Ginzburg-Landau Theory of Type II Superconductors

Size: px
Start display at page:

Download "Ginzburg-Landau Theory of Type II Superconductors"

Transcription

1 Ginzburg-Landau Theory of Type II Superconductors This interesting application of quantum field theory is reviewed by B. Rosenstein and D. Li, Ginzburg-Landau theory of type II superconductors in magnetic field, Rev. Mod. Phys. 82, (2010). G[Ψ, A] = G n [A] + d 2 r dz ( ) 2π D = + i A, Φ 0 [ ] D 2m DΨ 2 2m z Ψ 2 + a (T ) Ψ 2 + b (T ) c 2 Ψ 4 + d 3 (B H)2 r 8π Φ 0 = 2π c, e = 2e, a (T ) α (T e c2 T ), b (T ) const > 0, where H = Hẑ is the applied magnetic field, B = A is the magentic induction of the photon field, and Φ 0 is the Magnetic flux quantum of the superconducting vortices. PHY /25/2013,

2 Feynman Diagrams for High-Temperature Perturbation Series The nature of transition between the normal (high-temperature) and mixed phases was studied by G.J. Ruggeri and D.J. Thouless, Perturbation series for the critical behavior of type II superconductors near H c2, J. Phys. F: Metal Phys. 6, (1976). They evaluated the free energy to sixth order, computing = 80 Feynman diagrams in orders n = 2, 3, 4, 5, 6. The free energy was evaluated by others up to order n = 13. The 2010 URGE to Compute team, R.A. Dygert, M. Heavner, D. Pandey and M. Skvarch, counted n = 14, 15, see R.A. Dygert Honors Thesis. Order n = Diagrams PHY /25/2013

3 Exponentiation of Vacuum Diagrams Anharmonic oscillator ground state energy diagrams on the left. In QED and the Standard Model, the PHY /25/2013

4 vacuum diagrams contribute to the Casimir effect and the cosmological constant. In high-energy applications they exponentiate and do not contribute to scattering amplitudes as shown in Peskin-Schroeder 4.4 Feynman Diagrams. In condensed matter field theory, the vacuum (Fermi sea) diagrams are fundamentally important, as discussed in Altland-Simons 5.2 Ground state energy of the interacting electron gas: PHY /25/2013

5 Feynman Diagrams for Scalar Field Theory There are many software packages available to generate Feynman diagrams and compute probability amplitudes, see the Karlsruhe University Institut für Theoretische Teilchenphysik (TTP) Links to algebraic programs. These packages are based on special algorithms to handle the combinatorics of large numbers of diagrams. A good example of an algorithm to generate scalar field theory diagrams is given in the article by H. Kleinert, A. Pelster, B. Kastening, and M. Bachmann, Recursive graphical construction of Feynman diagrams and their multiplicities in φ 4 and φ 2 A theory, Phys. Rev. E 62, (2000). The important equations from this article are summarized below. Definitions and Notation Consider a self-interacting scalar field φ with N components in d Euclidean dimensions whose thermal fluctuations are controlled by the energy functional E[φ] = 1 G φ 1φ 2 + g V 1234 φ 1 φ 2 φ 3 φ 4 4! 12 with some coupling constant g. In this short-hand notation, the spatial and tensorial arguments of the field φ, the bilocal kernel G 1, and the quartic interaction V are indicated by simple number indices, for example 1234 PHY /25/2013

6 the index 1 x 1, α 1, and d d x 1, φ 1 φ α1 (x 1 ), G 1 12 G 1 α 1,α 2 (x 1, x 2 ), V 1234 V α1,α 2,α 3,α 4 (x 1, x 2, x 3, x 4 ). 1 α 1 The scalar field propagator, also called the kernel, is a functional matrix G 1, while V is a functional tensor, both being symmetric in their indices. The energy functional describes generically d-dimensional Euclidean φ 4 -theories. These are models for a family of universality classes of continuous phase transitions, such as the O(N)- symmetric φ 4 -theory which serves to derive the critical phenomena in dilute polymer solutions (N = 0), Ising- and Heisenberg-like magnets (N = 1, 3), and superfluids (N = 2). In all these cases, the energy functional is specified by G 1 α 1,α 2 (x 1, x 2 ) = δ α1,α 2 ( 2 x1 + m 2) δ(x 1 x 2 ), V α1,α 2,α 3,α 4 (x 1, x 2, x 3, x 4 ) = 1 3 δ α 1,α 2 δ α3,α 4 + δ α1,α 3 δ α2,α 4 + δ α1,α 4 δ α2,α 3 δ(x 1 x 2 )δ(x 1 x 3 )δ(x 1 x 4 ), where the mass m 2 is proportional to the temperature distance from the critical point. In the following G 1 and V are completely general, except for the symmetry with respect to their indices. By using natural units in which the Boltzmann constant k B times the temperature T equals unity, the PHY /25/2013

7 partition function is determined as a functional integral over the Boltzmann weight e E[φ] Z = Dφ e E[φ] and may be evaluated perturbatively as a power series in the coupling constant g. From this we obtain the negative free energy W = ln Z as an expansion ( ) p 1 g W = W (p). p! 4! p=0 The coefficients W (p) may be displayed as connected vacuum diagrams constructed from lines and vertices. Each line represents a free correlation function G 12, which is the functional inverse of the kernel G 1 in the energy functional, defined by G 12 G 1 23 = δ 13. The vertices represent an integral over the interaction V PHY /25/2013

8 Multiplicity of Vacuum Diagrams To construct all connected vacuum diagrams contributing to W (p) to each order p in perturbation theory, one connects p vertices with 4p legs in all possible ways according to Feynman s rules which follow from Wick s expansion of correlation functions into a sum of all pair contractions. This yields an increasing number of Feynman diagrams, each with a certain multiplicity which follows from combinatorics. In total there are 4! p p! ways of ordering the 4p legs of the p vertices. This number is reduced by permutations of the legs and the vertices which leave a vacuum diagram invariant. Denoting the number of self-, double, triple and fourfold connections with S, D, T, F, there are 2! S, 2! D, 3! T, 4! F leg permutations. An additional reduction arises from the number N of identical vertex permutations where the vertices remain attached to the lines emerging from them in the same way as before. The resulting multiplicity of a connected vacuum diagram in the φ 4 -theory is therefore given by the formula M E=0 φ 4 = 4! p p! 2! S+D 3! T 4! F N. The superscript E records that the number of external legs of the connected vacuum diagrams is zero. The diagrammatic representation of the coefficients W (p) in the expansion of the negative free energy W is displayed Table 1 up to five loops, and Table 2 lists diagrams and coefficients for corrections to the scalar field propagator. PHY /25/2013

9 PHY /25/2013

10 PHY /25/2013

11 Source Function Approach In the path integral method, a source function J(x) is coupled to the scalar field and modifies the free energy functional E[φ, J] = E[φ] J 1 φ 1. The functional integral for the generating functional Z[J] = Dφ e E[φ,J] is first explicitly calculated for a vanishing coupling constant g, yielding Z (0) [J] = exp 1 2 Tr ln G G 12 J 1 J 2, 2 where the trace of the logarithm of the kernel is defined by the series Tr ln G 1 ( 1) n+1 = G 1 12 n δ 12 G 1 n1 δ n1. n=1 1...n For non-vanishing coupling constant g, the generating functional Z[J] is expanded in powers of the quartic interaction V. Functional derivatives with respect to the current J then give the correlation functions of the theory. The original partition function can thus be obtained from the free generating functional by the formula Z = exp g 4! δ 4 V 1234 δj 1 δj 2 δj 3 δj 4 Z (0) [J]. J=0 PHY /25/2013

12 Expanding the exponential in a power series, we arrive at the perturbation expansion Z = 1 + g δ 4 V ! 1234 δj 1 δj 2 δj 3 δj ( ) 2 g δ 8 V 1234 V Z (0) [J], 2 4! δj 1 δj 2 δj 3 δj 4 δj 5 δj 6 δj 7 δj 8 J= in which the pth order contribution for the partition function requires the evaluation of 4p functional derivatives with respect to the current J. Kernel Approach The derivation of the perturbation expansion simplifies, if we use functional derivatives with respect to the kernel G 1 in the energy functional rather than with respect to the current J. This allows us to substitute the previous expression for the partition function by Z = exp g V 1234 δ 2 δg 1 12 δg 1 34 e W (0), where the zeroth order of the negative free energy has the diagrammatic representation W (0) = 1 2 Tr ln G PHY /25/2013

13 Expanding again the exponential in a power series, we obtain Z = 1 + g δ 2 V δg ( ) 2 g δ 4 V 1234 V δg δg 1 12 δg 1 Thus we need only half as many functional derivatives and gives δw (0) = 1 2 G δ 2 W (0) 12, = 1 4 G 13G 24 + G 14 G 23, δg 1 12 δg 1 12 δg 1 34 such that the partition function Z becomes Z = 1 + g 3 V 1234 G 12 G ( ) 2 g V 1234 V ! ! ] [9 G 12 G 34 G 56 G G 15 G 26 G 37 G G 12 G 35 G 46 G 78 This has the diagrammatic representation Z = 1 + g ( ) [ 2 g ! 2 4! 34 δg 1 56 δg ] +... e W (0) e W (0). e W (0). All diagrams in this expansion follow directly by successively cutting lines of the basic one-loop vacuum diagram. By going to the logarithm of the partition function Z, we find a diagrammatic expansion for the negative free energy W W = g 4! ( ) 2 g ! PHY /25/2013

1 Interaction of Quantum Fields with Classical Sources

1 Interaction of Quantum Fields with Classical Sources 1 Interaction of Quantum Fields with Classical Sources A source is a given external function on spacetime t, x that can couple to a dynamical variable like a quantum field. Sources are fundamental in the

More information

arxiv:hep-th/ v1 21 Jul 1999

arxiv:hep-th/ v1 21 Jul 1999 Recursive Graphical Construction of Feynman Diagrams and Their Multiplicities in φ - and in φ A-Theory Hagen Kleinert, Axel Pelster, Boris Kastening, and M. Bachmann Institut für Theoretische Physik, Freie

More information

E 1 G 1. Recursive graphical construction of Feynman diagrams and their multiplicities in 4 and 2 A theory

E 1 G 1. Recursive graphical construction of Feynman diagrams and their multiplicities in 4 and 2 A theory PHYSICAL REVIEW E VOLUME 62, NUMBER 2 AUGUST 2000 Recursive graphical construction of Feynman diagrams and their multiplicities in 4 and 2 A theory Hagen Kleinert, 1 Axel Pelster, 1 Boris Kastening, 2

More information

Recursive calculation of effective potential and variational resummation

Recursive calculation of effective potential and variational resummation JOURNAL OF MATHEMATICAL PHYSICS 46, 032101 2005 Recursive calculation of effective potential and variational resummation Sebastian F. Brandt and Hagen Kleinert Freie Universität Berlin, Institut für Theoretische

More information

4 4 and perturbation theory

4 4 and perturbation theory and perturbation theory We now consider interacting theories. A simple case is the interacting scalar field, with a interaction. This corresponds to a -body contact repulsive interaction between scalar

More information

Recursive graphical construction of Feynman diagrams and their weights in Ginzburg Landau theory

Recursive graphical construction of Feynman diagrams and their weights in Ginzburg Landau theory Physica A 3 2002 141 152 www.elsevier.com/locate/physa Recursive graphical construction of Feynman diagrams and their weights in Ginzburg Landau theory H. Kleinert a;, A. Pelster a, B. VandenBossche a;b

More information

Quantum Field Theory. Kerson Huang. Second, Revised, and Enlarged Edition WILEY- VCH. From Operators to Path Integrals

Quantum 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 information

Tadpole Summation by Dyson-Schwinger Equations

Tadpole Summation by Dyson-Schwinger Equations MS-TPI-96-4 hep-th/963145 Tadpole Summation by Dyson-Schwinger Equations arxiv:hep-th/963145v1 18 Mar 1996 Jens Küster and Gernot Münster Institut für Theoretische Physik I, Universität Münster Wilhelm-Klemm-Str.

More information

1 The Quantum Anharmonic Oscillator

1 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 information

Symmetry Factors of Feynman Diagrams for Scalar Fields

Symmetry Factors of Feynman Diagrams for Scalar Fields arxiv:0907.0859v2 [hep-ph] 15 Nov 2010 Symmetry Factors of Feynman Diagrams for Scalar Fields P. V. Dong, L. T. Hue, H. T. Hung, H. N. Long, and N. H. Thao Institute of Physics, VAST, P.O. Box 429, Bo

More information

A General Expression for Symmetry Factors of Feynman Diagrams. Abstract

A General Expression for Symmetry Factors of Feynman Diagrams. Abstract A General Expression for Symmetry Factors of Feynman Diagrams C.D. Palmer a and M.E. Carrington b,c a Department of Mathematics, Brandon University, Brandon, Manitoba, R7A 6A9 Canada b Department of Physics,

More information

Graviton 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 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 information

Euclidean path integral formalism: from quantum mechanics to quantum field theory

Euclidean 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 information

Vacuum Energy and Effective Potentials

Vacuum 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 information

cond-mat/ Mar 1995

cond-mat/ Mar 1995 Critical Exponents of the Superconducting Phase Transition Michael Kiometzis, Hagen Kleinert, and Adriaan M. J. Schakel Institut fur Theoretische Physik Freie Universitat Berlin Arnimallee 14, 14195 Berlin

More information

Interface Profiles in Field Theory

Interface Profiles in Field Theory Florian König Institut für Theoretische Physik Universität Münster January 10, 2011 / Forschungsseminar Quantenfeldtheorie Outline φ 4 -Theory in Statistical Physics Critical Phenomena and Order Parameter

More information

The path integral for photons

The 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 information

The Strong Interaction and LHC phenomenology

The Strong Interaction and LHC phenomenology The Strong Interaction and LHC phenomenology Juan Rojo STFC Rutherford Fellow University of Oxford Theoretical Physics Graduate School course Lecture 2: The QCD Lagrangian, Symmetries and Feynman Rules

More information

Ginzburg-Landau Theory of Phase Transitions

Ginzburg-Landau Theory of Phase Transitions Subedi 1 Alaska Subedi Prof. Siopsis Physics 611 Dec 5, 008 Ginzburg-Landau Theory of Phase Transitions 1 Phase Transitions A phase transition is said to happen when a system changes its phase. The physical

More information

DR.RUPNATHJI( DR.RUPAK NATH )

DR.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 information

NTNU Trondheim, Institutt for fysikk

NTNU Trondheim, Institutt for fysikk NTNU Trondheim, Institutt for fysikk Examination for FY3464 Quantum Field Theory I Contact: Michael Kachelrieß, tel. 998971 Allowed tools: mathematical tables 1. Spin zero. Consider a real, scalar field

More information

6.1 Quadratic Casimir Invariants

6.1 Quadratic Casimir Invariants 7 Version of May 6, 5 CHAPTER 6. QUANTUM CHROMODYNAMICS Mesons, then are described by a wavefunction and baryons by Φ = q a q a, (6.3) Ψ = ǫ abc q a q b q c. (6.4) This resolves the old paradox that ground

More information

REVIEW REVIEW. Quantum Field Theory II

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 information

Quantum Field Theory II

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 information

1 Equal-time and Time-ordered Green Functions

1 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 information

Quantum Field Theory 2 nd Edition

Quantum Field Theory 2 nd Edition Quantum Field Theory 2 nd Edition FRANZ MANDL and GRAHAM SHAW School of Physics & Astromony, The University of Manchester, Manchester, UK WILEY A John Wiley and Sons, Ltd., Publication Contents Preface

More information

Combinatorics of Feynman diagrams and algebraic lattice structure in QFT

Combinatorics of Feynman diagrams and algebraic lattice structure in QFT Combinatorics of Feynman diagrams and algebraic lattice structure in QFT Michael Borinsky 1 Humboldt-University Berlin Departments of Physics and Mathematics - Alexander von Humboldt Group of Dirk Kreimer

More information

A Brief Introduction to Linear Response Theory with Examples in Electromagnetic Response

A Brief Introduction to Linear Response Theory with Examples in Electromagnetic Response A Brief Introduction to Linear Response Theory with Examples in Electromagnetic Response Robert Van Wesep May 3, 2013 In order to gain information about any physical system, it is necessary to probe the

More information

Self-consistent Conserving Approximations and Renormalization in Quantum Field Theory at Finite Temperature

Self-consistent Conserving Approximations and Renormalization in Quantum Field Theory at Finite Temperature Self-consistent Conserving Approximations and Renormalization in Quantum Field Theory at Finite Temperature Hendrik van Hees in collaboration with Jörn Knoll Contents Schwinger-Keldysh real-time formalism

More information

Notes on Renormalization Group: ϕ 4 theory and ferromagnetism

Notes on Renormalization Group: ϕ 4 theory and ferromagnetism Notes on Renormalization Group: ϕ 4 theory and ferromagnetism Yi Zhou (Dated: November 4, 015) In this lecture, we shall study the ϕ 4 theory up to one-loop RG and discuss the application to ferromagnetic

More information

Contents. 1.1 Prerequisites and textbooks Physical phenomena and theoretical tools The path integrals... 9

Contents. 1.1 Prerequisites and textbooks Physical phenomena and theoretical tools The path integrals... 9 Preface v Chapter 1 Introduction 1 1.1 Prerequisites and textbooks......................... 1 1.2 Physical phenomena and theoretical tools................. 5 1.3 The path integrals..............................

More information

Lecture-05 Perturbation Theory and Feynman Diagrams

Lecture-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 information

arxiv:hep-ph/ v1 18 Nov 1996

arxiv:hep-ph/ v1 18 Nov 1996 TTP96-55 1 MPI/PhT/96-122 hep-ph/9611354 November 1996 arxiv:hep-ph/9611354v1 18 Nov 1996 AUTOMATIC COMPUTATION OF THREE-LOOP TWO-POINT FUNCTIONS IN LARGE MOMENTUM EXPANSION K.G. Chetyrkin a,b, R. Harlander

More information

QFT 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) 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 information

Review of scalar field theory. Srednicki 5, 9, 10

Review 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 information

A Superfluid Universe

A Superfluid Universe A Superfluid Universe Lecture 2 Quantum field theory & superfluidity Kerson Huang MIT & IAS, NTU Lecture 2. Quantum fields The dynamical vacuum Vacuumscalar field Superfluidity Ginsburg Landau theory BEC

More information

Variational Calculation of Eective Classical. November 12, Abstract

Variational Calculation of Eective Classical. November 12, Abstract Variational Calculation of Eective Classical Potential at T to Higher Orders H.Kleinert H.Meyer November, 99 Abstract Using the new variational approach proposed recently for a systematic improvement of

More information

Blobbed Topological Recursion for Tensor Models

Blobbed Topological Recursion for Tensor Models Quantum Gravity in Paris Institut Henri Poincaré, Paris 2017 Outline 1 2 3 The case for a tensor model 4 5 Matrix models for 2d quantum gravity Integrals of matrices with Feynman graphs = poly-angulations

More information

Quantum Field Theory. and the Standard Model. !H Cambridge UNIVERSITY PRESS MATTHEW D. SCHWARTZ. Harvard University

Quantum 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 information

Path Integrals in Quantum Field Theory C6, HT 2014

Path Integrals in Quantum Field Theory C6, HT 2014 Path Integrals in Quantum Field Theory C6, HT 01 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 information

Summary of free theory: one particle state: vacuum state is annihilated by all a s: then, one particle state has normalization:

Summary 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 information

An Introduction to Quantum Field Theory (Peskin and Schroeder) Solutions

An 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 information

SM, EWSB & Higgs. MITP Summer School 2017 Joint Challenges for Cosmology and Colliders. Homework & Exercises

SM, EWSB & Higgs. MITP Summer School 2017 Joint Challenges for Cosmology and Colliders. Homework & Exercises SM, EWSB & Higgs MITP Summer School 017 Joint Challenges for Cosmology and Colliders Homework & Exercises Ch!"ophe Grojean Ch!"ophe Grojean DESY (Hamburg) Humboldt University (Berlin) ( christophe.grojean@desy.de

More information

Loop corrections in Yukawa theory based on S-51

Loop 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 information

Renormalization of the Yukawa Theory Physics 230A (Spring 2007), Hitoshi Murayama

Renormalization of the Yukawa Theory Physics 230A (Spring 2007), Hitoshi Murayama Renormalization of the Yukawa Theory Physics 23A (Spring 27), Hitoshi Murayama We solve Problem.2 in Peskin Schroeder. The Lagrangian density is L 2 ( µφ) 2 2 m2 φ 2 + ψ(i M)ψ ig ψγ 5 ψφ. () The action

More information

The boundary state from open string fields. Yuji Okawa University of Tokyo, Komaba. March 9, 2009 at Nagoya

The boundary state from open string fields. Yuji Okawa University of Tokyo, Komaba. March 9, 2009 at Nagoya The boundary state from open string fields Yuji Okawa University of Tokyo, Komaba March 9, 2009 at Nagoya Based on arxiv:0810.1737 in collaboration with Kiermaier and Zwiebach (MIT) 1 1. Introduction Quantum

More information

Contact interactions in string theory and a reformulation of QED

Contact interactions in string theory and a reformulation of QED Contact interactions in string theory and a reformulation of QED James Edwards QFT Seminar November 2014 Based on arxiv:1409.4948 [hep-th] and arxiv:1410.3288 [hep-th] Outline Introduction Worldline formalism

More information

Maxwell s equations. electric field charge density. current density

Maxwell 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 information

Scattering amplitudes and the Feynman rules

Scattering 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 information

NTNU Trondheim, Institutt for fysikk

NTNU Trondheim, Institutt for fysikk FY3464 Quantum Field Theory II Final exam 0..0 NTNU Trondheim, Institutt for fysikk Examination for FY3464 Quantum Field Theory II Contact: Kåre Olaussen, tel. 735 9365/4543770 Allowed tools: mathematical

More information

Green s function of the Vector fields on DeSitter Background

Green s function of the Vector fields on DeSitter Background Green s function of the Vector fields on DeSitter Background Gaurav Narain The Institute for Fundamental Study (IF) March 10, 2015 Talk Based On Green s function of the Vector fields on DeSitter Background,

More information

Part I. Many-Body Systems and Classical Field Theory

Part 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 information

arxiv:cond-mat/ v1 4 Aug 2003

arxiv:cond-mat/ v1 4 Aug 2003 Conductivity of thermally fluctuating superconductors in two dimensions Subir Sachdev arxiv:cond-mat/0308063 v1 4 Aug 2003 Abstract Department of Physics, Yale University, P.O. Box 208120, New Haven CT

More information

TENTATIVE SYLLABUS INTRODUCTION

TENTATIVE 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 information

Theory of Elementary Particles homework VIII (June 04)

Theory 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 information

arxiv: v2 [math-ph] 4 Apr 2018

arxiv: v2 [math-ph] 4 Apr 2018 Enumeration of N-rooted maps using quantum field theory K. Gopala Krishna 1, Patrick Labelle, and Vasilisa Shramchenko 3 arxiv:1709.0100v [math-ph] 4 Apr 018 Abstract A one-to-one correspondence is proved

More information

Week 5-6: Lectures The Charged Scalar Field

Week 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 information

V.E. Rochev Institute for High Energy Physics, Protvino, Moscow region, Russia. 1 Introduction and General Consideration

V.E. Rochev Institute for High Energy Physics, Protvino, Moscow region, Russia. 1 Introduction and General Consideration A non-perturbative method of calculation of Green functions V.E. Rochev Institute for High Energy Physics, 142284 Protvino, Moscow region, Russia Abstract. A new method for non-perturbative calculation

More information

Critical Behavior I: Phenomenology, Universality & Scaling

Critical Behavior I: Phenomenology, Universality & Scaling Critical Behavior I: Phenomenology, Universality & Scaling H. W. Diehl Fachbereich Physik, Universität Duisburg-Essen, Campus Essen 1 Goals recall basic facts about (static equilibrium) critical behavior

More information

10 Thermal field theory

10 Thermal field theory 0 Thermal field theory 0. Overview Introduction The Green functions we have considered so far were all defined as expectation value of products of fields in a pure state, the vacuum in the absence of real

More information

The Fermion Bag Approach

The Fermion Bag Approach The Fermion Bag Approach Anyi Li Duke University In collaboration with Shailesh Chandrasekharan 1 Motivation Monte Carlo simulation Sign problem Fermion sign problem Solutions to the sign problem Fermion

More information

Final reading report: BPHZ Theorem

Final 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 information

Quantum Electrodynamics Test

Quantum 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 information

Vector mesons in the fireball

Vector mesons in the fireball Symmetries and Self-consistency Vector mesons in the fireball π,... ρ/ω γ e e + Hendrik van Hees Fakultät für Physik Universität Bielefeld Symmetries and Self-consistency p.1 Content Concepts Real time

More information

arxiv: v1 [math-ph] 5 Sep 2017

arxiv: v1 [math-ph] 5 Sep 2017 Enumeration of N-rooted maps using quantum field theory K. Gopala Krishna 1, Patrick Labelle, and Vasilisa Shramchenko 3 arxiv:1709.0100v1 [math-ph] 5 Sep 017 Abstract A one-to-one correspondence is proved

More information

Quantum Field Theory II

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 information

Green Functions in Many Body Quantum Mechanics

Green Functions in Many Body Quantum Mechanics Green Functions in Many Body Quantum Mechanics NOTE This section contains some advanced material, intended to give a brief introduction to methods used in many body quantum mechanics. The material at the

More information

The Ginzburg-Landau Theory

The Ginzburg-Landau Theory The Ginzburg-Landau Theory A normal metal s electrical conductivity can be pictured with an electron gas with some scattering off phonons, the quanta of lattice vibrations Thermal energy is also carried

More information

PRINCIPLES OF PHYSICS. \Hp. Ni Jun TSINGHUA. Physics. From Quantum Field Theory. to Classical Mechanics. World Scientific. Vol.2. Report and Review in

PRINCIPLES OF PHYSICS. \Hp. Ni Jun TSINGHUA. Physics. From Quantum Field Theory. to Classical Mechanics. World Scientific. Vol.2. Report and Review in LONDON BEIJING HONG TSINGHUA Report and Review in Physics Vol2 PRINCIPLES OF PHYSICS From Quantum Field Theory to Classical Mechanics Ni Jun Tsinghua University, China NEW JERSEY \Hp SINGAPORE World Scientific

More information

Statistical Mechanics

Statistical Mechanics Franz Schwabl Statistical Mechanics Translated by William Brewer Second Edition With 202 Figures, 26 Tables, and 195 Problems 4u Springer Table of Contents 1. Basic Principles 1 1.1 Introduction 1 1.2

More information

Abrikosov vortex lattice solution

Abrikosov vortex lattice solution Abrikosov vortex lattice solution A brief exploration O. Ogunnaike Final Presentation Ogunnaike Abrikosov vortex lattice solution Physics 295b 1 / 31 Table of Contents 1 Background 2 Quantization 3 Abrikosov

More information

Quantum Field Theory Example Sheet 4 Michelmas Term 2011

Quantum Field Theory Example Sheet 4 Michelmas Term 2011 Quantum Field Theory Example Sheet 4 Michelmas Term 0 Solutions by: Johannes Hofmann Laurence Perreault Levasseur Dave M. Morris Marcel Schmittfull jbh38@cam.ac.uk L.Perreault-Levasseur@damtp.cam.ac.uk

More information

Physics 444: Quantum Field Theory 2. Homework 2.

Physics 444: Quantum Field Theory 2. Homework 2. Physics 444: Quantum Field Theory Homework. 1. Compute the differential cross section, dσ/d cos θ, for unpolarized Bhabha scattering e + e e + e. Express your results in s, t and u variables. Compute the

More information

be stationary under variations in A, we obtain Maxwell s equations in the form ν J ν = 0. (7.5)

be stationary under variations in A, we obtain Maxwell s equations in the form ν J ν = 0. (7.5) Chapter 7 A Synopsis of QED We will here sketch the outlines of quantum electrodynamics, the theory of electrons and photons, and indicate how a calculation of an important physical quantity can be carried

More information

Phase Transitions in Condensed Matter Spontaneous Symmetry Breaking and Universality. Hans-Henning Klauss. Institut für Festkörperphysik TU Dresden

Phase Transitions in Condensed Matter Spontaneous Symmetry Breaking and Universality. Hans-Henning Klauss. Institut für Festkörperphysik TU Dresden Phase Transitions in Condensed Matter Spontaneous Symmetry Breaking and Universality Hans-Henning Klauss Institut für Festkörperphysik TU Dresden 1 References [1] Stephen Blundell, Magnetism in Condensed

More information

UNIVERSITY OF MARYLAND Department of Physics College Park, Maryland. PHYSICS Ph.D. QUALIFYING EXAMINATION PART II

UNIVERSITY OF MARYLAND Department of Physics College Park, Maryland. PHYSICS Ph.D. QUALIFYING EXAMINATION PART II UNIVERSITY OF MARYLAND Department of Physics College Park, Maryland PHYSICS Ph.D. QUALIFYING EXAMINATION PART II January 22, 2016 9:00 a.m. 1:00 p.m. Do any four problems. Each problem is worth 25 points.

More information

Continuum limit of fishnet graphs and AdS sigma model

Continuum limit of fishnet graphs and AdS sigma model Continuum limit of fishnet graphs and AdS sigma model Benjamin Basso LPTENS 15th Workshop on Non-Perturbative QCD, IAP, Paris, June 2018 based on work done in collaboration with De-liang Zhong Motivation

More information

Physics 127a: Class Notes

Physics 127a: Class Notes Physics 127a: Class Notes Lecture 15: Statistical Mechanics of Superfluidity Elementary excitations/quasiparticles In general, it is hard to list the energy eigenstates, needed to calculate the statistical

More information

Ising Lattice Gauge Theory with a Simple Matter Field

Ising Lattice Gauge Theory with a Simple Matter Field Ising Lattice Gauge Theory with a Simple Matter Field F. David Wandler 1 1 Department of Physics, University of Toronto, Toronto, Ontario, anada M5S 1A7. (Dated: December 8, 2018) I. INTRODUTION Quantum

More information

Introduction to particle physics Lecture 2

Introduction to particle physics Lecture 2 Introduction to particle physics Lecture 2 Frank Krauss IPPP Durham U Durham, Epiphany term 2009 Outline 1 Quantum field theory Relativistic quantum mechanics Merging special relativity and quantum mechanics

More information

For a complex order parameter the Landau expansion of the free energy for small would be. hc A. (9)

For a complex order parameter the Landau expansion of the free energy for small would be. hc A. (9) Physics 17c: Statistical Mechanics Superconductivity: Ginzburg-Landau Theory Some of the key ideas for the Landau mean field description of phase transitions were developed in the context of superconductivity.

More information

PAPER 51 ADVANCED QUANTUM FIELD THEORY

PAPER 51 ADVANCED QUANTUM FIELD THEORY MATHEMATICAL TRIPOS Part III Tuesday 5 June 2007 9.00 to 2.00 PAPER 5 ADVANCED QUANTUM FIELD THEORY Attempt THREE questions. There are FOUR questions in total. The questions carry equal weight. STATIONERY

More information

NTNU Trondheim, Institutt for fysikk

NTNU Trondheim, Institutt for fysikk NTNU Trondheim, Institutt for fysikk Examination for FY3464 Quantum Field Theory I Contact: Michael Kachelrieß, tel. 99890701 Allowed tools: mathematical tables Some formulas can be found on p.2. 1. Concepts.

More information

Phase transitions and critical phenomena

Phase transitions and critical phenomena Phase transitions and critical phenomena Classification of phase transitions. Discontinous (st order) transitions Summary week -5 st derivatives of thermodynamic potentials jump discontinously, e.g. (

More information

Contents. Appendix A Strong limit and weak limit 35. Appendix B Glauber coherent states 37. Appendix C Generalized coherent states 41

Contents. Appendix A Strong limit and weak limit 35. Appendix B Glauber coherent states 37. Appendix C Generalized coherent states 41 Contents Preface 1. The structure of the space of the physical states 1 1.1 Introduction......................... 1 1.2 The space of the states of physical particles........ 2 1.3 The Weyl Heisenberg algebra

More information

using D 2 D 2 D 2 = 16p 2 D 2

using 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 information

Notes on Renormalization Group: Berezinskii-Kosterlitz-Thouless (BKT) transition and Sine-Gordon model

Notes on Renormalization Group: Berezinskii-Kosterlitz-Thouless (BKT) transition and Sine-Gordon model Notes on Renormalization Group: Berezinskii-Kosterlitz-Thouless (BKT) transition and Sine-Gordon model Yi Zhou (Dated: December 4, 05) We shall discuss BKT transition based on +D sine-gordon model. I.

More information

Many-Body Problems and Quantum Field Theory

Many-Body Problems and Quantum Field Theory Philippe A. Martin Francois Rothen Many-Body Problems and Quantum Field Theory An Introduction Translated by Steven Goldfarb, Andrew Jordan and Samuel Leach Second Edition With 102 Figures, 7 Tables and

More information

LSZ reduction for spin-1/2 particles

LSZ 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 information

REVIEW REVIEW. A guess for a suitable initial state: Similarly, let s consider a final state: Summary of free theory:

REVIEW 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 information

Quantum Fields in Curved Spacetime

Quantum Fields in Curved Spacetime Quantum Fields in Curved Spacetime Lecture 3 Finn Larsen Michigan Center for Theoretical Physics Yerevan, August 22, 2016. Recap AdS 3 is an instructive application of quantum fields in curved space. The

More information

Finite Temperature Field Theory. Joe Schindler 2015

Finite Temperature Field Theory. Joe Schindler 2015 Finite Temperature Field Theory Joe Schindler 2015 Part 1: Basic Finite Temp Methods Preview Zero Temp Finite Temp Green Functions (vacuum expectation value) Generating Functional for GFs (real time) Preview

More information

Feynmann diagrams in a finite dimensional setting

Feynmann diagrams in a finite dimensional setting Feynmann diagrams in a finite dimensional setting Daniel Neiss Uppsala Universitet September 4, 202 Abstract This article aims to explain and justify the use of Feynmann diagrams as a computational tool

More information

Tensor Taylor series method for vacuum effects

Tensor Taylor series method for vacuum effects Tensor Taylor series method for vacuum effects M. W. Evans, H. Eckardt Civil List, A.I.A.S. and UPITEC (www.webarchive.org.uk, www.aias.us, www.atomicprecision.com, www.upitec.org) 3 Numerical and graphical

More information

Superinsulator: a new topological state of matter

Superinsulator: a new topological state of matter Superinsulator: a new topological state of matter M. Cristina Diamantini Nips laboratory, INFN and Department of Physics and Geology University of Perugia Coll: Igor Lukyanchuk, University of Picardie

More information

Topological insulator part I: Phenomena

Topological insulator part I: Phenomena Phys60.nb 5 Topological insulator part I: Phenomena (Part II and Part III discusses how to understand a topological insluator based band-structure theory and gauge theory) (Part IV discusses more complicated

More information

Heisenberg-Euler effective lagrangians

Heisenberg-Euler effective lagrangians Heisenberg-Euler effective lagrangians Appunti per il corso di Fisica eorica 7/8 3.5.8 Fiorenzo Bastianelli We derive here effective lagrangians for the electromagnetic field induced by a loop of charged

More information

What is a particle? Keith Fratus. July 17, 2012 UCSB

What 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 information

The Path Integral: Basics and Tricks (largely from Zee)

The Path Integral: Basics and Tricks (largely from Zee) The Path Integral: Basics and Tricks (largely from Zee) Yichen Shi Michaelmas 03 Path-Integral Derivation x f, t f x i, t i x f e H(t f t i) x i. If we chop the path into N intervals of length ɛ, then

More information

arxiv:hep-th/ v1 7 Feb 1992

arxiv:hep-th/ v1 7 Feb 1992 CP N 1 MODEL WITH A CHERN-SIMONS TERM arxiv:hep-th/9202026v1 7 Feb 1992 G. Ferretti and S.G. Rajeev Research Institute for Theoretical Physics Siltavuorenpenger 20 C SF-00170 Helsinki, Finland Abstract

More information