# WS18 Quantum Field Theory

(Physics Master Course TUM; TMP core module LMU/TUM)

**Prof. A. Weiler/ Dr. W. Dybalski, WS 2018/19**

The lecture is aimed at master students with an interest in theoretical physics. It is a crucial preparation for a master thesis in theoretical particle physics. The quantum field theory concepts discussed are however more widely applicable. The focus here will be on methods, rather than on phenomenology (as compared to the 'Theoretical particle physics' course).

**Preparatory course Oct 10-12,Wed&Fri 10am-2pm, Thu 9:30-3:30pm. Strongly advised for students who have not attended the "Relativity, Particles, Fields'' course at TUM in the summer or wish to refresh their theoretical physics background. More information at Preparatory Course below.**

Please register for the lecture on campus.tum.de, especially if you are a TMP TUM/LMU student!

## News

- 03.05: For those who participated in the second exam, there will be a discussion of the problems led by P. Vaudrevange on Monday, May 6, from 16:15 to 18:00 in room PH 3343 (Theory Library).
- 24.04: Results for the 2nd exam can be found here LMU_non_registered.pdf (only for people not registered on campus.tum.de)
- 19.03: Please remember to register for the
**second exam**(deadline 1.4.2019). - 13.03: The results of the
**first exam**have been posted on campus.tum.de after some administrative difficulties. Please wait for at least one day before contacting us since it might take time to appear online. The exam can be inspected next Thursday 12:00-13:00: please send an email to the exercise coordinators for an appointment. - 28.02: The
**second exam**will take place on Friday 12.04, from 9:00 to 12:00 in HS1, Physics Department. Please be there at 8:45. The instructions remain the same as for the first exam. - 26.02: The results of the
**exercise bonus**are available here . - 20.02: The
**first exam**will take place on Wednesday 27.02, from 9:00 to 12:00 in HS2, Physics Department. Please be there at 8:45. You can bring up to 4 pages (i.e. 2 two-sided A4 sheets) of handwritten notes and writing material, but no books or other notes. You will receive scratch paper from us. Please also bring your Student-ID card. If you intend to participate in the exam, please make sure that you are registered for it on TUMonline. - 14.02: We have uploaded a bonus sheet which will be discussed in the repetitorium on 20.02 from 13:30-16:00 in HS3.
- 05.02:
**Notice:**for Exercise Sheet 13, the tutorials of groups 2 and 3 will both be moved to Wed 06.02 from 16:00 - 18:00 in PH II 227, Seminarraum E20 (see link below for a map). - 15.01: the
**first exam**will take place on Wednesday 27.02 (February 27), from 9:00 to 12:00 in HS2, Physics Department. Further details will follow. - 23.12.: If you want to read ahead, please study chapter 7 of the relativity, particles fields script .
- 17.12: Please note that the exam dates currently indicated in TUMonline are only placeholders. The final dates will be announced in January (first exam just after the end of the course, second exam toward the end of the semester break).
- 11.12:
**Notice:**for Exercise Sheet 8 only, the tutorials of groups 2 and 3 will both be moved to Wed 19.12 from 16:00 - 18:00 in PH II 227, Seminarraum E20 (see link below for a map). - 21.11: Partial solution for Ex. 2, Sheet 3 has been posted.
- 05.11: In Exercise Sheet 2, the convention in Eq.(2) has been changed, which simplifies the notation.
- New starting times: Mon 12:00 (from 5/11), Wed 12:30 (since 24/10) due to request -- if this poses an issue for you, please let me know and we will need to vote on it.
- 20.10: We just posted the first exercise sheet, to be discussed in the central tutorial on 24.10.
- The first lecture will take place on the 15.10.
- The slot on Wednesday 2:15-4pm will be used for the central tutorial and additional lectures. The first central tutorial is on the 24.10 and the first Wed afternoon lecture is on the 31.10.

## Syllabus

- Lagrange formalism and canonical quantisation of the scalar field
- Path integral representation of quantum field theory
- Green functions and scattering processes
- Regularization and renormalization
- Effective quantum field theory and the computation of quantum loop corrections
- Renormalization group
- Symmetries and relativistic particles and quantum fields with spin
- Spinors
- Massless spin 1 fields and gauge redundancy

Insights into mathematical foundations of QFT will also be provided (Dr. Dybalski).

## Lectures

Monday 12:00-13:30 (HS3)

Wednesday 12:30-14:00 (HS3)

and every alternating week: Wednesday 14.15-16.00 (HS3)

## Central Tutorial

Every alternating week: Wednesday 14:15-16:00 (HS3)

**Exercises**

Rooms and schedule:

group # | time | location | tutor | email @tum.de |
---|---|---|---|---|

1 | Wed 16:00 - 18:00 | CH 26410 | Giovanni B. | giovanni.banelli |

Wed 16:00 - 18:00 | PH II 227, Seminarraum E20 | (backup room, not a regular tutorial) | ||

2 | Thu 12:00 - 14:00 | PH 2271 | Max R. | max.ruhdorfer |

3 | Mon 14:00 - 16:00 | C.3202 | Stefan S. | stefan.stelzl |

If obtained, the exercise bonus of 0.3 points is valid for the first attempt at the final exam. It only applies to a passed exam, in particular 4.3 is not upgraded to 4.0.

**Prerequisites:**

It is very helpful to have heard a course similar to Relativity, Particles, Fields at TUM. Please study the parts Prof. Weiler's Relativity, Particles, Fields script.pdf which haven't been covered in your previous lectures. See also information about the preparatory course below.

Note that RPF uses slightly different conventions, e.g. annihilation and creation operators a_{RPF} -> (2 E_p)^{1/2} a_{QFT} and particle states: |p>_{RPF} -> (2 E_p)^{1/2} |p>_{QFT} which affects the form of the expansion of Φ(x) etc.

**Literature:**

We will not follow a single book but it is strongly recommended that you read up on the topics discussed in the lecture in one or more of these books:

Schwartz - Quantum Field Theory and the Standard Model (*recommended*. Note that Schwartz uses slightly different conventions, e.g. annihilation and creation operators a_{schwartz} -> (2 E_p)^{1/2} a_{weiler} and particle states: |**p**>_{schwartz} -> (2 E_p)^{1/2} |**p**>_{weiler }which affects the form of the expansion of Φ(x) etc.)

S. Weinberg - Quantum Field Theory (1 & 2)

Peskin/Schroeder - An Introduction to Quantum Field Theory

Preskill - Lecture notes (on his Caltech website)

Cheng/Li - Gauge theory of elementary particle physics

and for the more mathematically oriented:

R. Haag - Local Quantum Physics, Fields, Particles, Algebras, Springer

J. Glimm/A. Jaffe - Quantum Physics, A Functional Integral Point of View, Springer

Sign conventions across QFT books, see here

## Lecture Notes

(Please contact Javi or Ennio for the password, it will be also announced during the lecture.)

**1** p1_p9.pdf (file:16/10/18, Introduction, Weinberg vs. Wilson, Basic paradigm)

**2** p9_p19.pdf (file:17/10/18, Noether, Dimensions, canonical quantization of free scalar fields)

**3** p20-p28.pdf (file:22/10, causality, energy-momentum tensor, zero-point energies IR vs.UV, particle states, FIXED conventions on p21 and p22)

**4 **p29-37.pdf (file: 24/10, complex scalar field, U(1), conserved charge, causality and anti-particles,interaction picture)

**m1 **math_lecture_1.pdf (W. Dybalski, 30/10 update: includes references now)

**5 **p38-p57.pdf (file:31/10, Dyson series, S-matrix, Path integral in QM, Hamilton form of path integral)

**6 **p57-p67.pdf (file:5/11, path int. for Green's functions, evaluating time ordered operator products on the vacuum, i \eps from boundary conditions and asymptotic fields, Lagrange form of path integral, functional determinant)

**7 **p67-p76.pdf (file:7/11, classical limit and ordering ambiguity, perturbative evaluation of n-point functions, generating functional, Feynman propagator, general result for free n-point Green functions.)

**m2 **Math_lecture_2.pdf (W. Dybalski)

**8** p77-p94.pdf (file:*update 22/11*, free complex scalar field & propagator, perturbative expansion of Green fn, position space Feynman rules, momentum space rules, vertex factors, composite operator Green fn, T* product and non-covariant contributions in operator formalism, connected Green fn)

**9** p95-p115.pdf (file:22/11, master formula of path integral transformations: Schwinger-Dyson and Ward Identities, contact terms, internal symmetries, local symmetries and current conservation in QFT, scattering theory, single particle states of interacting hamiltonian, Lippmann-Schwinger equation, QM example)

**m3/m4 **Math_lecture_3.pdf and Math_lecture_4.pdf (W. Dybalski)

**10 **p116-p127.pdf (scattering matrix, asymptotic fields, weak coupling limit, scattering cross-section and differential decay rate)

**11 **p128-p142.pdf (field strength renormalization, spectral representation of two-point function, mass renormalization, propagators can decay at most like 1/p^{2}, LSZ-reduction part 1)

**12 **p142-p151.pdf (LSZ-reduction part 2, amputated Green-fn, example computation of a scattering cross-section, kinematic limits)

**13 **p152-p170.pdf (regularization and renormalization, simplified 2->2 example, renormalized coupling, counter-terms, self-energy, Wick rotation, where do divergencies come from?, Feynman parameters, dimensional regularization, renormalized perturbation theory)

**14** p171-p179.pdf (renormalization schemes, the MSbar scheme, systematic renormalization, degree of divergence, computation of superficial degree of divergence, renormalizable and super-renormalizable couplings, renormalizable field theories, effective QFTs and non-renormalizable theories, predictive at low energy)

**m5/m6 **Math_lecture_5.pdf and Math_lecture_6.pdf (W. Dybalski)

**15** p179-p185.pdf (effective theories, example with heavy and light scalar fields, matching at tree-level, systematic expansion in 1/M, loop corrections, local/polynomial UV remainder)

**16** p186-p204.pdf (renormalization group,scattering example, scale dependence of coupling and mass, overview over possible scale dependence: trivial UV/IR fixed points, non-trivial fixed points, RGE flow with multiple couplings, flow of relevant/marginal/irrelevant couplings, Wilsonian RGE and phase transitions, RGE flow of Green-fn: Callan Symanzik equation, 2pt-fn example with non-trivial fixed point)

**17** p205-p214.pdf (Poincare representations, symmetries/groups/representations, Lorentz-invariance vs. unitarity, Poincare group ISO(1,3), proper orthochronos Lorentz group, Poincare algebra and Casimirs, classifying particle states)

**18** p215-p232.pdf (one particle basis states, massive particles with spin: little group SO(3) and reference momentum, Wigner functions, massless particles: ISO(2), continuous spin, helicity, CPT and helicity ±σ, parity and time reversal vs. little group, anti-unitary operators, internal symmetries and charge conjugation, relativistic fields and A=½(J+iK) and B=½(J-iK), complexified SO(1,3) = SU(2)_{A}xSU(2)_{B} )

**19** p232-p237.pdf (free fields transforming under Lorentz, creation and annihilation op.'s inherit transformation from 1-particle states, little group vs. matrix rep of Lorentz, scalar field)

**20** p238-p249.pdf (spin 1/2 particles and spinor fields, Weyl spinors (1/2,0) and (0,1/2), Dirac spinor, field operator and expansion coefficients, little group rotations in standard frame, boost to general momentum, neutral spin operator, spin-statistics: relativistic causality requires anti-commutation relations for spin 1/2, Majorana equation)

**21** p250-p285.pdf and spinor_summary.pdf (Field Lagrangian, Euler-Lagrange equations, equal-time anti-commutator relations, charged&massive: Dirac fermions as (1/2,0)+(0,1/2) Lorentzfields, mass-less spin 1/2 and helicity, Grassmann variables and fermionic path integral, Fermion Green function, Time ordering, Dirac propagator, Feynman rules, Compton scattering example)

**22** p286-p310.pdf (Vector fields and gauge redundancy, massive vector field, decoupling of scalar mode, little group unitary reps, path integral for massive spin 1, massless particles and gauge redundancy, covariant derivative, gauge fixing, QED)

## Central Tutorial

**1 **

## Exercise Sheets

Exercise coordination: Dr. Javi Serra (javi.serraATtum.de) and Dr. Ennio Salvioni (ennio.salvioniATtum.de)

**0** Exercise Sheet 0. You are encouraged to work on it but, for this time only, not required to hand in solutions; will be discussed at Central Tutorial of 24.10.

**1 **Exercise Sheet 1. [**hand-in: Ex1, a,b,d; Ex2, b,d,e.**]** **Solution due by 31.10 at 16:00, the parts required for hand-in will be announced on 30.10 at 8:00am; discussed at tutorials of 31.10, 31.10 and 05.11.

**2** Exercise Sheet 2. [**hand-in: Ex1, a,b,c; Ex2, c.**] Solution due by 07.11 at 16:00, the parts required for hand-in will be announced on 06.11 at 8:00am; discussed at tutorials of 07.11, 08.11 and 12.11.

**3** Exercise Sheet 3. [**hand-in: Ex1, b; Ex2, a; Ex2, b, parts (a) and (b).**] Solution due by 14.11 at 16:00, the parts required for hand-in will be announced on 13.11 at 8:00am; discussed at tutorials of 14.11, 15.11 and 19.11.

*Hint for Exercise 2 b) (a): you can use the ansatz f_z(a^\dagger) = exp [ A a^\dagger + B (a^\dagger)^2 ], where A, B are coefficients to be determined. Hint for Exercise 2 b) (b): decompose \hat\phi = \hat\phi_+ + \hat\phi_- (where \hat\phi_- depends only on annihilation operators) and first use \hat\phi_- instead of \hat\phi . *

**4** Exercise Sheet 4. [**hand-in: Ex2, parts a1, b, c; Ex3.**] Solution due by 21.11 at 16:00, the parts required for hand-in will be announced on 20.11 at 8:00am; discussed at tutorials of 21.11, 22.11 and 26.11.

**5** Exercise Sheet 5. [**hand-in: Ex1, b; Ex2.**] Solution due by 28.11 at 16:00, the parts required for hand-in will be announced on 27.11 at 8:00am; discussed at tutorials of 28.11, 29.11 and 03.12.

**6** Exercise Sheet 6. [**hand-in: Ex1, b; Ex2, b,c,d.**] Solution due by 05.12 at 16:00, the parts required for hand-in will be announced on 04.12 at 8:00am; discussed at tutorials of 05.12, 06.12 and 10.12.

**7** Exercise Sheet 7. [**hand-in: Ex1, b, d; Ex2.**] Solution due by 12.12 at 16:00, the parts required for hand-in will be announced on 11.12 at 8:00am; discussed at tutorials of 12.12, 13.12 and 17.12.

**8** Exercise Sheet 8. [**hand-in: Ex1.**] Solution due by 19.12 at 16:00, the parts required for hand-in will be announced on 18.12 at 8:00am; discussed at tutorials of 19.12 and 19.12.

**9** Exercise Sheet 9. [**hand-in: Ex1; Ex2.**] Solution due by 09.01 at 16:00, the parts required for hand-in will be announced on 08.01 at 8:00am; discussed at tutorials of 09.01, 10.01 and 14.01.

**10** Exercise Sheet 10. [**hand-in: Ex1; Ex2, a, c; Ex3.**] Solution due by 16.01 at 16:00, the parts required for hand-in will be announced on 15.01 at 8:00am; discussed at tutorials of 16.01, 17.01 and 21.01.

**11** Exercise Sheet 11. [**hand-in: Ex3, a, b, c, e; Ex4; Ex5.**] Solution due by 23.01 at 16:00, the parts required for hand-in will be announced on 22.01 at 8:00am; discussed at tutorials of 23.01, 24.01 and 28.01.

**12** Exercise Sheet 12. [**hand-in: Ex1, a7, a8, b5, b6; Ex2.**] Solution due by 30.01 at 16:00, the parts required for hand-in will be announced on 29.01 at 8:00am; discussed at tutorials of 30.01, 31.01 and 04.02.

*Note that a further hint has been added for Exercise 1.*

**13** Exercise Sheet 13. [**hand-in: Ex1; Ex2; Ex3.**] Solution due by 06.02 at 16:00, the parts required for hand-in will be announced on 05.02 at 8:00am; discussed at tutorials of 06.02 and 06.02.

**14** Bonus Sheet 14. This will be discussed in the central tutorial on 20.2. The sheet will be discussed in HS3 from 13:30 until 16:00.

## Preparatory Course

The preparatory course will review some basic concepts from the theory of continuous systems and advanced quantum mechanics to facilitate students with various backgrounds to deal with the rapid start of the course. Highly recommended.

Topics: Action principle in classical mechanics, Euler-Lagrange equations for continuous systems (vibrating string) and classical fields, quantization, Fock space, ...

The preparatory course will be taught by Dr. P. Vaudrevange*.*

Wed, 10.10.2018: 10:00-11:30 and 12:30-14:00, Hörsaal PH 3 TUM-PH

Thu,11.10.2018: 9:30-11:00 and 14:00-15:30, Hörsaal PH 3 TUM-PH

**Attention**: Friday’s lecture is at LMU right after the TMP introduction!

Fri, 12.10.2018: 10:00-11:30 and 12:30-14:00, A348/349

#### P. Vaudrevange's script:

part1.pdf, part2.pdf, and part3.pdf

You can also consult Prof. Weiler's Relativity, particles, fields script which covers similar topics.