynman prescription: • I Particles propagated into the future (retarded) I Antiparticles propagated into the past (advanced) D(x y) for x0 > y0 D (x y) = − = θ(x0 y0) 0 φ(x)φ(y) 0 F − D(y x) for x0 < y0 − h i − + θ(y0 x0) 0 φ(y)φ(x) 0 0 T [φ(x)φ(y)] 0 − h i ≡ h i Feynman propagator (time-ordered product, causal Green’s function) • p y d4p i x −! D (x y) = e−ip(x−y) = F − (2π)4 p2 m2 + i Z − 13/89J urgen¨ R. Reuter Theoretical Particle Physics DESY, 07/2015 i prescription Solution of wave equation by Fourier transform: • ( + m2)φ(x) = iδ4(x y) F:T: − −! i i φ~(p) = = p2 m2+i (p0 E +i)(p0 + E i) − − p p− 2 2 0 with Ep = + ~p + m p p Prescription tells you +ip0x0 • where to propagate in time: neg. frequ.: e E + i − P +E i P − 0 0 pos. frequ.: e−ip x 13/89J urgen¨ R. Reuter Theoretical Particle Physics DESY, 07/2015 i prescription Solution of wave equation by Fourier transform: • ( + m2)φ(x) = iδ4(x y) F:T: − −! i i φ~(p) = = p2 m2+i (p0 E + i)(p0+E i) − − p p − 2 2 0 with Ep = + ~p + m p p Prescription tells you +ip0x0 • where to propagate in time: neg. frequ.: e E + i − P +E i P − 0 0 pos. frequ.: e−ip x 13/89J urgen¨ R. Reuter Theoretical Particle Physics DESY, 07/2015 i prescription Solution of wave equation by Fourier transform: • ( + m2)φ(x) = iδ4(x y) F:T: − −! i i φ~(p) = = p2 m2+i (p0 E + i)(p0+E i) − − p p − 2 2 0 with Ep = + ~p + m p p Prescription tells you +ip0x0 • where to propagate in time: neg. frequ.: e E + i − P +E i P − 0 0 pos. frequ.: e−ip x 14/89J urgen¨ R. Reuter Theoretical Particle Physics DESY, 07/2015 Quantization of the Dirac field Spin-statistics theorem (Fierz/Luders/Pauli,¨ 1939/40): • I Spin 0; 1; 2;:::: bosons ) commutators 1 3 I Spin 2 ; 2 ;:::: fermions ) anticommutators equal-time anticommutators: (x) ; y(y) = δ(~x ~y)δ • α β − αβ Field operators: • Z X s s −ipx s y s +ipx (x) = dpf apu (p)e + bp v (p)e s Z X s y s +ipx s s −ipx (x) = dpf ap u (p)e + bpv (p)e s Feynman propagator (time-ordered product, causal Green’s function) • 0 (x) (y) 0 for x0 > y0 SF (x y) = − = 0 T (x) (y) 0 − 0 (y) (x) 0 for x0 < y0 − p y d4p i(=p + m) x S (x y) = e−ip(x−y) = F − (2π)4 p2 m2 + i Z − 15/89J urgen¨ R.