Quantum Field Theory II
|
|
- Camron Riley
- 5 years ago
- Views:
Transcription
1 Quantum Field Theory II PHYS-P 622 Radovan Dermisek, Indiana University Notes based on: M. Srednicki, Quantum Field Theory Chapters: 13, 14, 16-21, 26-28, 51, 52, 61-68, 44, 53, 69-74, 30-32, 84-86, 75, 87-89, 29 1
2 Review of scalar field theory Srednicki 5, 9, 10 2
3 The LSZ reduction formula based on S-5 In order to describe scattering experiments we need to construct appropriate initial and final states and calculate scattering amplitude. Summary of free theory: one particle state: vacuum state is annihilated by all a s: then, one particle state has normalization: normalization is Lorentz invariant! see e.g. Peskin & Schroeder, p. 23 3
4 Let s define a time-independent operator: that creates a particle localized in the momentum space near wave packet with width! and localized in the position space near the origin. (go back to position space by Furier transformation) is a state that evolves with time (in the Schrödinger picture), wave packet propagates and spreads out and so the particle is localized far from the origin in at. for in the past. In the interacting theory is a state describing two particles widely separated is not time independent 4
5 A guess for a suitable initial state: Similarly, let s consider a final state: we can normalize the wave packets so that where again and The scattering amplitude is then: 5
6 A useful formula: Integration by parts, surface term = 0, particle is localized, (wave packet needed). E.g. is 0 in free theory, but not in interacting one! 6
7 Thus we have: or its hermitian conjugate: The scattering amplitude: we put in time ordering (without changing anything) is then given as (generalized to n i- and n f-particles): 7
8 Lehmann-Symanzik-Zimmermann formula (LSZ) Note, initial and final states now have delta-function normalization, multiparticle generalization of. We expressed scattering amplitudes in terms of correlation functions! Now we need to learn how to calculate correlation functions in interacting quantum field theory. 8
9 Comments: we assumed that creation operators of free field theory would work comparably in the interacting theory... acting on ground state: we want, so that we can always shift the field by a constant is a Lorentz invariant number is a single particle state otherwise it would create a linear combination of the ground state and a single particle state so that 9
10 one particle state: is a Lorentz invariant number we want, since this is what it is in free field theory, correctly normalized one particle state. creates a we can always rescale (renormalize) the field by a constant so that. 10
11 multiparticle states: is a Lorentz invariant number in general, creates some multiparticle states. One can show that the overlap between a one-particle wave packet and a multiparticle wave packet goes to zero as time goes to infinity. see the discussion in Srednicki, p By waiting long enough we can make the multiparticle contribution to the scattering amplitude as small as we want. 11
12 Summary: Scattering amplitudes can be expressed in terms of correlation functions of fields of an interacting quantum field theory: provided that the fields obey: Lehmann-Symanzik-Zimmermann formula (LSZ) these conditions might not be consistent with the original form of lagrangian! 12
13 Consider for example: After shifting and rescaling we will have instead: 13
14 Path integral for interacting field Let s consider an interacting phi-cubed QFT: based on S-9 with fields satisfying: we want to evaluate the path integral for this theory: 14
15 it can be also written as: epsilon trick leads to additional factor; to get the correct normalization we require: and for the path integral of the free field theory we have found: 15
16 assumes thus in the case of: the perturbing lagrangian is: counterterm lagrangian in the limit we expect and we will find and 16
17 Let s look at Z( J ) (ignoring counterterms for now). Define: exponentials defined by series expansion: let s look at a term with particular values of P (propagators) and V (vertices): number of surviving sources, (after taking all derivatives) E (for external) is E = 2P - 3V 3V derivatives can act on 2P sources in (2P)! / (2P-3V)! different ways e.g. for V = 2, P = 3 there is 6! different terms 17
18 V = 2, E = 0 ( P = 3 ):!!!!!!!!!!!!!!!! " " " " " " 3! 3! x 1 x 2 2! 6 6 3! = 1 24 dx 1 dx 2 (iz g g) 2 1 i (x 1 x 2 ) 1 i (x 1 x 2 ) 1 i (x 1 x 2 ) symmetry factor 18
19 V = 2, E = 0 ( P = 3 ):!!!!!!!!!!!!!!!! " " " " " " 3! 3! 3! 2 x 1 x 2 2! 6 6 3! = 1 8 dx 1 dx 2 (iz g g) 2 1 i (x 1 x 1 ) 1 i (x 1 x 2 ) 1 i (x 1 x 1 ) symmetry factor 19
20 Feynman diagrams: a line segment stands for a propagator vertex joining three line segments stands for a filled circle at one end of a line segment stands for a source e.g. for V = 1, E = 1 What about those symmetry factors? symmetry factors are related to symmetries of Feynman diagrams... 20
21 Symmetry factors: we can rearrange three derivatives without changing diagram we can rearrange three vertices we can rearrange two sources we can rearrange propagators this in general results in overcounting of the number of terms that give the same result; this happens when some rearrangement of derivatives gives the same match up to sources as some rearrangement of sources; this is always connected to some symmetry property of the diagram; factor by which we overcounted is the symmetry factor 21
22 the endpoints of each propagator can be swapped and the effect is duplicated by swapping the two vertices propagators can be rearranged in 3! ways, and all these rearrangements can be duplicated by exchanging the derivatives at the vertices 22
23 23
24 24
25 25
26 26
27 27
28 28
29 29
30 All these diagrams are connected, but Z( J ) contains also diagrams that are products of several connected diagrams: e.g. for V = 4, E = 0 ( P = 6 ) in addition to connected diagrams we also have : and also: and also: 30
31 All these diagrams are connected, but Z( J ) contains also diagrams that are products of several connected diagrams: e.g. for V = 4, E = 0 ( P = 6 ) in addition to connected diagrams we also have : A general diagram D can be written as: the number of given C in D additional symmetry factor not already accounted for by symmetry factors of connected diagrams; it is nontrivial only if D contains identical C s: particular connected diagram 31
32 Now is given by summing all diagrams D: any D can be labeled by a set of n s thus we have found that is given by the exponential of the sum of connected diagrams. imposing the normalization means we can omit vacuum diagrams (those with no sources), thus we have: vacuum diagrams are omitted from the sum 32
33 If there were no counterterms we would be done: in that case, the vacuum expectation value of the field is: only diagrams with one source contribute: and we find: (the source is removed by the derivative) we used since we know which is not zero, as required for the LSZ; so we need counterterm 33
34 Including term in the interaction lagrangian results in a new type of vertex on which a line segment ends e.g. corresponding Feynman rule is: at the lowest order of g only contributes: in order to satisfy we have to choose: Note, must be purely imaginary so that Y is real; and, in addition, the integral over k is ultraviolet divergent. 34
35 to make sense out of it, we introduce an ultraviolet cutoff and in order to keep Lorentz-transformation properties of the propagator we make the replacement: the integral is now convergent: we will do this type of calculations later... and indeed, is purely imaginary. after choosing Y so that we can take the limit Y becomes infinite... we repeat the procedure at every order in g 35
36 e.g. at we have to sum up: and add to Y whatever term is needed to maintain... this way we can determine the value of Y order by order in powers of g. Adjusting Y so that means that the sum of all connected diagrams with a single source is zero! In addition, the same infinite set of diagrams with source replaced by ANY subdiagram is zero as well. Rule: ignore any diagram that, when a single line is cut, fall into two parts, one of which has no sources. = tadpoles 36
37 all that is left with up to 4 sources and 4 vertices is: 37
38 finally, let s take a look at the other two counterterms: we get it results in a new vertex at which two lines meet, the corresponding vertex factor or the Feynman rule is Summary: we have calculated in theory and expressed it as we used integration by parts for every diagram with a propagator there is additional one with this vertex where W is the sum of all connected diagrams with no tadpoles and at least two sources! 38
39 Scattering amplitudes and the Feynman rules based on S-10 We have found Z( J ) for the phi-cubed theory and now we can calculate vacuum expectation values of the time ordered products of any number of fields. Let s define exact propagator: short notation: thus we find: W contains diagrams with at least two sources
40 4-point function: we have dropped terms that contain Let s define connected correlation functions: does not correspond to any interaction; when plugged to LSZ, no scattering happens and plug these into LSZ formula. 40
41 at the lowest order in g only one diagram contributes: S = 8 derivatives remove sources in 4! possible ways, and label external legs in 3 distinct ways: each diagram occurs 8 times, which nicely cancels the symmetry factor. 41
42 General result for tree diagrams (no closed loops): each diagram with a distinct endpoint labeling has an overall symmetry factor 1. Let s finish the calculation of y z putting together factors for all pieces of Feynman diagrams we get: 42
43 For two incoming and two outgoing particles the LSZ formula is: and we have just written terms of propagators. The LSZ formula highly simplifies due to: in We find: 43
44 44
45 four-momentum is conserved in scattering process Let s define: scattering matrix element From this calculation we can deduce a set of rules for computing. 45
46 Feynman rules to calculate : for each incoming and outgoing particle draw an external line and label it with four-momentum and an arrow specifying the momentum flow draw all topologically inequivalent diagrams for internal lines draw arrows arbitrarily but label them with momenta so that momentum is conserved in each vertex assign factors: 1 for each external line for each internal line with momentum k for each vertex sum over all the diagrams and get 46
47 Additional rules for diagrams with loops: a diagram with L loops will have L internal momenta that are not fixed; integrate over all these momenta with measure divide by a symmetry factor include diagrams with counterterm vertex that connects two propagators, each with the same momentum k; the value of the vertex is now we are going to use to calculate cross section... 47
48 Lehmann-Källén form of the exact propagator based on S-13 What can we learn about the exact propagator from general principles? Let s define the exact propagator: The field is normalized so that Normalization of a one particle state in d-dimensions: The d-dimensional completeness statement: identity operator in one-particle subspace Lorentz invariant phase-space differential 48
49 Let s also define the exact propagator in the momentum space: In free field theory we found: it has an isolated pole at with residue one! What about the exact propagator in the interacting theory? 49
50 Let s insert the complete set of energy eigenstates between the two fields; for we have: ground state, 0 - energy one particle states multiparticle continuum of states specified by the total three momentum k and other parameters: relative momenta,..., denoted symbolically by n 50
51 51
52 Let s define the spectral density: then we have: 52
53 similarly: and we can plug them to the formula for time-ordered product: we get: was your homework or, in the momentum space: it has an isolated pole at Lehmann-Källén form of the exact propagator with residue one! 53
REVIEW REVIEW. Quantum Field Theory II
Quantum Field Theory II PHYS-P 622 Radovan Dermisek, Indiana University Notes based on: M. Srednicki, Quantum Field Theory Chapters: 13, 14, 16-21, 26-28, 51, 52, 61-68, 44, 53, 69-74, 30-32, 84-86, 75,
More informationQuantum Field Theory II
Quantum Field Theory II PHYS-P 622 Radovan Dermisek, Indiana University Notes based on: M. Srednicki, Quantum Field Theory Chapters: 13, 14, 16-21, 26-28, 51, 52, 61-68, 44, 53, 69-74, 30-32, 84-86, 75,
More informationReview of scalar field theory. Srednicki 5, 9, 10
Review of scalar field theory Srednicki 5, 9, 10 2 The LSZ reduction formula based on S-5 In order to describe scattering experiments we need to construct appropriate initial and final states and calculate
More informationSummary of free theory: one particle state: vacuum state is annihilated by all a s: then, one particle state has normalization:
The LSZ reduction formula based on S-5 In order to describe scattering experiments we need to construct appropriate initial and final states and calculate scattering amplitude. Summary of free theory:
More informationREVIEW REVIEW. A guess for a suitable initial state: Similarly, let s consider a final state: Summary of free theory:
LSZ reduction for spin-1/2 particles based on S-41 In order to describe scattering experiments we need to construct appropriate initial and final states and calculate scattering amplitude. Summary of free
More informationLSZ reduction for spin-1/2 particles
LSZ reduction for spin-1/2 particles based on S-41 In order to describe scattering experiments we need to construct appropriate initial and final states and calculate scattering amplitude. Summary of free
More informationScattering amplitudes and the Feynman rules
Scattering amplitudes and the Feynman rules based on S-10 We have found Z( J ) for the phi-cubed theory and now we can calculate vacuum expectation values of the time ordered products of any number of
More informationWe will also need transformation properties of fermion bilinears:
We will also need transformation properties of fermion bilinears: Parity: some product of gamma matrices, such that so that is hermitian. we easily find: 88 And so the corresponding bilinears transform
More informationMaxwell s equations. electric field charge density. current density
Maxwell s equations based on S-54 Our next task is to find a quantum field theory description of spin-1 particles, e.g. photons. Classical electrodynamics is governed by Maxwell s equations: electric field
More informationLoop corrections in Yukawa theory based on S-51
Loop corrections in Yukawa theory based on S-51 Similarly, the exact Dirac propagator can be written as: Let s consider the theory of a pseudoscalar field and a Dirac field: the only couplings allowed
More informationThe path integral for photons
The path integral for photons based on S-57 We will discuss the path integral for photons and the photon propagator more carefully using the Lorentz gauge: as in the case of scalar field we Fourier-transform
More informationSrednicki Chapter 9. QFT Problems & Solutions. A. George. August 21, Srednicki 9.1. State and justify the symmetry factors in figure 9.
Srednicki Chapter 9 QFT Problems & Solutions A. George August 2, 22 Srednicki 9.. State and justify the symmetry factors in figure 9.3 Swapping the sources is the same thing as swapping the ends of the
More informationWhat is a particle? Keith Fratus. July 17, 2012 UCSB
What is a particle? Keith Fratus UCSB July 17, 2012 Quantum Fields The universe as we know it is fundamentally described by a theory of fields which interact with each other quantum mechanically These
More informationPath integral in quantum mechanics based on S-6 Consider nonrelativistic quantum mechanics of one particle in one dimension with the hamiltonian:
Path integral in quantum mechanics based on S-6 Consider nonrelativistic quantum mechanics of one particle in one dimension with the hamiltonian: let s look at one piece first: P and Q obey: Probability
More informationBeta functions in quantum electrodynamics
Beta functions in quantum electrodynamics based on S-66 Let s calculate the beta function in QED: the dictionary: Note! following the usual procedure: we find: or equivalently: For a theory with N Dirac
More informationTwo particle elastic scattering at 1-loop
Two particle elastic scattering at 1-loop based on S-20 Let s use our rules to calculate two-particle elastic scattering amplitude in, theory in 6 dimensions including all one-loop corrections: For the
More informationQFT PS7: Interacting Quantum Field Theory: λφ 4 (30/11/17) The full propagator in λφ 4 theory. Consider a theory of a real scalar field φ
QFT PS7: Interacting Quantum Field Theory: λφ 4 (30/11/17) 1 Problem Sheet 7: Interacting Quantum Field Theory: λφ 4 Comments on these questions are always welcome. For instance if you spot any typos or
More informationTENTATIVE SYLLABUS INTRODUCTION
Physics 615: Overview of QFT Fall 2010 TENTATIVE SYLLABUS This is a tentative schedule of what we will cover in the course. It is subject to change, often without notice. These will occur in response to
More informationVacuum Energy and Effective Potentials
Vacuum Energy and Effective Potentials Quantum field theories have badly divergent vacuum energies. In perturbation theory, the leading term is the net zero-point energy E zeropoint = particle species
More information1 Equal-time and Time-ordered Green Functions
1 Equal-time and Time-ordered Green Functions Predictions for observables in quantum field theories are made by computing expectation values of products of field operators, which are called Green functions
More informationLoop Corrections: Radiative Corrections, Renormalization and All
Loop Corrections: Radiative Corrections, Renormalization and All That Michael Dine Department of Physics University of California, Santa Cruz Nov 2012 Loop Corrections in φ 4 Theory At tree level, we had
More informationQFT. Chapter 14: Loop Corrections to the Propagator
QFT Chapter 14: Loop Corrections to the Propagator Overview Here we turn to our next major topic: loop order corrections. We ll consider the effect on the propagator first. This has at least two advantages:
More informationQFT. Unit 11: Cross Sections and Decay Rates
QFT Unit 11: Cross Sections and Decay Rates Decays and Collisions n When it comes to elementary particles, there are only two things that ever really happen: One particle decays into stuff Two particles
More informationPhysics 218. Quantum Field Theory. Professor Dine. Green s Functions and S Matrices from the Operator (Hamiltonian) Viewpoint
Physics 28. Quantum Field Theory. Professor Dine Green s Functions and S Matrices from the Operator (Hamiltonian) Viewpoint Field Theory in a Box Consider a real scalar field, with lagrangian L = 2 ( µφ)
More informationObservables from Correlation Functions
Observables from Correlation Functions In this chapter we learn how to compute physical quantities from correlation functions beyond leading order in the perturbative expansion. We will not discuss ultraviolet
More informationDiagramology Types of Feynman Diagram
1. Pieces of Diagrams Diagramology Types of Feynman Diagram Tim Evans (2nd January 2018) Feynman diagrams 1 have four types of element:- Internal Vertices represented by a dot with some legs coming out.
More informationWeek 5-6: Lectures The Charged Scalar Field
Notes for Phys. 610, 2011. These summaries are meant to be informal, and are subject to revision, elaboration and correction. They will be based on material covered in class, but may differ from it by
More information1 The Quantum Anharmonic Oscillator
1 The Quantum Anharmonic Oscillator Perturbation theory based on Feynman diagrams can be used to calculate observables in Quantum Electrodynamics, like the anomalous magnetic moment of the electron, and
More information2P + E = 3V 3 + 4V 4 (S.2) D = 4 E
PHY 396 L. Solutions for homework set #19. Problem 1a): Let us start with the superficial degree of divergence. Scalar QED is a purely bosonic theory where all propagators behave as 1/q at large momenta.
More informationQuantum Field Theory Spring 2019 Problem sheet 3 (Part I)
Quantum Field Theory Spring 2019 Problem sheet 3 (Part I) Part I is based on material that has come up in class, you can do it at home. Go straight to Part II. 1. This question will be part of a take-home
More informationPart I. Many-Body Systems and Classical Field Theory
Part I. Many-Body Systems and Classical Field Theory 1. Classical and Quantum Mechanics of Particle Systems 3 1.1 Introduction. 3 1.2 Classical Mechanics of Mass Points 4 1.3 Quantum Mechanics: The Harmonic
More informationQuantum Electrodynamics Test
MSc in Quantum Fields and Fundamental Forces Quantum Electrodynamics Test Monday, 11th January 2010 Please answer all three questions. All questions are worth 20 marks. Use a separate booklet for each
More informationGraviton contributions to the graviton self-energy at one loop order during inflation
Graviton contributions to the graviton self-energy at one loop order during inflation PEDRO J. MORA DEPARTMENT OF PHYSICS UNIVERSITY OF FLORIDA PASI2012 1. Description of my thesis problem. i. Graviton
More informationLecture 6:Feynman diagrams and QED
Lecture 6:Feynman diagrams and QED 0 Introduction to current particle physics 1 The Yukawa potential and transition amplitudes 2 Scattering processes and phase space 3 Feynman diagrams and QED 4 The weak
More informationMaxwell s equations. based on S-54. electric field charge density. current density
Maxwell s equations based on S-54 Our next task is to find a quantum field theory description of spin-1 particles, e.g. photons. Classical electrodynamics is governed by Maxwell s equations: electric field
More information19 Entanglement Entropy in Quantum Field Theory
19 Entanglement Entropy in Quantum Field Theory So far we have discussed entanglement in ordinary quantum mechanics, where the Hilbert space of a finite region is finite dimensional. Now we will discuss
More informationAn Introduction to Quantum Field Theory (Peskin and Schroeder) Solutions
An Introduction to Quantum Field Theory (Peskin and Schroeder) Solutions Andrzej Pokraka February 5, 07 Contents 4 Interacting Fields and Feynman Diagrams 4. Creation of Klein-Gordon particles from a classical
More informationLecture notes for QFT I (662)
Preprint typeset in JHEP style - PAPER VERSION Lecture notes for QFT I (66) Martin Kruczenski Department of Physics, Purdue University, 55 Northwestern Avenue, W. Lafayette, IN 47907-036. E-mail: markru@purdue.edu
More informationQuantum Field Theory. Kerson Huang. Second, Revised, and Enlarged Edition WILEY- VCH. From Operators to Path Integrals
Kerson Huang Quantum Field Theory From Operators to Path Integrals Second, Revised, and Enlarged Edition WILEY- VCH WILEY-VCH Verlag GmbH & Co. KGaA I vh Contents Preface XIII 1 Introducing Quantum Fields
More informationQFT Perturbation Theory
QFT Perturbation Theory Ling-Fong Li (Institute) Slide_04 1 / 43 Interaction Theory As an illustration, take electromagnetic interaction. Lagrangian density is The combination L = ψ (x ) γ µ ( i µ ea µ
More informationThe Feynman Propagator and Cauchy s Theorem
The Feynman Propagator and Cauchy s Theorem Tim Evans 1 (1st November 2018) The aim of these notes is to show how to derive the momentum space form of the Feynman propagator which is (p) = i/(p 2 m 2 +
More informationContinuous symmetries and conserved currents
Continuous symmetries and conserved currents based on S-22 Consider a set of scalar fields, and a lagrangian density let s make an infinitesimal change: variation of the action: setting we would get equations
More informationlattice QCD and the hadron spectrum Jozef Dudek ODU/JLab
lattice QCD and the hadron spectrum Jozef Dudek ODU/JLab a black box? QCD lattice QCD observables (scattering amplitudes?) in these lectures, hope to give you a look inside the box 2 these lectures how
More informationQuantum Field theory. Kimmo Kainulainen. Abstract: QFT-II partial lecture notes. Keywords:.
Preprint typeset in JHEP style - PAPER VERSION Quantum Field theory Kimmo Kainulainen Department of Physics, P.O.Box 35 (YFL), FI-40014 University of Jyväskylä, Finland, and Helsinki Institute of Physics,
More informationAttempts at relativistic QM
Attempts at relativistic QM based on S-1 A proper description of particle physics should incorporate both quantum mechanics and special relativity. However historically combining quantum mechanics and
More informationusing D 2 D 2 D 2 = 16p 2 D 2
PHY 396 T: SUSY Solutions for problem set #4. Problem (a): Let me start with the simplest case of n = 0, i.e., no good photons at all and one bad photon V = or V =. At the tree level, the S tree 0 is just
More informationPHY 396 L. Solutions for homework set #20.
PHY 396 L. Solutions for homework set #. Problem 1 problem 1d) from the previous set): At the end of solution for part b) we saw that un-renormalized gauge invariance of the bare Lagrangian requires Z
More informationL = 1 2 µφ µ φ m2 2 φ2 λ 0
Physics 6 Homework solutions Renormalization Consider scalar φ 4 theory, with one real scalar field and Lagrangian L = µφ µ φ m φ λ 4 φ4. () We have seen many times that the lowest-order matrix element
More informationQuantum Field Theory
Quantum Field Theory PHYS-P 621 Radovan Dermisek, Indiana University Notes based on: M. Srednicki, Quantum Field Theory 1 Attempts at relativistic QM based on S-1 A proper description of particle physics
More informationA short Introduction to Feynman Diagrams
A short Introduction to Feynman Diagrams J. Bijnens, November 2008 This assumes knowledge at the level of Chapter two in G. Kane, Modern Elementary Particle Physics. This note is more advanced than needed
More informationQuantum Field Theory
Quantum Field Theory PHYS-P 621 Radovan Dermisek, Indiana University Notes based on: M. Srednicki, Quantum Field Theory 1 Attempts at relativistic QM based on S-1 A proper description of particle physics
More informationUnitarity, Dispersion Relations, Cutkosky s Cutting Rules
Unitarity, Dispersion Relations, Cutkosky s Cutting Rules 04.06.0 For more information about unitarity, dispersion relations, and Cutkosky s cutting rules, consult Peskin& Schröder, or rather Le Bellac.
More informationQuantum Algorithms for Quantum Field Theories
Quantum Algorithms for Quantum Field Theories Stephen Jordan Joint work with Keith Lee John Preskill Science, 336:1130 (2012) Jan 24, 2012 The full description of quantum mechanics for a large system with
More informationRepresentations of Lorentz Group
Representations of Lorentz Group based on S-33 We defined a unitary operator that implemented a Lorentz transformation on a scalar field: How do we find the smallest (irreducible) representations of the
More informationDR.RUPNATHJI( DR.RUPAK NATH )
11 Perturbation Theory and Feynman Diagrams We now turn our attention to interacting quantum field theories. All of the results that we will derive in this section apply equally to both relativistic and
More information1 Running and matching of the QED coupling constant
Quantum Field Theory-II UZH and ETH, FS-6 Assistants: A. Greljo, D. Marzocca, J. Shapiro http://www.physik.uzh.ch/lectures/qft/ Problem Set n. 8 Prof. G. Isidori Due: -5-6 Running and matching of the QED
More informationQuantum Field Theory
Quantum Field Theory PHYS-P 621 Radovan Dermisek, Indiana University Notes based on: M. Srednicki, Quantum Field Theory 1 Attempts at relativistic QM based on S-1 A proper description of particle physics
More informationOn particle production by classical backgrounds
On particle production by classical backgrounds arxiv:hep-th/0103251v2 30 Mar 2001 Hrvoje Nikolić Theoretical Physics Division, Rudjer Bošković Institute, P.O.B. 180, HR-10002 Zagreb, Croatia hrvoje@faust.irb.hr
More informationLight-Cone Quantization of Electrodynamics
Light-Cone Quantization of Electrodynamics David G. Robertson Department of Physics, The Ohio State University Columbus, OH 43210 Abstract Light-cone quantization of (3+1)-dimensional electrodynamics is
More informationSrednicki Chapter 62
Srednicki Chapter 62 QFT Problems & Solutions A. George September 28, 213 Srednicki 62.1. Show that adding a gauge fixing term 1 2 ξ 1 ( µ A µ ) 2 to L results in equation 62.9 as the photon propagator.
More informationPHYS 611, SPR 12: HOMEWORK NO. 4 Solution Guide
PHYS 611 SPR 12: HOMEWORK NO. 4 Solution Guide 1. In φ 3 4 theory compute the renormalized mass m R as a function of the physical mass m P to order g 2 in MS scheme and for an off-shell momentum subtraction
More informationCorrelation Functions in Perturbation Theory
Correlation Functions in Perturbation Theory Many aspects of quantum field theory are related to its n-point correlation functions F n (x 1,...,x n ) def = Ω TˆΦ H (x 1 ) ˆΦ H (x n ) Ω (1) or for theories
More informationTheory of Elementary Particles homework VIII (June 04)
Theory of Elementary Particles homework VIII June 4) At the head of your report, please write your name, student ID number and a list of problems that you worked on in a report like II-1, II-3, IV- ).
More informationPlan for the rest of the semester. ψ a
Plan for the rest of the semester ϕ ψ a ϕ(x) e iα(x) ϕ(x) 167 Representations of Lorentz Group based on S-33 We defined a unitary operator that implemented a Lorentz transformation on a scalar field: and
More informationEuclidean path integral formalism: from quantum mechanics to quantum field theory
: from quantum mechanics to quantum field theory Dr. Philippe de Forcrand Tutor: Dr. Marco Panero ETH Zürich 30th March, 2009 Introduction Real time Euclidean time Vacuum s expectation values Euclidean
More informationThe Quantum Theory of Finite-Temperature Fields: An Introduction. Florian Divotgey. Johann Wolfgang Goethe Universität Frankfurt am Main
he Quantum heory of Finite-emperature Fields: An Introduction Florian Divotgey Johann Wolfgang Goethe Universität Franfurt am Main Fachbereich Physi Institut für heoretische Physi 1..16 Outline 1 he Free
More informationOn Perturbation Theory, Dyson Series, and Feynman Diagrams
On Perturbation Theory, Dyson Series, and Feynman Diagrams The Interaction Picture of QM and the Dyson Series Many quantum systems have Hamiltonians of the form Ĥ Ĥ0 + ˆV where Ĥ0 is a free Hamiltonian
More informationQFT Perturbation Theory
QFT Perturbation Theory Ling-Fong Li Institute) Slide_04 1 / 44 Interaction Theory As an illustration, take electromagnetic interaction. Lagrangian density is The combination is the covariant derivative.
More informationParticle Physics 2018 Final Exam (Answers with Words Only)
Particle Physics 2018 Final Exam (Answers with Words Only) This was a hard course that likely covered a lot of new and complex ideas. If you are feeling as if you could not possibly recount all of the
More informationReview and Preview. We are testing QED beyond the leading order of perturbation theory. We encounter... IR divergences from soft photons;
Chapter 9 : Radiative Corrections 9.1 Second order corrections of QED 9. Photon self energy 9.3 Electron self energy 9.4 External line renormalization 9.5 Vertex modification 9.6 Applications 9.7 Infrared
More information2 Quantization of the scalar field
22 Quantum field theory 2 Quantization of the scalar field Commutator relations. The strategy to quantize a classical field theory is to interpret the fields Φ(x) and Π(x) = Φ(x) as operators which satisfy
More informationlattice QCD and the hadron spectrum Jozef Dudek ODU/JLab
lattice QCD and the hadron spectrum Jozef Dudek ODU/JLab the light meson spectrum relatively simple models of hadrons: bound states of constituent quarks and antiquarks the quark model empirical meson
More informationAn Introduction to. Michael E. Peskin. Stanford Linear Accelerator Center. Daniel V. Schroeder. Weber State University. Advanced Book Program
An Introduction to Quantum Field Theory Michael E. Peskin Stanford Linear Accelerator Center Daniel V. Schroeder Weber State University 4B Advanced Book Program TT Addison-Wesley Publishing Company Reading,
More informationChapter 7 -- Radiative Corrections: some formal developments. A quotation from Peskin & Schroeder, Chapter 7:
Chapter 7 -- Radiative Corrections: some formal developments A quotation from Peskin & Schroeder, Chapter 7: 7.1. Field-strength renormalization 7.2. The LSZ reduction formula 7.3. The optical theorem
More informationQuantum Field Theory. and the Standard Model. !H Cambridge UNIVERSITY PRESS MATTHEW D. SCHWARTZ. Harvard University
Quantum Field Theory and the Standard Model MATTHEW D. Harvard University SCHWARTZ!H Cambridge UNIVERSITY PRESS t Contents v Preface page xv Part I Field theory 1 1 Microscopic theory of radiation 3 1.1
More informationInteracting Quantum Fields C6, HT 2015
Interacting Quantum Fields C6, HT 2015 Uli Haisch a a Rudolf Peierls Centre for Theoretical Physics University of Oxford OX1 3PN Oxford, United Kingdom Please send corrections to u.haisch1@physics.ox.ac.uk.
More informationThe Hamiltonian operator and states
The Hamiltonian operator and states March 30, 06 Calculation of the Hamiltonian operator This is our first typical quantum field theory calculation. They re a bit to keep track of, but not really that
More informationConventions for fields and scattering amplitudes
Conventions for fields and scattering amplitudes Thomas DeGrand 1 1 Department of Physics, University of Colorado, Boulder, CO 80309 USA (Dated: September 21, 2017) Abstract This is a discussion of conventions
More information21 Renormalization group
Renormalization group. Renormalization and interpolation Probably because they describe point particles, quantum field theories are divergent. Unknown physics at very short distance scales, removes these
More informationPhysics 342 Lecture 17. Midterm I Recap. Lecture 17. Physics 342 Quantum Mechanics I
Physics 342 Lecture 17 Midterm I Recap Lecture 17 Physics 342 Quantum Mechanics I Monday, March 1th, 28 17.1 Introduction In the context of the first midterm, there are a few points I d like to make about
More informationRenormalization in Lorentz-violating field theory: external states
Renormalization in Lorentz-violating field theory: external states Robertus Potting CENTRA and Physics Department Faculdade de Ciências e Tecnologia Universidade do Algarve Faro, Portugal SME2015 Summer
More informationCOMPLEXITY OF QUANTUM FIELD THEORIES
COMPLEXITY OF QUANTUM FIELD THEORIES MAX ZIMET Abstract. Quantum field theories (QFTs) reconcile special relativity and quantum mechanics. We discuss the computational complexity of these theories. In
More informationTextbook Problem 4.2: We begin by developing Feynman rules for the theory at hand. The Hamiltonian clearly
PHY 396 K. Solutions for problem set #10. Textbook Problem 4.2: We begin by developing Feynman rules for the theory at hand. The Hamiltonian clearly decomposes into Ĥ = Ĥ0 + ˆV where and Ĥ 0 = Ĥfree Φ
More informationDonoghue, Golowich, Holstein Chapter 4, 6
1 Week 7: Non linear sigma models and pion lagrangians Reading material from the books Burgess-Moore, Chapter 9.3 Donoghue, Golowich, Holstein Chapter 4, 6 Weinberg, Chap. 19 1 Goldstone boson lagrangians
More informationPhase Transitions and Renormalization:
Phase Transitions and Renormalization: Using quantum techniques to understand critical phenomena. Sean Pohorence Department of Applied Mathematics and Theoretical Physics University of Cambridge CAPS 2013
More information1 Mathematical preliminaries
1 Mathematical preliminaries The mathematical language of quantum mechanics is that of vector spaces and linear algebra. In this preliminary section, we will collect the various definitions and mathematical
More informationManifestly diffeomorphism invariant classical Exact Renormalization Group
Manifestly diffeomorphism invariant classical Exact Renormalization Group Anthony W. H. Preston University of Southampton Supervised by Prof. Tim R. Morris Talk prepared for Asymptotic Safety seminar,
More informationPHY 2404S Lecture Notes
PHY 2404S Lecture Notes Michael Luke Winter, 2003 These notes are perpetually under construction. Please let me know of any typos or errors. Once again, large portions of these notes have been plagiarized
More informationFinal reading report: BPHZ Theorem
Final reading report: BPHZ Theorem Wang Yang 1 1 Department of Physics, University of California at San Diego, La Jolla, CA 92093 This reports summarizes author s reading on BPHZ theorem, which states
More informationWeek 11 Reading material from the books
Week 11 Reading material from the books Polchinski, Chapter 6, chapter 10 Becker, Becker, Schwartz, Chapter 3, 4 Green, Schwartz, Witten, chapter 7 Normalization conventions. In general, the most convenient
More informationPHY 396 K. Problem set #3. Due September 29, 2011.
PHY 396 K. Problem set #3. Due September 29, 2011. 1. Quantum mechanics of a fixed number of relativistic particles may be a useful approximation for some systems, but it s inconsistent as a complete theory.
More informationString Theory I GEORGE SIOPSIS AND STUDENTS
String Theory I GEORGE SIOPSIS AND STUDENTS Department of Physics and Astronomy The University of Tennessee Knoxville, TN 37996-2 U.S.A. e-mail: siopsis@tennessee.edu Last update: 26 ii Contents 4 Tree-level
More informationd 3 k In the same non-relativistic normalization x k = exp(ikk),
PHY 396 K. Solutions for homework set #3. Problem 1a: The Hamiltonian 7.1 of a free relativistic particle and hence the evolution operator exp itĥ are functions of the momentum operator ˆp, so they diagonalize
More informationLecture 11 Perturbative calculation
M.Krawczyk, AFZ Particles and Universe 11 1 Particles and Universe Lecture 11 Perturbative calculation Maria Krawczyk, Aleksander F. Żarnecki Faculty of Physics UW I.Theory of elementary particles description
More informationThe Quantum Theory of Fields. Volume I Foundations Steven Weinberg
The Quantum Theory of Fields Volume I Foundations Steven Weinberg PREFACE NOTATION x x xxv 1 HISTORICAL INTRODUCTION 1 1.1 Relativistic Wave Mechanics 3 De Broglie waves q Schrödinger-Klein-Gordon wave
More informationThe Non-commutative S matrix
The Suvrat Raju Harish-Chandra Research Institute 9 Dec 2008 (work in progress) CONTEMPORARY HISTORY In the past few years, S-matrix techniques have seen a revival. (Bern et al., Britto et al., Arkani-Hamed
More informationarxiv: v3 [hep-th] 2 May 2018
Cutkosky Rules for Superstring Field Theory Roji Pius a and Ashoke Sen b arxiv:1604.01783v3 [hep-th] 2 May 2018 a Perimeter Institute for Theoretical Physics Waterloo, ON N2L 2Y5, Canada b Harish-Chandra
More informationLecture-05 Perturbation Theory and Feynman Diagrams
Lecture-5 Perturbation Theory and Feynman Diagrams U. Robkob, Physics-MUSC SCPY639/428 September 3, 218 From the previous lecture We end up at an expression of the 2-to-2 particle scattering S-matrix S
More informationQFT. Unit 1: Relativistic Quantum Mechanics
QFT Unit 1: Relativistic Quantum Mechanics What s QFT? Relativity deals with things that are fast Quantum mechanics deals with things that are small QFT deals with things that are both small and fast What
More informationMonte Carlo simulation calculation of the critical coupling constant for two-dimensional continuum 4 theory
Monte Carlo simulation calculation of the critical coupling constant for two-dimensional continuum 4 theory Will Loinaz * Institute for Particle Physics and Astrophysics, Physics Department, Virginia Tech,
More information