What s up with those Feynman diagrams? an Introduction to Quantum Field Theories

Transcription

1 What s up with those Feynman diagrams? an Introduction to Quantum Field Theories Martin Nagel University of Colorado February 3, 2010 Martin Nagel (CU Boulder) Quantum Field Theories February 3, / 31

2 Outline What you should take away from this lecture: What do those Feynman diagrams mean? Where do they come from? B 0 b d W u, c, t s s s d φ K 0 Some appreciation for quantum gauge theories [ ˆX λ, ˆX µ] = cλµ ν ˆX ν Û(ɛ(x)) = exp( igɛ(x) ˆX) = cλµ ν δw µ = µ ɛ(x) g[ɛ(x) W µ ] X (R) λ Martin Nagel (CU Boulder) Quantum Field Theories February 3, / 31

3 Relativistiv Quantum Mechanics i t ψ = 2 2m 2 ψ + V ψ Schrödinger equation not Lorentz-invariant can t handle massless particles spin must be inserted by hand Martin Nagel (CU Boulder) Quantum Field Theories February 3, / 31

4 Klein-Gordon Ansatz: Use relativistic energy-momentum relation: E 2 = p 2 + m 2 Klein-Gordon equation: Problems: ( + m 2 ) φ(x, t) = 0, µ µ = 2 t 2 2 Negative energy solutions E = ±(p 2 + m 2 ) 1/2 Negative probability density describes only spinless particles Martin Nagel (CU Boulder) Quantum Field Theories February 3, / 31

5 Klein-Gordon Ansatz: Use relativistic energy-momentum relation: E 2 = p 2 + m 2 Klein-Gordon equation: Problems: ( + m 2 ) φ(x, t) = 0, µ µ = 2 t 2 2 Negative energy solutions E = ±(p 2 + m 2 ) 1/2 Negative probability density can both be fixed by Feynman interpretation: negative 4-momentum corresponds to antiparticles describes only spinless particles particle-antiparticle interpretation arises naturally from a complex Klein-Gordon field φ: correct descripton of bosons Martin Nagel (CU Boulder) Quantum Field Theories February 3, / 31

6 Dirac Dirac: linear in / t and in Ansatz: ψ(x, t) i = ( iα + βm)ψ(x, t) t demand a KG-type condition: 2 ψ/ t 2 = ( 2 + m 2 )ψ which yields the following anti-commutation relations: {α i, β} = 0 {α i, α j} = 2δ ij1 β 2 = 1 Martin Nagel (CU Boulder) Quantum Field Theories February 3, / 31

7 Dirac Dirac: linear in / t and in Ansatz: ψ(x, t) i = ( iα + βm)ψ(x, t) t demand a KG-type condition: 2 ψ/ t 2 = ( 2 + m 2 )ψ which yields the following anti-commutation relations: {α i, β} = 0 {α i, α j} = 2δ ij1 β 2 = 1 α i and β are Hermitian (H = α p + βm) Trα i = Trβ = 0 Eigenvalues of α i and β are ±1 dimensionality n of α i and β is even n = 2: {σ i} and 1 form a complete set Martin Nagel (CU Boulder) Quantum Field Theories February 3, / 31

8 Dirac Dirac: linear in / t and in Ansatz: ψ(x, t) i = ( iα + βm)ψ(x, t) t demand a KG-type condition: 2 ψ/ t 2 = ( 2 + m 2 )ψ which yields the following anti-commutation relations: {α i, β} = 0 {α i, α j} = 2δ ij1 β 2 = 1 α i and β are Hermitian (H = α p + βm) Trα i = Trβ = 0 Eigenvalues of α i and β are ±1 dimensionality n of α i and β is even n = 2: {σ i} and 1 form a complete set Martin Nagel (CU Boulder) Quantum Field Theories February 3, / 31

9 Dirac Dirac: linear in / t and in Ansatz: ψ(x, t) i = ( iα + βm)ψ(x, t) t demand a KG-type condition: 2 ψ/ t 2 = ( 2 + m 2 )ψ which yields the following anti-commutation relations: {α i, β} = 0 {α i, α j} = 2δ ij1 β 2 = 1 α i and β are Hermitian (H = α p + βm) Trα i = Trβ = 0 Eigenvalues of α i and β are ±1 dimensionality n of α i and β is even n = 2: {σ i} and 1 form a complete set Martin Nagel (CU Boulder) Quantum Field Theories February 3, / 31

10 Dirac Dirac: linear in / t and in Ansatz: ψ(x, t) i = ( iα + βm)ψ(x, t) t demand a KG-type condition: 2 ψ/ t 2 = ( 2 + m 2 )ψ which yields the following anti-commutation relations: {α i, β} = 0 {α i, α j} = 2δ ij1 β 2 = 1 α i and β are Hermitian (H = α p + βm) Trα i = Trβ = 0 Eigenvalues of α i and β are ±1 dimensionality n of α i and β is even n = 2: {σ i} and 1 form a complete set Martin Nagel (CU Boulder) Quantum Field Theories February 3, / 31

11 Dirac Dirac: linear in / t and in Ansatz: ψ(x, t) i = ( iα + βm)ψ(x, t) t demand a KG-type condition: 2 ψ/ t 2 = ( 2 + m 2 )ψ which yields the following anti-commutation relations: {α i, β} = 0 {α i, α j} = 2δ ij1 β 2 = 1 α i and β are Hermitian (H = α p + βm) Trα i = Trβ = 0 Eigenvalues of α i and β are ±1 dimensionality n of α i and β is even n = 2: {σ i} and 1 form a complete set Martin Nagel (CU Boulder) Quantum Field Theories February 3, / 31

12 Dirac Dirac: linear in / t and in Ansatz: ψ(x, t) i = ( iα + βm)ψ(x, t) t demand a KG-type condition: 2 ψ/ t 2 = ( 2 + m 2 )ψ which yields the following anti-commutation relations: {α i, β} = 0 {α i, α j} = 2δ ij1 β 2 = 1 α i and β are Hermitian (H = α p + βm) Trα i = Trβ = 0 Eigenvalues of α i and β are ±1 dimensionality n of α i and β is even n = 2: {σ i} and 1 form a complete set 9 >= >; 4 x 4 matrices Martin Nagel (CU Boulder) Quantum Field Theories February 3, / 31

13 Dirac equation I conventional representation: α i = 0 σi σ i 0 «β = «define γ matrices: Dirac equation: γ µ = (γ 0 = β, γ = βα) (i / m)ψ = 0, / γ µ µ Martin Nagel (CU Boulder) Quantum Field Theories February 3, / 31

14 Dirac equation I conventional representation: α i = 0 σi σ i 0 «β = «define γ matrices: Dirac equation: γ µ = (γ 0 = β, γ = βα) (i / m)ψ = 0, / γ µ µ Free-particle solutions, with E = ±(p 2 + m 2 ) 1/2 : ψ = φ χ «e ip x = ( u s (p)e ip+ x, positive 4-momentum p µ + = (+E, p) v s (p)e +ip+ x, negative 4-momentum p + 4-component Dirac spinors: u s (p) = φ s E + m σ p E+m φs «v s (p) = σ p E + m E+m χs χ s «, s = 1, 2 Martin Nagel (CU Boulder) Quantum Field Theories February 3, / 31

15 Dirac equation II two-component spinors φ s=1,2 and χ s=1,2 can be distinguished by helicity operator:! h(p) = σ p 0 p σ p 0 p spin degree of freedom arises naturally two-component spinors φ and χ are interchanged under parity need 4 components Martin Nagel (CU Boulder) Quantum Field Theories February 3, / 31

16 Dirac equation II two-component spinors φ s=1,2 and χ s=1,2 can be distinguished by helicity operator:! h(p) = σ p 0 p σ p 0 p spin degree of freedom arises naturally two-component spinors φ and χ are interchanged under parity need 4 components in chiral representation, the (split-up) Dirac equation becomes: Eφ = σ pφ + mχ Eχ = σ pχ + mφ φ and χ become helicity eigenstates for m 0 mass couples states of different helicity Dirac 4-current: j µ = ψγ µ ψ with ψ = ψ γ 0 ρ = j 0 is positive-definite still negative energy solutions Martin Nagel (CU Boulder) Quantum Field Theories February 3, / 31

17 E +m Interpretation of negative energy solutions positive-energy continuum E +m m negative-energy continuum E m Martin Nagel (CU Boulder) Quantum Field Theories February 3, / 31

18 E +m Interpretation of negative energy solutions positive-energy continuum E +m m negative-energy continuum E m Dirac positive-energy particle can cascade-down through negative-energy levels, without limit postulate vacuum negative-energy states filled with electrons: Dirac sea stable through Pauli exclusion principle infinite charge and energy, but only relative quantities observable missing negative-energy spin- electron = spin- hole with positive energy and charge positron not a single-particle theory: e + e pair creation through excitation QFT Martin Nagel (CU Boulder) Quantum Field Theories February 3, / 31

19 E +m m Dirac Interpretation of negative energy solutions positive-energy continuum E +m negative-energy continuum E m positive-energy particle can cascade-down through negative-energy levels, without limit postulate vacuum negative-energy states filled with electrons: Dirac sea stable through Pauli exclusion principle infinite charge and energy, but only relative quantities observable missing negative-energy spin- electron = spin- hole with positive energy and charge positron not a single-particle theory: e + e pair creation through excitation QFT Feynman Dirac interpretation fails for bosons (not subject to Pauli exclusion principle) particles with negative 4-momentum correspond to anti-particles with positive 4-momentum incoming state outgoing state The emission (absorption) of an anti-particle of 4-momentum p µ is physically equivalent to the absorption (emission) of a particle of 4-momentum p µ Martin Nagel (CU Boulder) Quantum Field Theories February 3, / 31

20 E +m m Dirac Interpretation of negative energy solutions positive-energy continuum E +m negative-energy continuum E m positive-energy particle can cascade-down through negative-energy levels, without limit postulate vacuum negative-energy states filled with electrons: Dirac sea stable through Pauli exclusion principle infinite charge and energy, but only relative quantities observable missing negative-energy spin- electron = spin- hole with positive energy and charge positron not a single-particle theory: e + e pair creation through excitation QFT Feynman Dirac interpretation fails for bosons (not subject to Pauli exclusion principle) particles with negative 4-momentum correspond to anti-particles with positive 4-momentum incoming state outgoing state The emission (absorption) of an anti-particle of 4-momentum p µ is physically equivalent to the absorption (emission) of a particle of 4-momentum p µ e (pi) e (pf ) e + ( pi) V V e + ( pf) Martin Nagel (CU Boulder) Quantum Field Theories February 3, / 31

21 Quantum field theories Field theories continuous system (field) with infinite degrees of freedom, localized interactions {q r (t), r = 1... N} N φ(x, t) Lagrange-Hamilton formalism: define Lagrangian, principle of least action Euler-Lagrange equations of motion canonically conjugate field momentum π = L/ φ field expansion in terms of appropriate normal modes, total energy = sum of individual mode energies φ(x, t) = Z d 3 k (2π) 3 2ω [a(k)ei(kx ωt) + a (k)e i(kx ωt) ] Martin Nagel (CU Boulder) Quantum Field Theories February 3, / 31

22 Quantum field theories Field theories continuous system (field) with infinite degrees of freedom, localized interactions {q r (t), r = 1... N} N φ(x, t) Lagrange-Hamilton formalism: define Lagrangian, principle of least action Euler-Lagrange equations of motion canonically conjugate field momentum π = L/ φ field expansion in terms of appropriate normal modes, total energy = sum of individual mode energies Quantization ˆφ(x, t) = Z promote fields to operators: φ ˆφ d 3 k (2π) 3 2ω [â(k)ei(kx ωt) + â (k)e i(kx ωt) ] Martin Nagel (CU Boulder) Quantum Field Theories February 3, / 31

23 Quantum field theories Field theories continuous system (field) with infinite degrees of freedom, localized interactions {q r (t), r = 1... N} N φ(x, t) Lagrange-Hamilton formalism: define Lagrangian, principle of least action Euler-Lagrange equations of motion canonically conjugate field momentum π = L/ φ field expansion in terms of appropriate normal modes, total energy = sum of individual mode energies Quantization ˆφ(x, t) = Z promote fields to operators: φ ˆφ postulate appropriate commutation relations: d 3 k (2π) 3 2ω [â(k)ei(kx ωt) + â (k)e i(kx ωt) ] [ ˆφ(x, t), ˆπ(y, t)] = iδ 3 (x y) [â(k), â (k )] = (2π) 3 δ 3 (k k ) [â(k), â(k )] = [â (k), â (k )] = 0 Martin Nagel (CU Boulder) Quantum Field Theories February 3, / 31

24 Gauge theories Gauge theories are characterized by close interrelation between symmetries, conservation laws, and dynamics. Gauge theories = dynamical theories based on local invariance principles Classical electrodynamics: A and V not uniquely defined for given E and B. Gauge invariance: gauge transformation (1) leaves field-strength tensor (2) and therefore Maxwell s equations (3) unchanged A µ A µ = A µ µ χ (1) F µν µ A ν ν A µ F µν F µν = F µν (2) µf µν = j ν (3) Remember: Maxwell modified Ampére s law by introducing the displacement current to save the continuity equation µj µ = 0, which is a statement of local charge conservation. Gauge invariance charge conservation? Martin Nagel (CU Boulder) Quantum Field Theories February 3, / 31

25 Gauge invariance in quantum mechanics Schrödinger equation in electromagnetic field: ψ(x, t) i = [ 1 t 2m ( i qa)2 + qv ]ψ(x, t) (4) But solution to (4) is no longer solution after (V, A) is replaced by (V, A ) Schrödinger equation is not gauge-invariant (but ψ itself is not an observable). Martin Nagel (CU Boulder) Quantum Field Theories February 3, / 31

26 Gauge invariance in quantum mechanics Schrödinger equation in electromagnetic field: ψ(x, t) i = [ 1 t 2m ( i qa)2 + qv ]ψ(x, t) (4) But solution to (4) is no longer solution after (V, A) is replaced by (V, A ) Schrödinger equation is not gauge-invariant (but ψ itself is not an observable). to make Schrödinger equation gauge-covariant, gauge transformation must be accompanied by a transformation of ψ A µ A µ = A µ µ χ ψ ψ = exp(iqχ)ψ Martin Nagel (CU Boulder) Quantum Field Theories February 3, / 31

27 Gauge invariance in quantum mechanics Schrödinger equation in electromagnetic field: ψ(x, t) i = [ 1 t 2m ( i qa)2 + qv ]ψ(x, t) (4) But solution to (4) is no longer solution after (V, A) is replaced by (V, A ) Schrödinger equation is not gauge-invariant (but ψ itself is not an observable). to make Schrödinger equation gauge-covariant, gauge transformation must be accompanied by a transformation of ψ and a replacement of µ : A µ A µ = A µ µ χ ψ ψ = exp(iqχ)ψ µ D µ µ + iqa µ probability density ψ 2 and current density ψ (Dψ) (Dψ) ψ remain invariant Martin Nagel (CU Boulder) Quantum Field Theories February 3, / 31

28 Gauge invariance in quantum mechanics Schrödinger equation in electromagnetic field: ψ(x, t) i = [ 1 t 2m ( i qa)2 + qv ]ψ(x, t) (4) But solution to (4) is no longer solution after (V, A) is replaced by (V, A ) Schrödinger equation is not gauge-invariant (but ψ itself is not an observable). to make Schrödinger equation gauge-covariant, gauge transformation must be accompanied by a transformation of ψ and a replacement of µ : A µ A µ = A µ µ χ ψ ψ = exp(iqχ)ψ µ D µ µ + iqa µ covariant derivative probability density ψ 2 and current density ψ (Dψ) (Dψ) ψ remain invariant Insistence on (local) gauge invariance gives rise to an interaction term Martin Nagel (CU Boulder) Quantum Field Theories February 3, / 31

29 The argument reversed: the gauge principle Start by demanding covariance under space-time dependent phase transformations: ψ(x, t) ψ (x, t) = exp[iqχ(x, t)]ψ(x, t) not possible for any free-particle relativistic wave equation introduces a new field A (to cancel the kinetic part of χ) which itself undergoes an exactly prescribed transformation when ψ ψ that interacts with matter field ψ in an exactly prescribed way A vector field such as A µ, necessary to guarantee local phase invariance, is called a gauge field Martin Nagel (CU Boulder) Quantum Field Theories February 3, / 31

30 Noether s theorem: Global gauge symmetries If the Lagrangian L of a system is invariant under a continuous transformation, there exists an associated conserved symmetry current µj µ = 0. Volume integral of µ = 0 component of a symmetry current is a symmetry operator. Martin Nagel (CU Boulder) Quantum Field Theories February 3, / 31

31 Noether s theorem: Global gauge symmetries If the Lagrangian L of a system is invariant under a continuous transformation, there exists an associated conserved symmetry current µj µ = 0. Volume integral of µ = 0 component of a symmetry current is a symmetry operator. Dirac Lagrangian L D = ψ(iγ µ µ m)ψ is invariant under global U(1) transformation ψ ψ = e iα ψ Symmetry current N µ ψ = ψγ µ ψ = j µ Symmetry operator and Hamiltonian: Z Z N ψ = ψ ψd 3 x = Z H D = d 3 p (2π) 3 d 3 p (2π) 3 X s=1,2 X s=1,2 [ap s ap s + bpb s p s ] E p[a s p a s p b s pb s p ] Martin Nagel (CU Boulder) Quantum Field Theories February 3, / 31

32 Noether s theorem: Global gauge symmetries If the Lagrangian L of a system is invariant under a continuous transformation, there exists an associated conserved symmetry current µj µ = 0. Volume integral of µ = 0 component of a symmetry current is a symmetry operator. Dirac Lagrangian L D = ψ(iγ µ µ m)ψ is invariant under global U(1) transformation ψ ψ = e iα ψ Symmetry current N µ ψ = ψγ µ ψ = j µ Symmetry operator and Hamiltonian: Z Z N ψ = ψ ψd 3 x = Z H D = d 3 p (2π) 3 postulate anti-commutation relations: d 3 p (2π) 3 X s=1,2 X s=1,2 [ap s ap s + bpb s p s ] E p[a s p a s p b s pb s p ] {a r p, a s q } = {b r p, b s q } = (2π) 3 δ 3 (p q)δ rs or {ψ α(x, t), ψ β (y, t)} = δ3 (x y)δ αβ i.e. quantum gauge theory enforces the spin-statistics connection! Pauli exclusion principle: identical fermions are antisymmetric under exchange of state labels two-fermion state: p 1, s 1 ; p 2, s 2 a s 1 p 1 a s 2 p 2 0 Martin Nagel (CU Boulder) Quantum Field Theories February 3, / 31

33 Local gauge symmetries I Martin Nagel (CU Boulder) Quantum Field Theories February 3, / 31

34 Local gauge symmetries II Demand invariance of Dirac Lagrangian L D = ψ(iγ µ µ m)ψ under local U(1) phase transformation ψ(x, t) ψ (x, t) = e iqχ(x,t) ψ(x, t) given the replacement µ D µ = µ + iqa µ The Lagrangian has gained an interaction term L µ D µ D L local D = L D + L int where L int = q ψγ µ ψ {z } Aµ Martin Nagel (CU Boulder) Quantum Field Theories February 3, / 31

35 Local gauge symmetries II Demand invariance of Dirac Lagrangian L D = ψ(iγ µ µ m)ψ under local U(1) phase transformation ψ(x, t) ψ (x, t) = e iqχ(x,t) ψ(x, t) given the replacement µ D µ = µ + iqa µ The Lagrangian has gained an interaction term L µ D µ D L local D = L D + L int where L int = q ψγ µ ψ {z } Aµ Global symmetries conserved symmetry current, symmetry operator conserved quantum number Martin Nagel (CU Boulder) Quantum Field Theories February 3, / 31

36 Local gauge symmetries II Demand invariance of Dirac Lagrangian L D = ψ(iγ µ µ m)ψ 1 µν FµνF 4 Maxwell Lagrangian under local U(1) phase transformation ψ(x, t) ψ (x, t) = e iqχ(x,t) ψ(x, t) given the replacement µ D µ = µ + iqa µ The Lagrangian has gained an interaction term Global symmetries L µ D µ D L local D = L D + L int where L int = q ψγ µ ψ {z } Aµ conserved symmetry current, symmetry operator conserved quantum number Local symmetries requires introduction of compensating gauge fields symmetry current becomes dynamical current that couples to gauge field prescribes a unique form of the interaction = j µ Martin Nagel (CU Boulder) Quantum Field Theories February 3, / 31

37 Perturbation theory split up Hamiltonian into free and interaction parts: Ĥ = Ĥ0 + Ĥ Heisenbeg picture: states (operators) are time-independent (time-dependent) Schrödinger picture: states (operators) are time-dependent (time-independent) Interaction picture: dâ I (t) time dependence of operators generated by Ĥ0: = dt i[âi (t), Ĥ0] time dependence of states generated by Ĥ : i d dt ψ(t) I = Ĥ I (t) ψ(t) I Martin Nagel (CU Boulder) Quantum Field Theories February 3, / 31

38 Perturbation theory split up Hamiltonian into free and interaction parts: Ĥ = Ĥ0 + Ĥ Heisenbeg picture: states (operators) are time-independent (time-dependent) Schrödinger picture: states (operators) are time-dependent (time-independent) Interaction picture: dâ I (t) time dependence of operators generated by Ĥ0: = dt i[âi (t), Ĥ0] time dependence of states generated by Ĥ : i d dt ψ(t) I = Ĥ I (t) ψ(t) I S-matrix = amplitude to find a particular final state f in time-evolved initial state i E E f ψ( ) I = Df Ŝ ψ( ) = Df Ŝ i I Martin Nagel (CU Boulder) Quantum Field Theories February 3, / 31

39 Perturbation theory split up Hamiltonian into free and interaction parts: Ĥ = Ĥ0 + Ĥ Heisenbeg picture: states (operators) are time-independent (time-dependent) Schrödinger picture: states (operators) are time-dependent (time-independent) Interaction picture: dâ I (t) time dependence of operators generated by Ĥ0: = dt i[âi (t), Ĥ0] time dependence of states generated by Ĥ : i d dt ψ(t) I = Ĥ I (t) ψ(t) I S-matrix = amplitude to find a particular final state f in time-evolved initial state i E E f ψ( ) I = Df Ŝ ψ( ) = Df Ŝ i Dyson expansion: X Ŝ = n=0 ( i) n n! with Ĥ I (t) = R Ĥ I (x, t)d3 x and time-ordering symbol Z Z... I d 4 x 1d 4 x 2... d 4 x nt {Ĥ I (x1)ĥ I (x2)... Ĥ I (xn)} T {Ĥ I (x1)ĥ I (x2)} = Ĥ I (x1)ĥ I (x2) for t1 > t2 = Ĥ I (x2)ĥ I (x1) for t1 < t2 Martin Nagel (CU Boulder) Quantum Field Theories February 3, / 31

40 QED H QED = L int = q ψγ µ ψa µ fermion field Z ψ(x) = Z ψ(x) = d 3 p 1 X p (2π) 3 2Ep d 3 p 1 X p (2π) 3 2Ep s s (apu s s (p)e ip x + bp s v s (p)e ip x ) (bp v s s (p)e ip x + ap s ū s (p)e ip x ) {a r p, a s q } = {b r p, b s q } = (2π) 3 δ 3 (p q)δ rs {ap, r aq} s = {ap r, aq s } = {ap r( ), bq s( ) } = {bp, r bq} s = {bp r, bq s } = 0 electromagnetic field Z A µ (x) = one-particle state: p, s p 2E pa s p 0 d 3 k (2π) 3 1 2ωk 3X (α λ k ɛ µ (k)e ik x λ + α λ λ=0 k ɛµ (k)eik x λ ) independent operators commute: [α λ( ) k, a p r( ) ] = [α λ( ) k, b p r( ) ] = 0 [α λ k, α λ k ] = g λλ (2π) 3 δ 3 (k k ) Martin Nagel (CU Boulder) Quantum Field Theories February 3, / 31

41 e + e scattering: 0th order 0th order term in Dyson expansion: 1 S (0) fi = f 1 i = D0 a s k ar p as k ar p 0 E (16E k E pe k E p ) 1/2 Martin Nagel (CU Boulder) Quantum Field Theories February 3, / 31

42 e + e scattering: 0th order 0th order term in Dyson expansion: 1 S (0) fi = f 1 i = D0 a s k ar p as k ar p 0 vacuum expectation value E (16E k E pe k E p ) 1/2 Martin Nagel (CU Boulder) Quantum Field Theories February 3, / 31

43 e + e scattering: 0th order 0th order term in Dyson expansion: 1 General strategy: S (0) fi = f 1 i = (anti-)commute a s to the left (anti-)commute a s to the right pick up δ-functions on the way and use 0 a i = a i 0 = 0 D0 a s k ar p as k ar p 0 vacuum expectation value E (16E k E pe k E p ) 1/2 Martin Nagel (CU Boulder) Quantum Field Theories February 3, / 31

44 e + e scattering: 0th order 0th order term in Dyson expansion: 1 General strategy: S (0) fi = f 1 i = (anti-)commute a s to the left (anti-)commute a s to the right pick up δ-functions on the way and use 0 a i = a i 0 = 0 D0 a s k ar p as k ar p 0 vacuum expectation value E (16E k E pe k E p ) 1/2... S (0) fi = 2E k (2π) 4 δ 3 (k k )2E p(2π) 3 δ 3 (p p ) e (p, s) e (p, s ) e + (k, r) e + (k, r ) Martin Nagel (CU Boulder) Quantum Field Theories February 3, / 31

45 e + e scattering: 1st order S (1) fi Z = ie d 4 x k p ψ(x)γ µ ψ(x)a µ(x) kp = 0 Note: only 1 electromagnetic field operator A µ no free photons in either initial or final states α λ k (α λ ) commutes all the way to the right (left) k Martin Nagel (CU Boulder) Quantum Field Theories February 3, / 31

46 e + e scattering: 1st order S (1) fi Z = ie d 4 x k p ψ(x)γ µ ψ(x)a µ(x) kp = 0 Note: only 1 electromagnetic field operator A µ no free photons in either initial or final states α λ k (α λ ) commutes all the way to the right (left) k amplitude vanishes! Martin Nagel (CU Boulder) Quantum Field Theories February 3, / 31

47 e + e scattering: 1st order S (1) fi Z = ie d 4 x k p ψ(x)γ µ ψ(x)a µ(x) kp = 0 Note: only 1 electromagnetic field operator A µ no free photons in either initial or final states α λ k (α λ ) commutes all the way to the right (left) k amplitude vanishes! Can generally decompose any free field into positive- and negative-frequency parts: φ + (x) = R d 3 p 1 a (2π) 3 2Ep pe ip x φ (x) = R d 3 p 1 a (2π) 3 2Ep pe +ip x so that φ(x) = φ + (x) + φ (x) and 0 φ (x) = φ + (x) 0 = 0 Martin Nagel (CU Boulder) Quantum Field Theories February 3, / 31

48 ψ(x)γ µ ψ(x)a µ(x) three-fields-at-a-point interaction Martin Nagel (CU Boulder) Quantum Field Theories February 3, / 31

49 Bremsstrahlung kλ p s ψ(x)γ µ ψ(x)a µ(x) p s three-fields-at-a-point interaction γ(k, λ) e (p, s) e (p, s ) Martin Nagel (CU Boulder) Quantum Field Theories February 3, / 31

50 Bremsstrahlung kλ p s ψ(x)γ µ ψ(x)a µ(x) p s = 0 since p µ p µ + k µ three-fields-at-a-point interaction γ(k, λ) e (p, s) e (p, s ) Martin Nagel (CU Boulder) Quantum Field Theories February 3, / 31

51 Bremsstrahlung kλ p s ψ(x)γ µ ψ(x)a µ(x) p s = 0 since p µ p µ + k µ three-fields-at-a-point interaction γ(k, λ) e (p, s) e (p, s ) isolated diagram, takes place in vacuum for bremsstrahlung to occur, we need a nucleus to absorb some of the momentum additional interaction needed 2nd order process: Martin Nagel (CU Boulder) Quantum Field Theories February 3, / 31

52 Bremsstrahlung kλ p s ψ(x)γ µ ψ(x)a µ(x) p s = 0 since p µ p µ + k µ three-fields-at-a-point interaction γ(k, λ) e (p, s) e (p, s ) isolated diagram, takes place in vacuum for bremsstrahlung to occur, we need a nucleus to absorb some of the momentum additional interaction needed 2nd order process: e (p, s) γ(k, λ) γ e (p, s ) Martin Nagel (CU Boulder) Quantum Field Theories February 3, / 31

53 e + e scattering: 2nd order ZZ S (2) fi = ( ie)2 2! d 4 x 1d 4 x 2 k s, p r T [ ψ(x 1)γ µ ψ(x 1)A µ(x 1) ψ(x 2)γ ν ψ(x 2)A ν(x 2)] k s, p r (16E k E pe k E p ) 1/2 16 operators, 12 out of those inside time-ordering symbol 6 momentum integrals we need a more efficient formalism! Martin Nagel (CU Boulder) Quantum Field Theories February 3, / 31

54 Wick s theorem Wick s theorem T {â 1â 2... â n} = N{â 1â 2... â n + all possible contractions} T {â 1â 2â 3â 4} = N{â 1â 2â 3â 4 + â 1â 2â 3â 4 + â 1â 2â 3â 4 + â 1â 2â 3â 4 + â 1â 2â 3â 4 + +â 1â 2â 3â 4 + â 1â 2â 3â 4 + â 1â 2â 3â 4 + â 1â 2â 3â 4 + â 1â 2â 3â 4} Martin Nagel (CU Boulder) Quantum Field Theories February 3, / 31

55 Wick s theorem Wick s theorem T {â 1â 2... â n} = N{â 1â 2... â n + all possible contractions} T {â 1â 2â 3â 4} = N{â 1â 2â 3â 4 + â 1â 2â 3â 4 + â 1â 2â 3â 4 + â 1â 2â 3â 4 + â 1â 2â 3â 4 + +â 1â 2â 3â 4 + â 1â 2â 3â 4 + â 1â 2â 3â 4 + â 1â 2â 3â 4 + â 1â 2â 3â 4} normal ordering symbol N() places operators in normal order, i.e. all a(k) s to the right of all a (k ) s, so that 0 N(any number of operators) 0 = 0 Martin Nagel (CU Boulder) Quantum Field Theories February 3, / 31

56 Wick s theorem Wick s theorem T {â 1â 2... â n} = N{â 1â 2... â n + all possible contractions} T {â 1â 2â 3â 4} = N{â 1â 2â 3â 4 + â 1â 2â 3â 4 + â 1â 2â 3â 4 + â 1â 2â 3â 4 + â 1â 2â 3â 4 + +â 1â 2â 3â 4 + â 1â 2â 3â 4 + â 1â 2â 3â 4 + â 1â 2â 3â 4 + â 1â 2â 3â 4} normal ordering symbol N() places operators in normal order, i.e. all a(k) s to the right of all a (k ) s, so that 0 N(any number of operators) 0 = 0 decompose field φ into positive- and negative-frequency parts: φ + (x) a(k)e ikx φ (x) a (k)e +ikx contraction of two fields: j ˆφ+ (x), φ (y) φ(x)φ(y) ˆφ+ (y), φ (x) = 0 φ(x)φ(y) 0 for x 0 > y 0 = 0 φ(y)φ(x) 0 for x 0 < y 0 ff = 0 T φ(x)φ(y) 0 Feynman propagator j ff y to x = amplitude for (KG, Dirac,...) particle to propagate from x to y = momentum-space Green s function for (KG, Dirac,...) differential operator Martin Nagel (CU Boulder) Quantum Field Theories February 3, / 31

57 e + e scattering: 2nd order Wick s theorem for fermions time- and normal-ordered products and contractions pick up one minus sign for each fermion interchange: T (ψ 1ψ 2ψ 3ψ 4) = ( 1) 3 ψ 3ψ 1ψ 4ψ 2 if x 0 3 > x 0 1 > x 0 4 > x 0 2 N(a pa qa r ) = ( 1) 2 a r a pa q = ( 1) 3 a r a qa p ( ψ(x) ψ(y) {ψ + (x), ψ (y)} for x 0 > y 0 { ψ + (y), ψ (x)} for x 0 < y 0 ψ(x)ψ(y) = ψ(x) ψ(y) = 0 N(ψ 1ψ 2 ψ 3 ψ 4) = ψ 1 ψ 3N(ψ 2 ψ 4) ZZ d 4 x 1d 4 x 2 k s, p r T [ ψ(x 1)γ µ ψ(x 1)A µ(x 1) ψ(x 2)γ ν ψ(x 2)A ν(x 2)] k s, p r external states are not 0 not-fully-contracted terms do not necessarily vanish! Martin Nagel (CU Boulder) Quantum Field Theories February 3, / 31

58 e + e scattering: 2nd order An uncontracted ψ operator inside N-product has 2 terms: ψ + on the far right and ψ on the far left. One contribution for each way of anti-commuting the a of ψ + past an initial-state a, and for each way of anti-commuting the a of ψ past a final-state a. Martin Nagel (CU Boulder) Quantum Field Theories February 3, / 31

59 e + e scattering: 2nd order An uncontracted ψ operator inside N-product has 2 terms: ψ + on the far right and ψ on the far left. One contribution for each way of anti-commuting the a of ψ + past an initial-state a, and for each way of anti-commuting the a of ψ past a final-state a. define contractions of field operators with external states: ψ(x) p s = ψ + (x) p s = e ip x u s (p) 0 p s ψ(x) = p s ψ (x) = 0 e +ip x ū s (p) ψ(x) k s = ψ + (x) k s = e ik x v s (k) 0 k s ψ(x) = k s ψ (x) = 0 e +ik x v s (k)...incoming fermion...outgoing fermion...incoming anti-fermion...outgoing anti-fermion Martin Nagel (CU Boulder) Quantum Field Theories February 3, / 31

60 e + e scattering: 2nd order An uncontracted ψ operator inside N-product has 2 terms: ψ + on the far right and ψ on the far left. One contribution for each way of anti-commuting the a of ψ + past an initial-state a, and for each way of anti-commuting the a of ψ past a final-state a. define contractions of field operators with external states: ψ(x) p s = ψ + (x) p s = e ip x u s (p) 0 p s ψ(x) = p s ψ (x) = 0 e +ip x ū s (p) ψ(x) k s = ψ + (x) k s = e ik x v s (k) 0 k s ψ(x) = k s ψ (x) = 0 e +ik x v s (k)...incoming fermion...outgoing fermion...incoming anti-fermion...outgoing anti-fermion An uncontracted A operator inside N-product vanishes (no photons in external states) two A s must be contracted internally Martin Nagel (CU Boulder) Quantum Field Theories February 3, / 31

61 operators fully contracted internally p, k ( ψγ µ ψa µ) x( ψγ ν ψa ν) y p, k p, k ( ψγ µ ψa µ) x( ψγ ν ψa ν) y p, k Martin Nagel (CU Boulder) Quantum Field Theories February 3, / 31

62 operators fully contracted internally p, k ( ψγ µ ψa µ) x( ψγ ν ψa ν) y p, k p, k ( ψγ µ ψa µ) x( ψγ ν ψa ν) y p, k y γ x y γ x e (p, s) e (p, s ) e + (k, r) e + (k, r ) Martin Nagel (CU Boulder) Quantum Field Theories February 3, / 31

63 operators fully contracted internally p, k ( ψγ µ ψa µ) x( ψγ ν ψa ν) y p, k p, k ( ψγ µ ψa µ) x( ψγ ν ψa ν) y p, k y γ x y γ x Exponentiation of the disconnected diagrams: disconnected pieces ( vacuum bubbles ) exponentiate to an overall phase factor giving the shift of the energy of the interacting vacuum state (as opposed to the free-theory vacuum state) no sensible contribution to S-matrix element! Martin Nagel (CU Boulder) Quantum Field Theories February 3, / 31

64 external states partially contracted p, k ( ψγ µ ψa µ) x( ψγ ν ψa ν) y p, k p, k ( ψγ µ ψa µ) x( ψγ ν ψa ν) y p, k e (p, s) γ e (p, s ) e (p, s) y e (p, s ) y x x e + (k, r) e + (k, r ) e + (k, r) e + (k, r ) + x y + (p, s) (k, r), (p, s ) (k, r ) + x y + (p, s) (k, r), (p, s ) (k, r ) Martin Nagel (CU Boulder) Quantum Field Theories February 3, / 31

65 external states partially contracted p, k ( ψγ µ ψa µ) x( ψγ ν ψa ν) y p, k p, k ( ψγ µ ψa µ) x( ψγ ν ψa ν) y p, k e (p, s) γ e (p, s ) e (p, s) y e (p, s ) y x x e + (k, r) e + (k, r ) e + (k, r) e + (k, r ) + x y + (p, s) (k, r), (p, s ) (k, r ) + x y + (p, s) (k, r), (p, s ) (k, r ) diagrams with loops connected to only one external line have nothing to do with the scattering process itself; they are a property of the external states in the interacting theory, i.e. they represent corrections to the no-scattering diagram taken care of by Renormalization Martin Nagel (CU Boulder) Quantum Field Theories February 3, / 31

66 external states fully contracted p, k ( ψγ µ ψa µ) x( ψγ ν ψa ν) y p, k p, k ( ψγ µ ψa µ) x( ψγ ν ψa ν) y p, k e (p, s) y e (p, s ) e (p, s) e (p, s ) γ y γ x e + (k, r) x e + (k, r ) e + (k, r) e + (k, r ) + x y + x y Martin Nagel (CU Boulder) Quantum Field Theories February 3, / 31

67 external states fully contracted p, k ( ψγ µ ψa µ) x( ψγ ν ψa ν) y p, k p, k ( ψγ µ ψa µ) x( ψγ ν ψa ν) y p, k e (p, s) y e (p, s ) e (p, s) e (p, s ) γ y γ x e + (k, r) x e + (k, r ) e + (k, r) e + (k, r ) + x y + x y only fully connected diagrams, in which all external lines are connected to each other, contribute in a non-trivial way to the S-matrix! Martin Nagel (CU Boulder) Quantum Field Theories February 3, / 31

68 external states fully contracted p, k ( ψγ µ ψa µ) x( ψγ ν ψa ν) y p, k p, k ( ψγ µ ψa µ) x( ψγ ν ψa ν) y p, k e (p, s) y e (p, s ) e (p, s) e (p, s ) γ y γ x e + (k, r) x e + (k, r ) e + (k, r) e + (k, r ) + x y + x y only fully connected diagrams, in which all external lines are connected to each other, contribute in a non-trivial way to the S-matrix! Note: vertices are labeled for illustration purposes only! momentum-space propagators carry no implied order of emission/absorption; both time-ordered processes are always included! Martin Nagel (CU Boulder) Quantum Field Theories February 3, / 31

69 external states fully contracted p, k ( ψγ µ ψa µ) x( ψγ ν ψa ν) y p, k p, k ( ψγ µ ψa µ) x( ψγ ν ψa ν) y p, k but those photons are not real, they are virtual! e (p, s) y e (p, s ) e (p, s) e (p, s ) γ y γ x e + (k, r) x e + (k, r ) e + (k, r) e + (k, r ) + x y + x y only fully connected diagrams, in which all external lines are connected to each other, contribute in a non-trivial way to the S-matrix! Note: vertices are labeled for illustration purposes only! momentum-space propagators carry no implied order of emission/absorption; both time-ordered processes are always included! Martin Nagel (CU Boulder) Quantum Field Theories February 3, / 31

70 Virtual particles space-time integral enforces 4-momentum conservation at each vertex photon cannot have the right energy : photon is virtual or off-mass-shell, with q 2 m 2 γ = 0 due to energy-time uncertainty relation, a virtual photon can live for a time t E Martin Nagel (CU Boulder) Quantum Field Theories February 3, / 31

71 Virtual particles space-time integral enforces 4-momentum conservation at each vertex photon cannot have the right energy : photon is virtual or off-mass-shell, with q 2 m 2 γ = 0 due to energy-time uncertainty relation, a virtual photon can live for a time t E In quantum field theories, particles interact by exchanging virtual field quanta, which mediate the force a given order of perturbation series corresponds to the same number of verteces, where a field quantum is emitted or absorbed replacing t r/c and E mc 2 for massive field quanta gives the effective range of the associated force (Yukawa) r mc Martin Nagel (CU Boulder) Quantum Field Theories February 3, / 31

72 Feynman rules for QED fermion propagator: photon propagator: vertex: p p µ = i( /p + m) p 2 + m 2 + iɛ = igµν p 2 + iɛ = iqγ µ incoming photon: outgoing photon: incoming fermion: outgoing fermion: incoming antifermion: outgoing antifermion: p p p p p p = ɛ µ(p) = ɛ µ(p) = u s (p) = ū s (p) = v s (p) = v s (p) Martin Nagel (CU Boulder) Quantum Field Theories February 3, / 31

73 QED Non-Abelian gauge theories Strong and weak forces are built on generalizations of gauge principle Martin Nagel (CU Boulder) Quantum Field Theories February 3, / 31

74 Non-Abelian gauge theories QED = Abelian U(1) gauge theory invariant under local phase transformations of a single field: ψ ψ = exp(iqχ)ψ different transformations commute: e iqχ1 e iqχ 2 = e iqχ2 e iqχ 1 Strong and weak forces are built on generalizations of gauge principle Martin Nagel (CU Boulder) Quantum Field Theories February 3, / 31

75 Non-Abelian gauge theories QED = Abelian U(1) gauge theory invariant under local phase transformations of a single field: ψ ψ = exp(iqχ)ψ different transformations commute: e iqχ1 e iqχ 2 = e iqχ2 e iqχ 1 Strong and weak forces = non-abelian gauge theories local phase transformations involve more than one field they do not commute Martin Nagel (CU Boulder) Quantum Field Theories February 3, / 31

76 Non-Abelian gauge theories QED = Abelian U(1) gauge theory invariant under local phase transformations of a single field different transformations commute ψ ψ = exp(iqχ)ψ µ D µ = µ + iqa µ A µ A µ = A µ µ χ A µ transforms independent of q photon is neutral Strong and weak forces = non-abelian gauge theories local phase transformations involve more than one field they do not commute Martin Nagel (CU Boulder) Quantum Field Theories February 3, / 31

77 Non-Abelian gauge theories QED = Abelian U(1) gauge theory invariant under local phase transformations of a single field different transformations commute ψ ψ = exp(iqχ)ψ µ D µ = µ + iqa µ A µ A µ = A µ µ χ A µ transforms independent of q photon is neutral Strong and weak forces = non-abelian gauge theories local phase transformations involve more than one field they do not commute gauge fields belong to regular (adjoint) representation of gauge group coupling constant enters transformation law of gauge fields through non-vanishing commutator, i.e. gauge fields carry non-abelian charge degrees of freedom Martin Nagel (CU Boulder) Quantum Field Theories February 3, / 31

78 Non-Abelian gauge theories QED = Abelian U(1) gauge theory invariant under local phase transformations of a single field different transformations commute ψ ψ = exp(iqχ)ψ µ D µ = µ + iqa µ A µ A µ = A µ µ χ A µ transforms independent of q photon is neutral Strong and weak forces = non-abelian gauge theories local phase transformations involve more than one field they do not commute gauge fields belong to regular (adjoint) representation of gauge group coupling constant enters transformation law of gauge fields through non-vanishing commutator, i.e. gauge fields carry non-abelian charge degrees of freedom the field quanta will necessarily interact with themselves g g g g g g g Martin Nagel (CU Boulder) Quantum Field Theories February 3, / 31

79 QED = Abelian U(1) gauge theory Non-Abelian gauge theories invariant under local phase transformations of a single field different transformations commute ψ ψ = exp(iqχ)ψ µ D µ = µ + iqa µ A µ A µ = A µ µ χ A µ transforms independent of q photon is neutral Strong and weak forces = non-abelian gauge theories local phase transformations involve more than one field they do not commute gauge fields belong to regular (adjoint) representation of gauge group coupling constant enters transformation law of gauge fields through non-vanishing commutator, i.e. gauge fields carry non-abelian charge degrees of freedom the field quanta will necessarily interact with themselves All three dynamical theories in the Standard Model are based on a local gauge principle: SU(3) c SU(2) L U(1) Y {z } strong electroweak = (weak isospin hypercharge) Martin Nagel (CU Boulder) Quantum Field Theories February 3, / 31

80 Summary Feynman diagrams formalism that allows convenient organization and visualization of QED processes in perturbation series Feynman diagrams = representations of momentum-space amplitudes each diagram represents a specific contribution to the scattering matrix, by a precise mathematical correspondance ( Feynman rules ) contributions to S-matrix elements only from fully connected, amputated diagrams! disconnected diagrams (vacuum bubbles) represent the evolution of the free-theory vacuum state into the interacting-theory vacuum state external line corrections represent the evolution of the free-theory single-particle state into the interacting-theory single-particle state Quantum gauge theories gauge principle: demand invariance of Lagrangian under local phase transformations of matter fields requires introduction of compensating gauge fields which couple to matter fields in an exactly determined way Martin Nagel (CU Boulder) Quantum Field Theories February 3, / 31