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Quantum Field Theory II Quantum Field Theory II Babis Anastasiou Institute for Theoretical Physics, ETH Zurich, 8093 Zurich, Switzerland E-mail: [email protected] March 4, 2018 Contents 1 Path integral quantization in Quantum Mechanics 4 1.1 The propagator . .4 1.2 The path integral . .7 1.3 An \adventurous" transition . 11 1.4 Functional differentiation . 15 1.5 Vacuum to vacuum transitions . 17 1.6 The simple harmonic oscillator . 19 2 Path integrals and scalar fields 23 2.1 Functional Integration . 25 2.2 Path integrals and interacting fields . 30 2.3 Perturbation theory for −λφ4=4! interactions . 32 2.4 Fermionic path integrals . 35 3 Non-abelian gauge theories 43 3.1 Gauge invariance . 43 3.2 Non-abelian (global) SU(N) transformations . 45 3.3 Local non-abelian gauge symmetries . 48 1 4 Quantization of non-abelian gauge theories 55 4.1 Perturbative QCD . 64 5 BRST symmetry 69 5.1 Application to the free electromagnetic field . 74 6 Quantum effective action and the effective potential 79 6.1 The quantum effective action as a generating functional . 81 6.2 The effective potential . 88 7 Symmetries of the path integral and the effective action 92 7.1 Slavnov-Taylor identities . 92 7.2 Symmetry constraints on the effective action . 93 7.3 Contraints on the effective action from BRST symmetry trans- formations of the classical action . 95 7.4 Slavnov-Taylor identities in QED . 97 8 Spontaneous symmetry breaking 102 8.1 Goldstone theorem . 102 8.2 General broken global symmetries . 106 8.3 Spontaneous symmetry breaking of local gauge symmetries . 108 9 Renormalization: counting the degree of ultraviolet diver- gences 110 9.1 Subdivergences . 114 9.2 Cancelation of superficial divergences with counterterms . 115 9.3 Nested and overlapping divergences . 118 10 Proof of renormalizability for non-abelian gauge theories 120 Abstract The subject of the course is modern applications of quantum field theory with emphasis on the quantization of non-Abelian gauge theo- ries. The following topics are discussed: • Classical gauge transformations. • Quantization for fermionic and bosonic fields and perturbation theory with path-integrals is developed. 2 • Quantization of non-Abelian gauge-theories. The Fadeev-Popov method. BRST symmetry. • The quantum effective action and the effective potential. • Classical symmetries of the effective action. Slavnov-Taylor iden- tities. The Zinn-Justin equation. • Physical interpretation of the effective action. • Spontaneous symmetry breaking. Goldstone theorem. Sponta- neous symmetry breaking for theories with local gauge invari- ance. • Power-counting and ultraviolet infinities in field theories. Renor- malizable Lagrangians. • Renormalization and symmetries of non-Abelian gauge theories. Renormalization group evolution. • Infrared divergences. Landau equations. Coleman-Norton phys- ical picture of infrared divergences. Soft and collinear singulari- ties. 3 1 Path integral quantization in Quan- tum Mechanics We have carried out a quantization program for simple field theories in the course of QFT I by means of \canonical quantization" (imposing commutation and anti-commutation relations on fields). Here we will quantise gauge invariant field theories with a different method, us- ing a formalism based on path integrals. The formalism is somewhat imperative to develop. While the canonical formalism can be suc- cessfully applied to Quantum Electrodynamics, it is not understood how it can be applied to the gauge field theories which describe the unified strong and electroweak interactions. As a warm-up we revisit quantum mechanics, formulating quantisation using path integrals. 1.1 The propagator We consider a quantum mechanical state j i which satisfies the Schr¨odinger equation: i¯h@t j i = H^ j i : (1) The solution of this equation i − H^ (t2−t1) j (t2)i = e h¯ j (t1)i ; (2) determines the evolution of this state from an initial moment t1 to a later moment t2. The wave function (x; t) ≡ hxj (t)i is then i − H^ (t2−t1) (x; t2) = hxj e h¯ j (t1)i (3) The sum of the Hamiltonian eigenstates H^ jβi = Eβ jβi (4) is unitary, 1 = X jβi hβj : (5) β Inserting it in the rhs of the evolution equation we find 0 1 i − H^ (t2−t1) X (x; t2) = hxj e h¯ @ jβi hβjA j (t1)i β i X − Eβ (t2−t1) = e h¯ hxj βi hβj (t1)i : (6) β 4 We now insert another unit operator Z 3 0 0 0 1 = d x x x ; (7) obtaining i Z X − Eβ (t2−t1) 3 0 0 0 (x; t2) = e h¯ hxj βi hβj d x x x j (t1)i β 8 9 Z < i = 3 0 X − Eβ (t2−t1) 0 0 = d x e h¯ hxj βi hβj x x ; t1 : (8) : β ; In other words, if we know the wave-function at one time, we can determine it fully at a later time by integrating it, Z 3 0 0 0 (x; t2) = d x K(x; x ; t2 − t1) x ; t1 ; (9) with a kernel i 0 X − Eβ (t2−t1) 0 K(x; x ; t2 − t1) = e h¯ hxj βi hβj x ; (10) β which depends on the Hamiltonian of the system and the elapsed time t2 − t1 during the evolution of the quantum state. This integration kernel is called the \propagator". For t2 = t1 the propagator is a delta function 0 X 0 0 3 0 K(x; x ; t1 − t1) = e hxj βi hβj x = δ (x − x ): (11) β Exercise: Prove that the propagator K(x; x0; t−t0) satisfies the Schr¨odinger equation in the variables x; t for times t > t0. Exercise: Prove that 1 r Z 2 π dxe−ax = : −∞ a Exercise: Compute the propagator K(x; x0; t − t0) for • a free particle, • the simple harmonic oscillator. 5 The propagator is the amplitude for a particle measured at position 0 jx i at time t1 to propagate to a new position jxi at time t2 > t1. We can verify this easily. Consider a particle measured at a position x0. After time t2 − t1 it will be evolved to a new state − i H^ (t −t ) 0 e h¯ 2 1 x : (12) The probability amplitude to be measured at a position x is − i H^ (t −t ) 0 hxj e h¯ 2 1 x 0 1 − i H^ (t −t ) X 0 = hxj e h¯ 2 1 @ jβi hβjA x β X − i E (t −t ) 0 = e h¯ β 2 1 hxj βi hβj x β 0 = K(x; x ; t2 − t1): (13) We can attempt to compute the propagator for a transition which takes a very small time δt ! 0. 0 − i Hδt^ 0 K(x; x ; δt) = hxj e h¯ x i 0 2 = hxj 1 − Hδt^ x + O δt ¯h Z i 3 0 2 = hxj 1 − Hδt^ d p jpi hpj x + O δt ¯h Z i = d3p hxj pi hpj x0 − δt hxj H^ jpi hpj x0 + O δt2 ¯h (14) We now specialise to Hamiltonian operators of the form H^ = f1(^p) + f2(^x). Then hxj H^ jpi = H hxj pi ; (15) where H is not an operator anymore but the classical Hamiltonian, i.e. a real-valued function of position and momentum. Recall that the position and momentum states are related via a Fourier transform, i px e h¯ hxj pi = p : (16) 2π¯h For simplicity let us consider one only dimension; the three-dimensional case is a faithful repetition of the same steps. Then we find for the 6 propagator at small time intervals: Z 0 dp i p(x−x0) i 2 K(x; x ; δt) = e h¯ 1 − Hδt + O δt 2π¯h ¯h Z dp i = exp p x − x0 − Hδt + O δt2 (17): 2π¯h ¯h An interesting form for the propagator for small time intervals arises when the Hamiltonian is of the form p2 H = + V (x): (18) 2m Then, ( )! Z dp i p2 K(x; x0; δt) = exp p x − x0 − δt − V (x)δt + O δt2 2π¯h ¯h 2m ( )! Z dp i δt x − x0 2 1 x − x0 2 = exp − p − m + δt m − V (x)δt + O δt2 2π¯h ¯h 2m δt 2 δt ( )! Z dp~ iδt p~2 1 x − x0 2 = exp − + m − V (x) + O δt2 ; (19) 2π¯h ¯h 2m 2 δt where in the final step we performed a trivial change of integration variable. Our final result reads: ( )! 1 iδt 1 x − x0 2 K(x; x0; δt) ' exp m − V (x) ; (20) N(δt) ¯h 2 δt with ! 1 Z dp −iδtp2 m 1=2 = exp = (21) N(δt) 2π¯h 2m¯h 2πi¯hδt 1.2 The path integral We consider now the transition from an initial position (xi; ti) to a final position (xf ; tf ), which has a propability amplitude given by the propagator: K(xf ; xi; tf − ti): We can take a \snapshot" at an intermediate time t1 during this tran- sition: ti < t1 < tf : 7 If taking the \snapshot" particle is a measurement of the position x1 of the particle at the moment t1, then the amplitude for the full transition will be: K(xf ; xi; tf − ti) = K(xf ; x1; tf − t1)K(x1; xi; t1 − ti): (22) If taking the \spanshot" is only a thought experiment and we don't actually determine the position x1 of the particle at t1 with a real measurement, we should integrate over all probability amplitudes for the particle to have performed this transition via any point. We then have: Z 1 K(xf ; xi; tf − ti) = dx1K(xf ; x1; tf − t1)K(x1; xi; t1 − ti): (23) −∞ We are allowed decide to take several \snapshots" during the transi- tion from xi to xf , in times ti < t1 < t2 < : : : < tn < tf .
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