Functional Quantization
|
|
- Chad Claud Carroll
- 5 years ago
- Views:
Transcription
1 Functional Quantization In quantum mechanics of one or several particles, we may use path integrals to calculate the transition matrix elements as out Ût out t in in D[allx i t] expis[allx i t] Ψ out allx i@t out Ψ in allx in 1 where the paths x i t are independent of each other and are not subject to any boundary conditions, and the action depends on all particles paths as S[allx i t] t out t in Ldt, L i m i ẋ i Vallx i. Let s generalize formula 1 to the quantum field theory. In classical field theory, a finite set of dynamical variables x i becomes a field Φx where x is now a continuous label rather than a dynamical variable of its own. Thus, a set of all the particles paths x i t becomes a field configuration Φx,t, with action functional where the Lagrangian density is t out S[Φx,t] dt d 3 xl 3 t in L 1 µφ µ Φ m Φ λ 4 Φ4 4 for a scalar field, or something more complicated for other kinds of fields. In the quantum field theory, the path integral D[allx i t] i D[x i t] 5 generalizes to the functional integral over the field configurations D[Φx,t] 6 A careful definition of this functional integral involves discretization of both time and space 1
2 to a 4D lattice of N TV/a 4 spacetime points x n and then taking a zero-lattice-spacing limit D[Φx,t] lim a 0 Normalization factor n dφx n. 7 I shall address this issue later this week. For now, let s focus on using the functional integral. The QFT analogue of eq. 1 for the transition matrix element is out Ût out t in in D[Φx,t] expis[φx,t] Ψ out [Φx@t out]ψ in [Φx@t in ]. 8 Note that in the field theory s Hilbert space, the wave-functions Ψ in and Ψ out depend on the entire 3-space field configuration Φx for a fixed time t t in or t t out. In terms of such wave-functions, the out in bracket is given by a 3-space analogue of the function integrals over Φx at a fixed time, out in D[Φx]Ψ out [Φx] Ψ in [Φx]. 9 Now, let s go to the Heisenberg picture and consider the matrix element of the quantum field ˆΦ H x 1,t 1 at some point x µ 1 x 1,t 1. The states and the operators in Heisenberg and Schrödinger pictures are related to each other via the evolution operator Ût t 1, hence out ˆΦ H x 1,t 1 in out Ût out t 1 ˆΦ S x 1 Ût 1 t in in D[Φ 1 x] out Ût out t 1 Φ 1 Φ 1 x 1 Φ 1 Ût 1 t in in D[Φx,t for the whole time] Ψ out [Φx@t out] exp is[φ]fromt 1 tot out Φx 1,t 1 exp is[φ]fromt in tot 1 Ψin [Φx@t in ] D[Φx,t for the whole time] Ψ out[φx@t out ] Ψ in [Φx@t in ] exp is[φ] for the whole time Φx 1,t 1. Likewise, matrix elements of time-ordered products of several fields ˆΦ H x n,t n ˆΦ H x 1,t 1 10
3 for t n > > t 1 are given by functional integrals out ˆΦ H x n,t n ˆΦ H x 1,t 1 in D[Φx,t for the whole time] Ψ out[φx@t out ] Ψ in [Φx@t in ] exp is[φ] for the whole time Φx n,t n Φx 1,t Now, let s take the time interval between the in and out states to infinity in both directions, or rather let s take the limit t out + 1 iǫ, t in 1 iǫ. 1 As we saw back in January cf. class notes, in this limit out e iĥtout Ω c-number factor, e +iĥtin in Ω c-number factor, 13 where Ω is the vacuum state of the quantum field theory. Therefore, we may obtain the correlation functions of n quantum fields G n x 1,x,...,x n Ω TˆΦ H x 1 ˆΦ H x n Ω 14 from the functional integrals G n x 1,x,...,x n const D[Φx] exp is[φ] i d 4 xl Φx 1 Φx n. Note that the action integral S d 4 xl for each field configuration Φx is taken over the whole Minkowski space time, so it s Lorentz invariant, and the correlations functions 15 are also Lorentz invariant. 15 Strictly speaking, the limit 1 implies that all the time coordinates x 0 of all fields in 15 live on the slightly tilted time axis that runs from 1 iǫ to + 1 iǫ. Consequently, the action has a complexified version of the Lorentz symmetry, but after analytic continuation of all the correlation functions back to the real axis, we recover the usual Lorentz symmetry SO3, 1. Alternatively, we may analytically continue to the imaginary times; this gives us SO4 symmetry in the Euclidean spacetime. 3
4 We do not know the constant factor in front of the functional integral 15 and we do not know the overall normalization of the functional integral. But we can get rid of both of these unknown factors by taking ratios of correlation functions. In particular, using G 0 Ω Ω 1 16 we can write all the other correlation functions as normalization-independent ratios G n x 1,x,...,x n Ω TˆΦ H x 1 ˆΦ H x n Ω Ω Ω D[Φx] exp i D[Φx] exp i d 4 xl Φx 1 Φx n. d 4 xl 17 In a few pages, we shall learn how to calculate such ratios in perturbation theory and rederive the Feynman rules. But before we do the perturbation theory, let s see how functional integrals work for the free scalar field. Functional Integrals for the Free Scalar Field The free scalar field has a quadratic action functional S[Φx] d 4 x L 1 µφ µ Φ 1 m Φ 1 d 4 xφx +m Φx. 18 Hence, the functional integral Z D[Φx] exp is[φx] 19 is a kind of a Gaussian integral, generalization of the ordinary Gaussian integral ZA dξ 1 dξ dξ N exp 1 to a continuous family of variables, {ξ j } D[Φx]. N j,k1 A jk ξ j ξ k 0 4
5 Lemma 1: The ordinary Gaussian integral 0 evaluates to ZA πn/ deta. 1 Note: A jk is a symmetric N N matrix, real or complex. To make the integral 0 converge, the real part ReA jk of the A matrix should be positive definite. Proof: Any symmetric matrix can be diagonalized, thus A jk ξ j ξ k k jk B k η k where η 1,...,η N are some independent linear combinations of the ξ 1,...,ξ N. Changing integration variables from the ξ k to η k carries a Jacobian J det ξj ; 3 η k by linearity of the ξ η transform, this Jacobian is constant. Therefore, Z J d N ηexp 1 B k ηk J J N k1 N k1 k dη k exp B k η k π B k πn/ J detb 4 where B is the diagonal matrix B jk jk B k. This matrix is related to A according to B ξ A η ξ, 5 η hence and therefore ξ detb deta det η Z π N/ deta J 6 J detb πn/ deta. 1 5
6 Lemma concerns the rations of Gaussian integrals for the same matrix A: d N ξ exp 1 ij A ijξ i ξ j ξ k ξ l A 1 d N ξ exp 1 ij A kl. 7 ijξ i ξ j Proof: d N ξexp 1 A ij ξ i ξ j ξ k ξ l A kl ij d N ξexp 1 A ij ξ i ξ j, 8 ij hence d N ξ exp 1 d N ξ exp 1 ij A ijξ i ξ j ξ k ξ l log ij A ijξ i ξ A j kl [ d ] ξexp N 1 ij A ijξ i ξ j log πn/ A kl deta + logdeta A kl A 1 kl. 9 Both lemmas apply to the Gaussian functional integrals such as 19. For example, consider the free scalar propagator G x,y. Eq. 17 gives us a formula for this propagator in terms of a ratio of two Gaussian functional integrals G x,y D[Φx] exp i d 4 xφ +m Φ ΦxΦy D[Φx] exp i d 4 xφ +m Φ. 30 The role of the matrix A here is played by the differential operator i +m. This operator is purely anti-hermitian, which corresponds to a purely imaginary matrix A. To make the Gaussian integrals converge, we add an infinitesimal real part, A ij A ij + ǫ ij, or i + m i +m +ǫ. Consequently, lemma tells us that G x,y x 1 d 4 i +m +ǫ y p ie ipx y π 4 p m +iǫ. 31 Note that the rule A A+ǫ we use to make the Gaussina integral converge for an imaginary or anhti-hermitian matrix A immediately leads us to the Feynman propagator 31 rather than some other Green s function. 6
7 To obtain correlation functions G n for n > we need analogues of Lemma for Gaussian integrals involving more than two ξs outside the exponential. For example, for four ξs we have d N ξ exp 1 ij A ijξ i ξ j ξ k ξ l ξ m ξ n 3 d N ξ exp 1 ij A ijξ i ξ j A 1 kl A 1 mn + A 1 km A 1 ln + A 1 kn A 1 lm. Applying this formula to the Gaussian functional integral for the free scalar fields, we obtain for the 4 point function G 4 x,y,z,w G x y G z w + G x z G y w + G x w G y z Likewise, for higher even numbers of ξs, d N ξ exp 1 ij A ijξ i ξ j ξ 1 ξ N d N ξ exp 1 ij A ijξ i ξ j A 1 34 indexpair pairs pairings and similarly G free N x 1,...,x N D[Φx] exp i d 4 xφ +m Φ Φx 1 Φx N D[Φx] exp i d 4 xφ +m Φ pairings pairs. 35 7
8 Feynman Rules for the Interacting Field A self-interacting scalar field has action S[Φx] S free [Φx] λ d 4 xφ 4 x. 36 4! Expanding the exponential e is in powers of the coupling λ, we have exp is[φx] exp is free [Φx] n0 n 1 iλ d 4 xφ x n! 4! This expansion gives us the perturbation theory for the functional integral of the interacting field, D[Φx] exp is[φx] Φx 1 Φx k D[Φx] exp is free [Φx] i d 4 xφ +m Φ n0 iλ n n!4! n d 4 z 1 d 4 z n Φx 1 Φx k n0 n 1 iλ d 4 xφ 4 x n! 4! i D[Φx] exp d 4 xφ +m Φ Φx 1 Φx k Φ 4 z 1 Φ 4 z n using eq. 35 for the free-field functional integrals [ ] i D[Φx] exp d 4 xφ +m Φ iλ n n!4! n d 4 z 1 d 4 z n products of k +4n n0 [ D[Φx] exp n0 pairings i d 4 xφ +m Φ ] free propagators Feynman diagrams with k external and n internal vertices This is how the Feynman rules emerge from the functional integral formalism
9 At this stage, we are summing over all the Feynman diagrams, connected or disconnected, and even the vacuum bubbles are allowed. Also, the functional integral 38 carries an overall factor Z 0 i D[Φx] exp d 4 xφ +m Φ which multiplies the whole perturbation series. Fortunately, this pesky factor as well as all the vacuum bubbles cancel out when we divide eq. 38 by a similar functional integral for k 0 to get the correlation function G k. Indeed, 39 G k x 1,...,x k D[Φx] exp is[φx] Φx1 Φx k D[Φx] exp is[φx] Z 0 all Feynman diagrams with k external vertices Z 0 all Feynman diagrams without external vertices Feynman diagrams with k external vertices and no vacuum bubbles. 40 Generating Functional In the functional integral formalism there is a compact way of summarizing all the connected correlation functions in terms of a single generating functional. Let s add to the scalar field s Lagrangian L a linear source term, LΦ, Φ LΦ, Φ + J Φ 41 where Jx is an arbitrary function of x. Now let s calculate the partition function for Φx as a functional of the source Jx, Z[Jx] D[Φx] exp i d 4 xl+jφ D[Φx] exp is[φx] exp i d 4 xjxφx. 4 To make use of this partition function, we need functional i.e., variational derivatives /Jx. Here is how they work: JyΦyd 4 y Φx, 43 Jx 9
10 µ Jy A µ yd 4 y µ A µ x, 44 Jx Jx exp i JyΦyd 4 y iφx exp i JyΦyd 4 y, 45 etc., etc. Applying these derivatives to the partition function 4, we have and similarly Jx 1 Jx Z[J] D[Φy] exp i d 4 yl+jφ iφx 46 Jx Jx k Z[J] D[Φy] exp i d 4 yl+jφ i k Φx 1 Φx Φx k. Thus, the partition function 4 generates all the functional integrals of the scalar theory: k, D[Φy] exp i d 4 yl Φx 1 Φx k i k Jx 1 47 Jx k Z[J]. Jy 0 Consequently, all the correlation functions 40 follows from the derivatives of the Z[J] according to G k x 1,...,x k ik Z[J] 48 Jx 1 Jx k Z[J]. 49 Jy 0 In terms of the Feynman diagrams, the correlation functions 49 contain both connected and disconnected graphs; only the vacuum bubbles are excluded. To obtain the connected correlation functions G conn k x 1,...,x k connected Feynman diagrams only 50 we should take the /Jx derivatives of log Z[J] rather than Z[J] itself, G conn k x 1,...,x k i k Jx 1 Jx k logz[jy]. 51 Jy 0 In other words, log Z[Jx] is the generating functional for for all the connected correlation functions of the quantum field theory. 10
11 Proof: For the sake of generality, let s allow for G 1 x Φx 0. For k 1, G conn 1 G 1 and eq. 51 is equivalent to eq. 49. For k we use logz[j] JxJy 1 Z Z JxJy 1 Z Z Jx 1 Z Z Jy. 5 Let smultiplythisformulaby i, setj 0aftertaking thederivatives, andapplyeqs. 49 to the right hand side; then i logz[j] Jx Jy G x,y G 1 x G 1 y G conn x,y. 53 J 0 The k 1 and k cases serve as a base of a proof by induction in k. Now we need to prove that is eqs. 51 hold true for all l < k that they also hold true for k itself. A k-point correlation function has a cluster expansion in terms of connected correlation functions, G k x 1,...,x k G conn k x 1,...,x k + cluster decompositions clusters G conn cluster where each cluster is proper subset of x 1,...,x n and we sum over decompositions of the whole set x 1,...,x n into a union of disjoint clusters. For example, for k 3 54 G 3 x,y,z G conn 3 x,y,z + G 1 x G 1 y G 1 z + G conn x,y G 1 z + G conn x,z G 1 y + G conn y,z G 1 x two similar. 55 There is a similar expansion of derivatives of Z in terms of derivatives of logz, 1 k Z Z Jx 1 Jx k k logz Jx 1 Jx k + clusters l logz, Jx 1 Jx l x 1,...,x l cluster decompositions 56 11
12 Now let s set Jy 0 after taking all the derivatives in this expansion. On the left hand side, this gives us i k G k x 1,...,x k by eq. 49. On the right hand side, for each cluster l logz Jx 1 Jx l il G conn l x 1,...,x l 57 by the induction assumption. Altogether, we have G n x 1,...,x n i k k logz Jx 1 Jx k + J 0 cluster decompositions cluster G conn cluster, 58 and comparing this formula to eq. 54 we immediately see that G conn n x 1,...,x n i k k logz Jx 1 Jx k, 51 J 0 quod erat demonstrandum. We may summarize all the equations 51 in a single formula which makes manifest the role of log Z as a generating functional for the connected correlation functions, logz[jx] logz[0] + k1 i k d 4 x 1 d 4 x k G conn k x 1,...,x k Jx 1 Jx k. 59 k! To see how this works, consider the free theory as an example. A free scalar field has only one connected correlation function, namely G conn x y G F x y; all the other G conn k require interactions and vanish for the free field. Hence, the partition function of the free field should have form logz free [Jx] logz free [0] + i d 4 d 4 yg F x y JxJy + nothing else. 60 To check this formula, let s calculate the partition function. For the free field Φ, L free + JΦ 1 Φ +m Φ +JΦ 1 Φ +m Φ + 1 J +m 1 J 61 1
13 up to a total derivative where Φ Φ +m 1 J. In other words, S free [Φx,Jx] def L+JΦd 4 x S free [Φ x,j ] + i d 4 d 4 yjxg F x yjy 6 where Φ x Φx i d 4 yg F x yjy. 63 In the path integral, D[Φx] D[Φ x], hence Z free [Jx] exp exp D[Φx] exp is free [Φx,Jx] D[Φ x] exp d 4 x is free [Φ x,j ] exp 1 d 4 yjxg F x yjy Z free [0]. 1 1 d 4 x d 4 yjxg F x yjy d 4 x d 4 yjxg F x yjy D[Φ x] exp is free [Φ x,j ] Taking the log of both sides of this formula gives us eq. 60, which confirms that logz free [J] is indeed the generating functional for the correlation functions of the free field
Functional Quantization
Functonal Quantzaton In quantum mechancs of one or several partcles, we may use path ntegrals to calculate transton matrx elements as out Ût out t n n = D[allx t] exps[allx t] Ψ outallx @t out Ψ n allx
More informationPath 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 informationdx dt dy dt x(t) y 1 t x 1 t
PHY 396. Solutions for homework set #19. Problem 1a): The difference between a circle and a straight line is that on a circle the path of a particle going from point x 1 to point x 2 does not need to be
More informationPHY 396 L. Solutions for homework set #21.
PHY 396 L. Solutions for homework set #21. Problem 1a): The difference between a circle and a straight line is that on a circle the path of a particle going from point x 0 to point x does not need to be
More informationSection 4: The Quantum Scalar Field
Physics 8.323 Section 4: The Quantum Scalar Field February 2012 c 2012 W. Taylor 8.323 Section 4: Quantum scalar field 1 / 19 4.1 Canonical Quantization Free scalar field equation (Klein-Gordon) ( µ µ
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 informationQuantum Field Theory II
Quantum Field Theory II T. Nguyen PHY 391 Independent Study Term Paper Prof. S.G. Rajeev University of Rochester April 2, 218 1 Introduction The purpose of this independent study is to familiarize ourselves
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 informationThe 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 informationAdvanced Quantum Mechanics
Advanced Quantum Mechanics Rajdeep Sensarma! sensarma@theory.tifr.res.in Lecture #22 Path Integrals and QM Recap of Last Class Statistical Mechanics and path integrals in imaginary time Imaginary time
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 information4 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 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 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 informationObservables in the GBF
Observables in the GBF Max Dohse Centro de Ciencias Matematicas (CCM) UNAM Campus Morelia M. Dohse (CCM-UNAM Morelia) GBF: Observables GBF Seminar (14.Mar.2013) 1 / 36 References:. [RO2011] R. Oeckl: Observables
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 informationPhysics 582, Problem Set 4 Solutions
Physics 582, Problem Set 4 Solutions TAs: Hart Goldman and Ramanjit Sohal Fall 2018 1. PATH INTEGRAL FOR A PARTICLE IN A DOUBLE POTENTIAL WELL 1. In real time, q 0, T/2 q 0, T/2 = q 0 e i HT q 0 = where
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 informationClassical field theory 2012 (NS-364B) Feynman propagator
Classical field theory 212 (NS-364B Feynman propagator 1. Introduction States in quantum mechanics in Schrödinger picture evolve as ( Ψt = Û(t,t Ψt, Û(t,t = T exp ı t dt Ĥ(t, (1 t where Û(t,t denotes the
More informationQuantization of Scalar Field
Quantization of Scalar Field Wei Wang 2017.10.12 Wei Wang(SJTU) Lectures on QFT 2017.10.12 1 / 41 Contents 1 From classical theory to quantum theory 2 Quantization of real scalar field 3 Quantization of
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 informationQCD on the lattice - an introduction
QCD on the lattice - an introduction Mike Peardon School of Mathematics, Trinity College Dublin Currently on sabbatical leave at JLab HUGS 2008 - Jefferson Lab, June 3, 2008 Mike Peardon (TCD) QCD on the
More informationFeynman s path integral approach to quantum physics and its relativistic generalization
Feynman s path integral approach to quantum physics and its relativistic generalization Jürgen Struckmeier j.struckmeier@gsi.de, www.gsi.de/ struck Vortrag im Rahmen des Winterseminars Aktuelle Probleme
More information8.821 String Theory Fall 2008
MIT OpenCourseWare http://ocw.mit.edu 8.8 String Theory Fall 008 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms. 8.8 F008 Lecture 0: CFTs in D > Lecturer:
More informationSchwinger-Dyson Equation(s)
Mobolaji Williams mwilliams@physics.harvard.edu x First Version: uly 8, 016 Schwinger-Dyson Equations In these notes we derive the Schwinger-Dyson equation, a functional differential equation whose solution
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 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 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 informationPath Integrals in Quantum Mechanics and Quantum Field Theory C6, MT 2017
Path Integrals in Quantum Mechanics and Quantum Field Theory C6, MT 017 Joseph Conlon a a Rudolf Peierls Centre for Theoretical Physics University of Oxford OX1 3PN Oxford, United Kingdom Please send corrections
More informationLecture 10: A (Brief) Introduction to Group Theory (See Chapter 3.13 in Boas, 3rd Edition)
Lecture 0: A (Brief) Introduction to Group heory (See Chapter 3.3 in Boas, 3rd Edition) Having gained some new experience with matrices, which provide us with representations of groups, and because symmetries
More information2 Lie Groups. Contents
2 Lie Groups Contents 2.1 Algebraic Properties 25 2.2 Topological Properties 27 2.3 Unification of Algebra and Topology 29 2.4 Unexpected Simplification 31 2.5 Conclusion 31 2.6 Problems 32 Lie groups
More informationLECTURE 3: Quantization and QFT
LECTURE 3: Quantization and QFT Robert Oeckl IQG-FAU & CCM-UNAM IQG FAU Erlangen-Nürnberg 14 November 2013 Outline 1 Classical field theory 2 Schrödinger-Feynman quantization 3 Klein-Gordon Theory Classical
More informationIntroduction to Path Integrals
Introduction to Path Integrals Consider ordinary quantum mechanics of a single particle in one space dimension. Let s work in the coordinate space and study the evolution kernel Ut B, x B ; T A, x A )
More informationProblem 1(a): The scalar potential part of the linear sigma model s Lagrangian (1) is. 8 i φ2 i f 2) 2 βλf 2 φ N+1,
PHY 396 K. Solutions for problem set #10. Problem 1a): The scalar potential part of the linear sigma model s Lagrangian 1) is Vφ) = λ 8 i φ i f ) βλf φ N+1, S.1) where the last term explicitly breaks the
More informationOn angular momentum operator in quantum field theory
On angular momentum operator in quantum field theory Bozhidar Z. Iliev Short title: Angular momentum operator in QFT Basic ideas: July October, 2001 Began: November 1, 2001 Ended: November 15, 2001 Initial
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 informationPHY 396 K. Problem set #7. Due October 25, 2012 (Thursday).
PHY 396 K. Problem set #7. Due October 25, 2012 (Thursday. 1. Quantum mechanics of a fixed number of relativistic particles does not work (except as an approximation because of problems with relativistic
More information2. The Schrödinger equation for one-particle problems. 5. Atoms and the periodic table of chemical elements
1 Historical introduction The Schrödinger equation for one-particle problems 3 Mathematical tools for quantum chemistry 4 The postulates of quantum mechanics 5 Atoms and the periodic table of chemical
More informationPreliminaries: what you need to know
January 7, 2014 Preliminaries: what you need to know Asaf Pe er 1 Quantum field theory (QFT) is the theoretical framework that forms the basis for the modern description of sub-atomic particles and their
More information2.4 Parity transformation
2.4 Parity transformation An extremely simple group is one that has only two elements: {e, P }. Obviously, P 1 = P, so P 2 = e, with e represented by the unit n n matrix in an n- dimensional representation.
More informationADVANCED TOPICS IN THEORETICAL PHYSICS II Tutorial problem set 2, (20 points in total) Problems are due at Monday,
ADVANCED TOPICS IN THEORETICAL PHYSICS II Tutorial problem set, 15.09.014. (0 points in total) Problems are due at Monday,.09.014. PROBLEM 4 Entropy of coupled oscillators. Consider two coupled simple
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 informationQuantum 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 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 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 informationFinite Temperature Field Theory
Finite Temperature Field Theory Dietrich Bödeker, Universität Bielefeld 1. Thermodynamics (better: thermo-statics) (a) Imaginary time formalism (b) free energy: scalar particles, resummation i. pedestrian
More informationHeisenberg-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 informationNTNU 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 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 informationFinite temperature QFT: A dual path integral representation
A dual path integral representation I. Roditi Centro Brasileiro de Pesquisas Físicas February 20, 2009 1 I. Roditi (CBPF) Collaborators: 2 I. Roditi (CBPF) Collaborators: C. Cappa Ttira 2 I. Roditi (CBPF)
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 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 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 informationTextbook Problem 4.2: We begin by developing Feynman rules for the theory at hand. The Hamiltonian clearly decomposes into Ĥ = Ĥ0 + ˆV where
PHY 396 K. Solutions for problem set #11. Textbook Problem 4.2: We begin by developing Feynman rules for the theory at hand. The Hamiltonian clearly decomposes into Ĥ = Ĥ0 + ˆV where Ĥ 0 = Ĥfree Φ + Ĥfree
More informationIntroduction to Group Theory
Chapter 10 Introduction to Group Theory Since symmetries described by groups play such an important role in modern physics, we will take a little time to introduce the basic structure (as seen by a physicist)
More informationADVANCED QUANTUM FIELD THEORY
Imperial College London MSc EXAMINATION May 2016 This paper is also taken for the relevant Examination for the Associateship ADVANCED QUANTUM FIELD THEORY For Students in Quantum Fields and Fundamental
More informationNTNU 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 informationSPACETIME FROM ENTANGLEMENT - journal club notes -
SPACETIME FROM ENTANGLEMENT - journal club notes - Chris Heinrich 1 Outline 1. Introduction Big picture: Want a quantum theory of gravity Best understanding of quantum gravity so far arises through AdS/CFT
More informationT H E K M S - C O N D I T I O N
T H E K M S - C O N D I T I O N martin pauly January 3, 207 Abstract This report develops the relation between periodicity in imaginary time and finite temperature for quantum field theories, given by
More information2.3 Calculus of variations
2.3 Calculus of variations 2.3.1 Euler-Lagrange equation The action functional S[x(t)] = L(x, ẋ, t) dt (2.3.1) which maps a curve x(t) to a number, can be expanded in a Taylor series { S[x(t) + δx(t)]
More information1 Quantum fields in Minkowski spacetime
1 Quantum fields in Minkowski spacetime The theory of quantum fields in curved spacetime is a generalization of the well-established theory of quantum fields in Minkowski spacetime. To a great extent,
More informationTadpole 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 informationNumerical Methods in Quantum Field Theories
Numerical Methods in Quantum Field Theories Christopher Bell 2011 NSF/REU Program Physics Department, University of Notre Dame Advisors: Antonio Delgado, Christopher Kolda 1 Abstract In this paper, preliminary
More informationPath integration in relativistic quantum mechanics
Path integration in relativistic quantum mechanics Ian H. Redmount and Wai-Mo Suen McDonnell Center for the Space Sciences Washington University, Department of Physics St. Louis, Missouri 63130 4899 USA
More informationTo make this point clear, let us rewrite the integral in a way that emphasizes its dependence on the momentum variable p:
3.2 Construction of the path integral 101 To make this point clear, let us rewrite the integral in a way that emphasizes its dependence on the momentum variable p: q f e iĥt/ q i = Dq e i t 0 dt V (q)
More informationPerturbative Quantum Field Theory and Vertex Algebras
ε ε ε x1 ε ε 4 ε Perturbative Quantum Field Theory and Vertex Algebras Stefan Hollands School of Mathematics Cardiff University = y Magnify ε2 x 2 x 3 Different scalings Oi O O O O O O O O O O O 1 i 2
More informationApplications of AdS/CFT correspondence to cold atom physics
Applications of AdS/CFT correspondence to cold atom physics Sergej Moroz in collaboration with Carlos Fuertes ITP, Heidelberg Outline Basics of AdS/CFT correspondence Schrödinger group and correlation
More informationIntroduction to Instantons. T. Daniel Brennan. Quantum Mechanics. Quantum Field Theory. Effects of Instanton- Matter Interactions.
February 18, 2015 1 2 3 Instantons in Path Integral Formulation of mechanics is based around the propagator: x f e iht / x i In path integral formulation of quantum mechanics we relate the propagator to
More informationds/cft Contents Lecturer: Prof. Juan Maldacena Transcriber: Alexander Chen August 7, Lecture Lecture 2 5
ds/cft Lecturer: Prof. Juan Maldacena Transcriber: Alexander Chen August 7, 2011 Contents 1 Lecture 1 2 2 Lecture 2 5 1 ds/cft Lecture 1 1 Lecture 1 We will first review calculation of quantum field theory
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 information14 Renormalization 1: Basics
4 Renormalization : Basics 4. Introduction Functional integrals over fields have been mentioned briefly in Part I devoted to path integrals. In brief, time ordering properties and Gaussian properties generalize
More informationAnalytic continuation of functional renormalization group equations
Analytic continuation of functional renormalization group equations Stefan Flörchinger (CERN) Aachen, 07.03.2012 Short outline Quantum effective action and its analytic continuation Functional renormalization
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 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 information(K + L)(c x) = K(c x) + L(c x) (def of K + L) = K( x) + K( y) + L( x) + L( y) (K, L are linear) = (K L)( x) + (K L)( y).
Exercise 71 We have L( x) = x 1 L( v 1 ) + x 2 L( v 2 ) + + x n L( v n ) n = x i (a 1i w 1 + a 2i w 2 + + a mi w m ) i=1 ( n ) ( n ) ( n ) = x i a 1i w 1 + x i a 2i w 2 + + x i a mi w m i=1 Therefore y
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 informationIntroduction to Quantum Field Theory
Introduction to Quantum Field Theory John Cardy Hilary Term 2012 Version 19/1/12 Abstract These notes are intended to supplement the lecture course Field Theory in Condensed Matter and are not intended
More informationQuantum Field Theory III
Quantum Field Theory III Prof. Erick Weinberg March 9, 0 Lecture 5 Let s say something about SO(0. We know that in SU(5 the standard model fits into 5 0(. In SO(0 we know that it contains SU(5, in two
More informationwhere P a is a projector to the eigenspace of A corresponding to a. 4. Time evolution of states is governed by the Schrödinger equation
1 Content of the course Quantum Field Theory by M. Srednicki, Part 1. Combining QM and relativity We are going to keep all axioms of QM: 1. states are vectors (or rather rays) in Hilbert space.. observables
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 informationL8. Basic concepts of stress and equilibrium
L8. Basic concepts of stress and equilibrium Duggafrågor 1) Show that the stress (considered as a second order tensor) can be represented in terms of the eigenbases m i n i n i. Make the geometrical representation
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 informationVectors, metric and the connection
Vectors, metric and the connection 1 Contravariant and covariant vectors 1.1 Contravariant vectors Imagine a particle moving along some path in the 2-dimensional flat x y plane. Let its trajectory be given
More informationWeek 1. 1 The relativistic point particle. 1.1 Classical dynamics. Reading material from the books. Zwiebach, Chapter 5 and chapter 11
Week 1 1 The relativistic point particle Reading material from the books Zwiebach, Chapter 5 and chapter 11 Polchinski, Chapter 1 Becker, Becker, Schwartz, Chapter 2 1.1 Classical dynamics The first thing
More informationarxiv: 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 informationIntroduction to string theory 2 - Quantization
Remigiusz Durka Institute of Theoretical Physics Wroclaw / 34 Table of content Introduction to Quantization Classical String Quantum String 2 / 34 Classical Theory In the classical mechanics one has dynamical
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 informationPhD in Theoretical Particle Physics Academic Year 2017/2018
July 10, 017 SISSA Entrance Examination PhD in Theoretical Particle Physics Academic Year 017/018 S olve two among the four problems presented. Problem I Consider a quantum harmonic oscillator in one spatial
More informationMathematical Methods for Physics
Mathematical Methods for Physics Peter S. Riseborough June 8, 8 Contents Mathematics and Physics 5 Vector Analysis 6. Vectors................................ 6. Scalar Products............................
More informationClassical Field Theory
Classical Field Theory Asaf Pe er 1 January 12, 2016 We begin by discussing various aspects of classical fields. We will cover only the bare minimum ground necessary before turning to the quantum theory,
More information2. As we shall see, we choose to write in terms of σ x because ( X ) 2 = σ 2 x.
Section 5.1 Simple One-Dimensional Problems: The Free Particle Page 9 The Free Particle Gaussian Wave Packets The Gaussian wave packet initial state is one of the few states for which both the { x } and
More informationREVIEW 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 informationCombinatorial Hopf algebras in particle physics I
Combinatorial Hopf algebras in particle physics I Erik Panzer Scribed by Iain Crump May 25 1 Combinatorial Hopf Algebras Quantum field theory (QFT) describes the interactions of elementary particles. There
More information10 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 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 informationAppendix A Notational Conventions
Appendix A Notational Conventions Throughout the book we use Einstein s implicit summation convention: repeated indices in an expression are automatically summed over. We work in natural units where the
More informationSYMPLECTIC MANIFOLDS, GEOMETRIC QUANTIZATION, AND UNITARY REPRESENTATIONS OF LIE GROUPS. 1. Introduction
SYMPLECTIC MANIFOLDS, GEOMETRIC QUANTIZATION, AND UNITARY REPRESENTATIONS OF LIE GROUPS CRAIG JACKSON 1. Introduction Generally speaking, geometric quantization is a scheme for associating Hilbert spaces
More informationReview and Notation (Special relativity)
Review and Notation (Special relativity) December 30, 2016 7:35 PM Special Relativity: i) The principle of special relativity: The laws of physics must be the same in any inertial reference frame. In particular,
More informationSpecial classical solutions: Solitons
Special classical solutions: Solitons by Suresh Govindarajan, Department of Physics, IIT Madras September 25, 2014 The Lagrangian density for a single scalar field is given by L = 1 2 µφ µ φ Uφ), 1) where
More information