Problem 1(a): As discussed in class, Euler Lagrange equations for charged fields can be written in a manifestly covariant form as L (D µ φ) L

Size: px
Start display at page:

Download "Problem 1(a): As discussed in class, Euler Lagrange equations for charged fields can be written in a manifestly covariant form as L (D µ φ) L"

Transcription

1 PHY 396 K. Solutions for problem set #. Problem a: As discussed in class, Euler Lagrange equations for charged fields can be written in a manifestly covariant form as D µ D µ φ φ = 0. S. In particularly, for φ = Φ, we have D µ Φ = Dµ Φ, Φ = m Φ, which gives us D µ D µ Φ + m Φ = 0. S. Likewise, for φ = Φ we have D µ Φ = Dµ Φ, Φ = m Φ, and therefore D µ D µ Φ + m Φ = 0. S.3 As for the vector fields A ν, the Lagrangian depends on µ A ν only through F µν, which gives us the usual Maxwell equation µ F µν = J ν where J ν A ν. S.4 To obtain the current J ν, we notice that the covariant derivatives of the charged fields Φ and

2 Φ depend on the gauge field: D µ Φ A ν = iqδ ν µφ, D µ Φ A ν = iqδ ν µφ. S.5 Consequently, J ν = D ν Φ iqφ = iq ΦD ν Φ Φ D ν Φ. D ν Φ iqφ S.6 Note that all derivatives on the last line here are gauge-covariant, which makes the current J ν gauge invariant. In a non-covariant form, J ν = iqφ D ν Φ iqφ D ν Φ q Φ Φ A ν. S.7 To prove the conservation of this current, we use the Leibniz rule for covariant derivatives, D ν XY = XD ν Y + Y D ν X. Thus, µ Φ D µ Φ = D µ Φ D µ Φ = D µ Φ D µ Φ + Φ D µ D µ Φ, µ ΦD µ Φ = D µ ΦD µ Φ = D µ ΦD µ Φ + Φ D µ D µ Φ, S.8 and hence in light of eq. S.6for the current, νj ν = iq D ν ΦD ν Φ + ΦD ν D ν Φ = iqφ D Φ iqφ D Φ by equations of motion = iqφ m Φ iqφ m Φ = 0. + iq D ν Φ D ν Φ + Φ D ν D ν Φ S.9

3 Problem b: According to the Noether theorem, where T µν Noether = µ A λ ν A λ + µ Φ ν Φ + = T µν µν Noether EM + T Noether matter µ Φ ν Φ g µν L S.0 similar to the free EM fields, and T µν Noether EM = F µλ ν A λ + 4 gµν F κλ F κλ S. T µν Noether matter = Dµ Φ ν Φ + D µ Φ ν Φ g µν D λ Φ D λ Φ m Φ Φ. S. Both terms on the second line of eq. S.0 lack µ ν symmetry and gauge invariance and thus need λ K λµν corrections for some K λµν = K µλν. We would like to show that the same K λµν = F µλ A ν we used to improve the free electromagnetic stress-energy tensor will now improve both the T µν µν EM and T mat at the same time! Indeed, to improve the scalar fields stress-energy tensor we need T µν matter T µν µν true matter T Noether matter = D µ Φ D ν Φ ν Φ + D µ ΦD ν Φ ν Φ = D µ Φ iqa ν Φ + D µ Φ iqa ν Φ S.3 = A ν iqφ D µ Φ iqφd µ Φ = A ν J µ, while the improvement of the EM stress-energy requires cf. previous homework. T µν EM = F µλ F ν λ ν A λ = +F µλ λa ν = λ F λµ A ν + A ν J µ. S.4 Altogether, to symmetrize the whole stress-energy tensor, we need T µν tot T µν µν true total T Noether total = λ F µλ A ν K λµν. 3

4 Problem c: Because the fields Φx and Φ x have opposite electric charges, their product is neutral and therefore µ Φ Φ = D µ Φ Φ = D µ Φ Φ + Φ D µ Φ. Similarly, µ D µ Φ D ν Φ = D µ D µ Φ D ν Φ + D µ Φ D µ D ν Φ = m Φ D ν Φ + D µ Φ D ν D µ Φ + iqf µν Φ S.5 where we have applied the field equation D µ D µ + m Φ x = 0 to the first term on the right hand side and used [D µ, D ν ]Φ = iqf µν Φ to expand the second term. Likewise, µ D µ Φ D ν Φ = D µ D µ Φ D ν Φ + D µ Φ D µ D ν Φ = m Φ D ν Φ + D µ Φ D ν D µ Φ iqf µν Φ S.6 and µ [ g ] µν D λ Φ D λ Φ m Φ Φ = ν D λ Φ D λ Φ + m ν Φ Φ = D ν D µ Φ D µ Φ D µ Φ D ν D µ Φ + m Φ D ν Φ + m Φ D ν Φ. Together, the left hand sides of eqs. S.5, S.6 and S.7 comprise µ T µν mat S.7 cf. eq. 7. On the other hand, combining the right hand sides of these three equations results in massive cancellation of all terms except those containing the gauge field strength tensor F µν. Thus, µ T µν mat = D µ Φ iqf µν Φ + D µ Φ iqf µν Φ = F µν iqφ D µ Φ iqφ D µ Φ S.8 = F µν J ν. And since in the previous homework we have shown µ T µν EM = F µν J ν, it follows that the total stress tensor 4 is conserved, µ T µν = 0. 4

5 Problem a: According to the time-independent Maxwell equations 0, the divergences Ê and ˆB vanish as operators and therefore should commute with any other operator in the theory. On the other hand, the commutation relations of the Ê and ˆB follow directly from those of the Ê and ˆB field themselves: [ Êx, whateverx ] = [Êi x i x, same whateverx ], [ ˆBx, whateverx ] = [ ˆBi x i x, same whateverx ]. S.9 Thus to make sure that the fields commutation relations are consistent with eqs. 0, we must verify that x i [Êi x, Êj x ] = 0, x i [Êi x, ˆB j x ] = 0, x i [ ˆBi x, Êj x ] = 0, x i [ ˆBi x, ˆB j x ] = 0, Plugging in the commutation relations into these formulæ, we immediately see that S.0 x i [Êi x, Êj x ] = x i [ ˆBi x, ˆB j x ] = 0 S. while x i [Êi x, ˆB j x ] = x i i hc ɛ ijk x k δ3 x x = 0 S. because ɛ ijk x i x k 0. Likewise, x i [ ˆBi x, Êj x ] = x i +i hc ɛ jik = i hc ɛ jik x i x k δ3 x x x k δ3 x x = 0. S.3 5

6 Problem b: According to eqs., at equal times [ ˆB i, ˆB x ] = 0 while [ ˆB i, Ê x ] = Êj x [ ˆB i x, Êj x ] = Êj x +i hcɛ jik x k δ3 x x, hence [ ˆB i x, Ĥ] = d 3 x +i hcɛ jik Ê j x x k δ3 x x = i hc ɛ jik k Ê j x, S.4 S.5 or in vector notations, [ˆBx, Ĥ] = i hc Ê. S.6 Consequently, in the Heisenberg picture c t ˆBx, t = Êx, t..a Likewise, [Êi, Ê x ] = 0 while [Êi, ˆB x ] = ˆB j x [Êi x, ˆB j x ] = ˆB j x i hcɛ ijk x k δ3 x x, hence [Êi x, Ĥ] = d 3 x i hcɛ ijk ˆBj x x k δ3 x x = i hc ɛ ijk ˆBj k x, S.7 S.8 or in vector notations, [Êx, Ĥ] = +i hc ˆB. S.9 Consequently, in the Heisenberg picture c t ˆBx, t = + Êx, t..b 6

7 Problem 3a: In the Hamiltonian formalism for the classical fields Φx and Φ x, the canonical conjugate fields are Πx = 0 Φ = 0Φx and Π x = 0 Φ = 0Φ x. S.30 The canonical conjugation implies canonical Poisson brackets between the classical fields Φx and Π x, and likewise Φ x and Πx and hence the canonical commutation relation between their quantum counterparts: In the Schrödinger picture [ˆΦx, ˆΦx ] = [ˆΦx, ˆΦ x ] = [ˆΦ x, ˆΦ x ] = 0, [ˆΠx, ˆΠx ] = [ˆΠx, ˆΠ x ] = [ˆΠ x, ˆΠ x ] = 0, [ˆΦx, ˆΠx ] = [ˆΦ x, ˆΠ x ] = 0, S.3 [ˆΦx, ˆΠ x ] = [ˆΦ x, ˆΠx ] = iδ 3 x x. In the Heisenberg picture, we have similar commutation relations for equal times t = t ; for un-equal times, the formulæ are much more complicated. Classically, the Hamiltonian density is H = Π 0 Φ + Π 0 Φ L = Π Π + Φ Φ + m Φ Φ, S.3 so the quantum theory s Hamiltonian is obviously as in eq. 5. Problem 3b: Fourier transforming the canonical commutation relations S.3 results in [ˆΦ p, ˆΦ p ] = [ˆΦ p, ˆΦ p ] = [ˆΦ p, ˆΦ p ] = 0, [ˆΠ p, ˆΠ p ] = [ˆΠ p, ˆΠ p ] = [ˆΠ p, ˆΠ p ] = 0, [ˆΦ p, ˆΠ p ] = [ˆΦ p, ˆΠ p ] = 0, S.33 [ˆΦ p, ˆΠ p ] = [ˆΦ p, ˆΠ p ] = δ p,p. 7

8 Consequently, [â p, â p ] = [â p, ˆb p ] = [ˆb p, ˆb p ] = 0 S.34 because all the ˆΦ p and all the ˆΠ p operators commute with each other, and likewise [ˆb p, ˆb p ] = [ˆb p, â p ] = [â p, â p ] = 0 S.35 because all the ˆΦ p and all ˆΠ p operators commute with each other too. Less obviously [â p, ˆb p ] = E p E p = δ p, p E p E p E p E p Ep E p 0 + ie p iδ p, p + ie p iδ p, p 0 = 0, S.36 and similarly [â p, ˆb p ] = 0. Finally, [â p, â p ] = [ˆb p, ˆb p ] = E p E p 0 ie p iδ p,p + ie p iδ p,p + 0 E p E p = δ p,p E p + E p E p E p = δ p,p. S.37 Problem 3c: First, Fourier-transforming the free Hamiltonian 5 gives us Ĥ free = p ˆΠ p ˆΠp + E p ˆΦ p ˆΦ p. S.38 Second, we reverse the definitions 7 to obtain ˆΦ p = â p + ˆb p Ep and ˆΠp = Ep i ˆbp â p. S.39 8

9 Third, we calculate E p ˆΦ p ˆΦ p + ˆΠ p ˆΠ p = E p â p + ˆb p âp + ˆb p = E p â pâ p + ˆb pˆb p = E p â pâ p + ˆb pˆb p + Finally, we put eqs. S.40 and S.38 together and derive Ĥ free = E ˆΦ p ˆΦ p p + ˆΠ p ˆΠ p p = E p â pâ p + ˆb pˆb p + p = E p â pâ p + E pˆb pˆbp + const. p + E p â p ˆb p âp ˆb p S.40 S.4 Problem 3d: We begin by Fourier transforming the charge operator ˆQ = d 3 x ˆρx = d 3 x {ˆΠ i x, ˆΦx } {ˆΠx, i ˆΦ x }, S.4 which yields ˆQ = p i {ˆΦ p, ˆΠ } p i {ˆΦp, ˆΠ } p. S.43 Next, we re-express the anticommutators here in terms of the creation and annihilator operators according to eqs. S.39. After simple algebra we find i {ˆΦ p, ˆΠ } p = â pâ p ˆb pˆb p + â pˆb p â pˆb p i, {ˆΦp, ˆΠ p} = â pâ p ˆb pˆb p â pˆb p + â pˆb p, S.44 and therefore ˆQ = p â pâ p ˆb pˆb p = p â pâ p ˆb pˆb p. 9

10 Problem 3e: According to eq., classically T 0i = 0 Φ i Φ + 0 Φ i Φ = Π i Φ Π i Φ, S.45 and hence in the quantum theory ˆP mech = d 3 x {ˆΠ, ˆΦ } {ˆΠ, ˆΦ }. S.46 Fourier-transforming this formula, we arrive at ˆP mech = p ip {ˆΠ p, ˆΦ } ip p + {ˆΠp, ˆΦ } p, S.47 and hence in light of eqs. S.44, ˆP mech = p p â pâ p p ˆb pˆb p = p p â pâ p + p ˆb pˆb p. 3 0

Problem 1(a): As discussed in class, Euler Lagrange equations for charged fields can be written in a manifestly covariant form as L (D µ φ) L

Problem 1(a): As discussed in class, Euler Lagrange equations for charged fields can be written in a manifestly covariant form as L (D µ φ) L PHY 396 K. Solutions for problem set #. Problem 1a: As discussed in class, Euler Lagrange equations for charged fields can be written in a manifestly covariant form as D µ D µ φ φ = 0. S.1 In particularly,

More information

d 2 Area i K i0 ν 0 (S.2) when the integral is taken over the whole space, hence the second eq. (1.12).

d 2 Area i K i0 ν 0 (S.2) when the integral is taken over the whole space, hence the second eq. (1.12). PHY 396 K. Solutions for prolem set #. Prolem 1a: Let T µν = λ K λµ ν. Regardless of the specific form of the K λµ ν φ, φ tensor, its antisymmetry with respect to its first two indices K λµ ν K µλ ν implies

More information

etc., etc. Consequently, the Euler Lagrange equations for the Φ and Φ fields may be written in a manifestly covariant form as L Φ = m 2 Φ, (S.

etc., etc. Consequently, the Euler Lagrange equations for the Φ and Φ fields may be written in a manifestly covariant form as L Φ = m 2 Φ, (S. PHY 396 K. Solutions for problem set #3. Problem 1a: Let s start with the scalar fields Φx and Φ x. Similar to the EM covariant derivatives, the non-abelian covariant derivatives may be integrated by parts

More information

Problem 1(a): At equal times or in the Schrödinger picture the quantum scalar fields ˆΦ a (x) and ˆΠ a (x) satisfy commutation relations

Problem 1(a): At equal times or in the Schrödinger picture the quantum scalar fields ˆΦ a (x) and ˆΠ a (x) satisfy commutation relations PHY 396 K. Solutions for homework set #7. Problem 1a: At equal times or in the Shrödinger iture the quantum salar fields ˆΦ a x and ˆΠ a x satisfy ommutation relations ˆΦa x, ˆΦ b y 0, ˆΠa x, ˆΠ b y 0,

More information

PHY 396 K. Problem set #7. Due October 25, 2012 (Thursday).

PHY 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 information

From Particles to Fields

From Particles to Fields From Particles to Fields Tien-Tsan Shieh Institute of Mathematics Academic Sinica July 25, 2011 Tien-Tsan Shieh (Institute of MathematicsAcademic Sinica) From Particles to Fields July 25, 2011 1 / 24 Hamiltonian

More information

PHY 396 K. Solutions for problems 1 and 2 of set #5.

PHY 396 K. Solutions for problems 1 and 2 of set #5. PHY 396 K. Solutions for problems 1 and of set #5. Problem 1a: The conjugacy relations  k,  k,, Ê k, Ê k, follow from hermiticity of the Âx and Êx quantum fields and from the third eq. 6 for the polarization

More information

d 3 k In the same non-relativistic normalization x k = exp(ikk),

d 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 information

PHY 396 K. Problem set #3. Due September 29, 2011.

PHY 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 information

Vector Fields. It is standard to define F µν = µ ϕ ν ν ϕ µ, so that the action may be written compactly as

Vector Fields. It is standard to define F µν = µ ϕ ν ν ϕ µ, so that the action may be written compactly as Vector Fields The most general Poincaré-invariant local quadratic action for a vector field with no more than first derivatives on the fields (ensuring that classical evolution is determined based on the

More information

Maxwell s equations. based on S-54. electric field charge density. current density

Maxwell 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 information

An Introduction to Quantum Field Theory

An Introduction to Quantum Field Theory An Introduction to Quantum Field Theory Hartmut Wittig Theory Group Deutsches Elektronen-Synchrotron, DESY Notkestrasse 85 22603 Hamburg Germany Lectures presented at the School for Young High Energy Physicists,

More information

PHY 396 K. Solutions for problem set #6. Problem 1(a): Starting with eq. (3) proved in class and applying the Leibniz rule, we obtain

PHY 396 K. Solutions for problem set #6. Problem 1(a): Starting with eq. (3) proved in class and applying the Leibniz rule, we obtain PHY 396 K. Solutions for problem set #6. Problem 1(a): Starting with eq. (3) proved in class and applying the Leibniz rule, we obtain γ κ γ λ, S µν] = γ κ γ λ, S µν] + γ κ, S µν] γ λ = γ κ( ig λµ γ ν ig

More information

Quantum Field Theory I Examination questions will be composed from those below and from questions in the textbook and previous exams

Quantum Field Theory I Examination questions will be composed from those below and from questions in the textbook and previous exams Quantum Field Theory I Examination questions will be composed from those below and from questions in the textbook and previous exams III. Quantization of constrained systems and Maxwell s theory 1. The

More information

d 2 Area i K i0 ν 0 (S.2) d 3 x t 0ν

d 2 Area i K i0 ν 0 (S.2) d 3 x t 0ν PHY 396 K. Solutions for prolem set #. Prolem 1: Let T µν = λ K λµ ν. Regrdless of the specific form of the K λµ ν φ, φ tensor, its ntisymmetry with respect to its first two indices K λµ ν K µλ ν implies

More information

CHAPTER 1. SPECIAL RELATIVITY AND QUANTUM MECHANICS

CHAPTER 1. SPECIAL RELATIVITY AND QUANTUM MECHANICS CHAPTER 1. SPECIAL RELATIVITY AND QUANTUM MECHANICS 1.1 PARTICLES AND FIELDS The two great structures of theoretical physics, the theory of special relativity and quantum mechanics, have been combined

More information

Problem 1(a): Since the Hamiltonian (1) is a function of the particles momentum, the evolution operator has a simple form in momentum space,

Problem 1(a): Since the Hamiltonian (1) is a function of the particles momentum, the evolution operator has a simple form in momentum space, PHY 396 K. Solutions for problem set #5. Problem 1a: Since the Hamiltonian 1 is a function of the particles momentum, the evolution operator has a simple form in momentum space, exp iĥt d 3 2π 3 e itω

More information

3 Quantization of the Dirac equation

3 Quantization of the Dirac equation 3 Quantization of the Dirac equation 3.1 Identical particles As is well known, quantum mechanics implies that no measurement can be performed to distinguish particles in the same quantum state. Elementary

More information

7 Quantized Free Dirac Fields

7 Quantized Free Dirac Fields 7 Quantized Free Dirac Fields 7.1 The Dirac Equation and Quantum Field Theory The Dirac equation is a relativistic wave equation which describes the quantum dynamics of spinors. We will see in this section

More information

ADVANCED 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 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 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

Prancing Through Quantum Fields

Prancing Through Quantum Fields November 23, 2009 1 Introduction Disclaimer Review of Quantum Mechanics 2 Quantum Theory Of... Fields? Basic Philosophy 3 Field Quantization Classical Fields Field Quantization 4 Intuitive Field Theory

More information

Attempts at relativistic QM

Attempts 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 information

Classical and Quantum Mechanics of a Charged Particle Moving in Electric and Magnetic Fields

Classical and Quantum Mechanics of a Charged Particle Moving in Electric and Magnetic Fields Classical Mechanics Classical and Quantum Mechanics of a Charged Particle Moving in Electric and Magnetic Fields In this section I describe the Lagrangian and the Hamiltonian formulations of classical

More information

MSci EXAMINATION. Date: XX th May, Time: 14:30-17:00

MSci EXAMINATION. Date: XX th May, Time: 14:30-17:00 MSci EXAMINATION PHY-415 (MSci 4242 Relativistic Waves and Quantum Fields Time Allowed: 2 hours 30 minutes Date: XX th May, 2010 Time: 14:30-17:00 Instructions: Answer THREE QUESTIONS only. Each question

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

PROBLEM SET 1 SOLUTIONS

PROBLEM SET 1 SOLUTIONS MASSACHUSETTS INSTITUTE OF TECHNOLOGY Physics Department 8.323: Relativistic Quantum Field Theory I Prof.Alan Guth February 29, 2008 PROBLEM SET 1 SOLUTIONS Problem 1: The energy-momentum tensor for source-free

More information

2P + E = 3V 3 + 4V 4 (S.2) D = 4 E

2P + 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 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

σ 2 + π = 0 while σ satisfies a cubic equation λf 2, σ 3 +f + β = 0 the second derivatives of the potential are = λ(σ 2 f 2 )δ ij, π i π j

σ 2 + π = 0 while σ satisfies a cubic equation λf 2, σ 3 +f + β = 0 the second derivatives of the potential are = λ(σ 2 f 2 )δ ij, π i π j PHY 396 K. Solutions for problem set #4. Problem 1a: The linear sigma model has scalar potential V σ, π = λ 8 σ + π f βσ. S.1 Any local minimum of this potential satisfies and V = λ π V σ = λ σ + π f =

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

Canonical Quantization

Canonical Quantization Canonical Quantization March 6, 06 Canonical quantization of a particle. The Heisenberg picture One of the most direct ways to quantize a classical system is the method of canonical quantization introduced

More information

PHY 396 K. Solutions for problem set #7.

PHY 396 K. Solutions for problem set #7. PHY 396 K. Solution for problem et #7. Problem 1a: γ µ γ ν ±γ ν γ µ where the ign i + for µ ν and otherwie. Hence for any product Γ of the γ matrice, γ µ Γ 1 nµ Γγ µ where n µ i the number of γ ν µ factor

More information

Classical field theory 2012 (NS-364B) Feynman propagator

Classical 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 information

3P1a Quantum Field Theory: Example Sheet 1 Michaelmas 2016

3P1a Quantum Field Theory: Example Sheet 1 Michaelmas 2016 3P1a Quantum Field Theory: Example Sheet 1 Michaelmas 016 Corrections and suggestions should be emailed to B.C.Allanach@damtp.cam.ac.uk. Starred questions may be handed in to your supervisor for feedback

More information

Physics 582, Problem Set 3 Solutions

Physics 582, Problem Set 3 Solutions Physics 582, Problem Set 3 Solutions TAs: Hart Goldman and Ramanjit Sohal Fall 208 Contents. Spin Waves in a Quantum Heisenberg Antiferromagnet 2. The Two-Component Complex Scalar Field 5. SPIN WAVES IN

More information

MASSACHUSETTS INSTITUTE OF TECHNOLOGY Physics Department 8.323: Relativistic Quantum Field Theory I PROBLEM SET 2

MASSACHUSETTS INSTITUTE OF TECHNOLOGY Physics Department 8.323: Relativistic Quantum Field Theory I PROBLEM SET 2 MASSACHUSETTS INSTITUTE OF TECHNOLOGY Physics Department 8.323: Relativistic Quantum Field Theory I PROBLEM SET 2 REFERENCES: Peskin and Schroeder, Chapter 2 Problem 1: Complex scalar fields Peskin and

More information

1 Quantum fields in Minkowski spacetime

1 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 information

An Introduction to Quantum Field Theory

An Introduction to Quantum Field Theory An Introduction to Quantum Field Theory Owe Philipsen Institut für Theoretische Physik Universität Münster Wilhelm-Klemm-Str.9, 48149 Münster, Germany Lectures presented at the School for Young High Energy

More information

QUANTUM FIELD THEORY. Kenzo INOUE

QUANTUM FIELD THEORY. Kenzo INOUE QUANTUM FIELD THEORY Kenzo INOUE September 0, 03 Contents Field and Lorentz Transformation. Fields........................................... Lorentz Transformation..................................3

More information

PHY 396 K. Problem set #11, the last set this semester! Due December 1, 2016.

PHY 396 K. Problem set #11, the last set this semester! Due December 1, 2016. PHY 396 K. Problem set #11, the last set this semester! Due December 1, 2016. In my notations, the A µ and their components A a µ are the canonically normalized vector fields, while the A µ = ga µ and

More information

1 Free real scalar field

1 Free real scalar field 1 Free real scalar field The Hamiltonian is H = d 3 xh = 1 d 3 x p(x) +( φ) + m φ Let us expand both φ and p in Fourier series: d 3 p φ(t, x) = ω(p) φ(t, x)e ip x, p(t, x) = ω(p) p(t, x)eip x. where ω(p)

More information

Electric and Magnetic Forces in Lagrangian and Hamiltonian Formalism

Electric and Magnetic Forces in Lagrangian and Hamiltonian Formalism Electric and Magnetic Forces in Lagrangian and Hamiltonian Formalism Benjamin Hornberger 1/26/1 Phy 55, Classical Electrodynamics, Prof. Goldhaber Lecture notes from Oct. 26, 21 Lecture held by Prof. Weisberger

More information

The Klein-Gordon equation

The Klein-Gordon equation Lecture 8 The Klein-Gordon equation WS2010/11: Introduction to Nuclear and Particle Physics The bosons in field theory Bosons with spin 0 scalar (or pseudo-scalar) meson fields canonical field quantization

More information

Quantization of Scalar Field

Quantization 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 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

Quantum Field Theory II

Quantum 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 information

The Dirac Field. Physics , Quantum Field Theory. October Michael Dine Department of Physics University of California, Santa Cruz

The Dirac Field. Physics , Quantum Field Theory. October Michael Dine Department of Physics University of California, Santa Cruz Michael Dine Department of Physics University of California, Santa Cruz October 2013 Lorentz Transformation Properties of the Dirac Field First, rotations. In ordinary quantum mechanics, ψ σ i ψ (1) is

More information

What s up with those Feynman diagrams? an Introduction to Quantum Field Theories

What s up with those Feynman diagrams? an Introduction to Quantum Field Theories What s up with those Feynman diagrams? an Introduction to Quantum Field Theories Martin Nagel University of Colorado February 3, 2010 Martin Nagel (CU Boulder) Quantum Field Theories February 3, 2010 1

More information

Week 1, solution to exercise 2

Week 1, solution to exercise 2 Week 1, solution to exercise 2 I. THE ACTION FOR CLASSICAL ELECTRODYNAMICS A. Maxwell s equations in relativistic form Maxwell s equations in vacuum and in natural units (c = 1) are, E=ρ, B t E=j (inhomogeneous),

More information

Preliminaries: what you need to know

Preliminaries: 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 information

Relativistic Waves and Quantum Fields

Relativistic Waves and Quantum Fields Relativistic Waves and Quantum Fields (SPA7018U & SPA7018P) Gabriele Travaglini December 10, 2014 1 Lorentz group Lectures 1 3. Galileo s principle of Relativity. Einstein s principle. Events. Invariant

More information

Lecture notes for the 2016 HEP School for Experimental High Energy Physics Students University of Lancaster, 4-16 September, 2016

Lecture notes for the 2016 HEP School for Experimental High Energy Physics Students University of Lancaster, 4-16 September, 2016 Technical Report RAL-TR-2016-007 Lecture notes for the 2016 HEP School for Experimental High Energy Physics Students University of Lancaster, 4-16 September, 2016 Lecturers: A Banfi, DG Cerdeño, C Englert,

More information

PHY 396 K. Solutions for homework set #9.

PHY 396 K. Solutions for homework set #9. PHY 396 K. Solutions for homework set #9. Problem 2(a): The γ 0 matrix commutes with itself but anticommutes with the space-indexed γ 1,2,3. At the same time, the parity reflects the space coordinates

More information

Hamiltonian Field Theory

Hamiltonian Field Theory Hamiltonian Field Theory August 31, 016 1 Introduction So far we have treated classical field theory using Lagrangian and an action principle for Lagrangian. This approach is called Lagrangian field theory

More information

Quantum Field Theory

Quantum 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 information

Lecture notes for FYS610 Many particle Quantum Mechanics

Lecture notes for FYS610 Many particle Quantum Mechanics UNIVERSITETET I STAVANGER Institutt for matematikk og naturvitenskap Lecture notes for FYS610 Many particle Quantum Mechanics Note 20, 19.4 2017 Additions and comments to Quantum Field Theory and the Standard

More information

General Relativity in a Nutshell

General Relativity in a Nutshell General Relativity in a Nutshell (And Beyond) Federico Faldino Dipartimento di Matematica Università degli Studi di Genova 27/04/2016 1 Gravity and General Relativity 2 Quantum Mechanics, Quantum Field

More information

Problem Set No. 3: Canonical Quantization Due Date: Wednesday October 19, 2018, 5:00 pm. 1 Spin waves in a quantum Heisenberg antiferromagnet

Problem Set No. 3: Canonical Quantization Due Date: Wednesday October 19, 2018, 5:00 pm. 1 Spin waves in a quantum Heisenberg antiferromagnet Physics 58, Fall Semester 018 Professor Eduardo Fradkin Problem Set No. 3: Canonical Quantization Due Date: Wednesday October 19, 018, 5:00 pm 1 Spin waves in a quantum Heisenberg antiferromagnet In this

More information

Continuous Symmetries and Conservation Laws. Noether s Theorem

Continuous Symmetries and Conservation Laws. Noether s Theorem As far as the laws of mathematics refer to reality, they are not certain; and as far as they are certain, they do not refer to reality. Albert Einstein (1879-1955) 3 Continuous Symmetries and Conservation

More information

Introduction to gauge theory

Introduction to gauge theory Introduction to gauge theory 2008 High energy lecture 1 장상현 연세대학교 September 24, 2008 장상현 ( 연세대학교 ) Introduction to gauge theory September 24, 2008 1 / 72 Table of Contents 1 Introduction 2 Dirac equation

More information

221A Miscellaneous Notes Continuity Equation

221A Miscellaneous Notes Continuity Equation 221A Miscellaneous Notes Continuity Equation 1 The Continuity Equation As I received questions about the midterm problems, I realized that some of you have a conceptual gap about the continuity equation.

More information

Scalar Electrodynamics. The principle of local gauge invariance. Lower-degree conservation

Scalar Electrodynamics. The principle of local gauge invariance. Lower-degree conservation . Lower-degree conservation laws. Scalar Electrodynamics Let us now explore an introduction to the field theory called scalar electrodynamics, in which one considers a coupled system of Maxwell and charged

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

Quantization of a Scalar Field

Quantization of a Scalar Field Quantization of a Scalar Field Required reading: Zwiebach 0.-4,.4 Suggested reading: Your favorite quantum text Any quantum field theory text Quantizing a harmonic oscillator: Let s start by reviewing

More information

Exercises Symmetries in Particle Physics

Exercises Symmetries in Particle Physics Exercises Symmetries in Particle Physics 1. A particle is moving in an external field. Which components of the momentum p and the angular momentum L are conserved? a) Field of an infinite homogeneous plane.

More information

where P a is a projector to the eigenspace of A corresponding to a. 4. Time evolution of states is governed by the Schrödinger equation

where 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 information

We start by recalling electrodynamics which is the first classical field theory most of us have encountered in theoretical physics.

We start by recalling electrodynamics which is the first classical field theory most of us have encountered in theoretical physics. Quantum Field Theory I ETH Zurich, HS12 Chapter 6 Prof. N. Beisert 6 Free Vector Field Next we want to find a formulation for vector fields. This includes the important case of the electromagnetic field

More information

The Hamiltonian operator and states

The 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 information

As usual, these notes are intended for use by class participants only, and are not for circulation. Week 7: Lectures 13, 14.

As usual, these notes are intended for use by class participants only, and are not for circulation. Week 7: Lectures 13, 14. As usual, these notes are intended for use by class participants only, and are not for circulation. Week 7: Lectures 13, 14 Majorana spinors March 15, 2012 So far, we have only considered massless, two-component

More information

Under evolution for a small time δt the area A(t) = q p evolves into an area

Under evolution for a small time δt the area A(t) = q p evolves into an area Physics 106a, Caltech 6 November, 2018 Lecture 11: Hamiltonian Mechanics II Towards statistical mechanics Phase space volumes are conserved by Hamiltonian dynamics We can use many nearby initial conditions

More information

Week 7 Lecture: Concepts of Quantum Field Theory (QFT)

Week 7 Lecture: Concepts of Quantum Field Theory (QFT) Week 7 Lecture: Concepts of Quantum Field Theory QFT Andrew Forrester February, 008 Deriving the Klein-Gordon Equation with a Physical Model This Week s Questions/Goals How do you derive the Klein-Gordon

More information

Fermionic Algebra and Fock Space

Fermionic Algebra and Fock Space Fermionic Algebra and Fock Space Earlier in class we saw how the harmonic-oscillator-like bosonic commutation relations [â α,â β ] = 0, [ ] â α,â β = 0, [ ] â α,â β = δ α,β (1) give rise to the bosonic

More information

Quantum Field Theory Notes. Ryan D. Reece

Quantum Field Theory Notes. Ryan D. Reece Quantum Field Theory Notes Ryan D. Reece November 27, 2007 Chapter 1 Preliminaries 1.1 Overview of Special Relativity 1.1.1 Lorentz Boosts Searches in the later part 19th century for the coordinate transformation

More information

THE QFT NOTES 5. Badis Ydri Department of Physics, Faculty of Sciences, Annaba University, Annaba, Algeria. December 9, 2011

THE QFT NOTES 5. Badis Ydri Department of Physics, Faculty of Sciences, Annaba University, Annaba, Algeria. December 9, 2011 THE QFT NOTES 5 Badis Ydri Department of Physics, Faculty of Sciences, Annaba University, Annaba, Algeria. December 9, 2011 Contents 1 The Electromagnetic Field 2 1.1 Covariant Formulation of Classical

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

Non Abelian Higgs Mechanism

Non Abelian Higgs Mechanism Non Abelian Higgs Mechanism When a local rather than global symmetry is spontaneously broken, we do not get a massless Goldstone boson. Instead, the gauge field of the broken symmetry becomes massive,

More information

PHY 396 L. Solutions for homework set #20.

PHY 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 information

(a p (t)e i p x +a (t)e ip x p

(a p (t)e i p x +a (t)e ip x p 5/29/3 Lecture outline Reading: Zwiebach chapters and. Last time: quantize KG field, φ(t, x) = (a (t)e i x +a (t)e ip x V ). 2Ep H = ( ȧ ȧ(t)+ 2E 2 E pa a) = p > E p a a. P = a a. [a p,a k ] = δ p,k, [a

More information

Mandl and Shaw reading assignments

Mandl and Shaw reading assignments Mandl and Shaw reading assignments Chapter 2 Lagrangian Field Theory 2.1 Relativistic notation 2.2 Classical Lagrangian field theory 2.3 Quantized Lagrangian field theory 2.4 Symmetries and conservation

More information

Fermionic Algebra and Fock Space

Fermionic Algebra and Fock Space Fermionic Algebra and Fock Space Earlier in class we saw how harmonic-oscillator-like bosonic commutation relations [â α,â β ] = 0, [ ] â α,â β = 0, [ ] â α,â β = δ α,β (1) give rise to the bosonic Fock

More information

Quantum Reduced Loop Gravity: matter fields coupling

Quantum Reduced Loop Gravity: matter fields coupling Quantum Reduced Loop Gravity: matter fields coupling Jakub Bilski Department of Physics Fudan University December 5 th, 2016 Jakub Bilski (Fudan University QRLG: matter fields coupling December 5 th, 2016

More information

2 Quantization of the scalar field

2 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 information

Part III Quantum Field Theory

Part III Quantum Field Theory Part III Quantum Field Theory Based on lectures by B. Allanach Notes taken by Dexter Chua Michaelmas 2016 These notes are not endorsed by the lecturers, and I have modified them (often significantly) after

More information

Quantum Mechanics I Physics 5701

Quantum Mechanics I Physics 5701 Quantum Mechanics I Physics 5701 Z. E. Meziani 02/24//2017 Physics 5701 Lecture Commutation of Observables and First Consequences of the Postulates Outline 1 Commutation Relations 2 Uncertainty Relations

More information

Supergravity in Quantum Mechanics

Supergravity in Quantum Mechanics Supergravity in Quantum Mechanics hep-th/0408179 Peter van Nieuwenhuizen C.N. Yang Institute for Theoretical Physics Stony Brook University Erice Lectures, June 2017 Vienna Lectures, Jan/Feb 2017 Aim of

More information

PHY 396 K. Problem set #5. Due October 9, 2008.

PHY 396 K. Problem set #5. Due October 9, 2008. PHY 396 K. Problem set #5. Due October 9, 2008.. First, an exercise in bosonic commutation relations [â α, â β = 0, [â α, â β = 0, [â α, â β = δ αβ. ( (a Calculate the commutators [â αâ β, â γ, [â αâ β,

More information

Lattice Gauge Theory and the Maxwell-Klein-Gordon equations

Lattice Gauge Theory and the Maxwell-Klein-Gordon equations Lattice Gauge Theory and the Maxwell-Klein-Gordon equations Tore G. Halvorsen Centre of Mathematics for Applications, UiO 19. February 2008 Abstract In this talk I will present a discretization of the

More information

H =Π Φ L= Φ Φ L. [Φ( x, t ), Φ( y, t )] = 0 = Φ( x, t ), Φ( y, t )

H =Π Φ L= Φ Φ L. [Φ( x, t ), Φ( y, t )] = 0 = Φ( x, t ), Φ( y, t ) 2.2 THE SPIN ZERO SCALAR FIELD We now turn to applying our quantization procedure to various free fields. As we will see all goes smoothly for spin zero fields but we will require some change in the CCR

More information

arxiv: v2 [math-ph] 10 Aug 2011

arxiv: v2 [math-ph] 10 Aug 2011 Classical mechanics in reparametrization-invariant formulation and the Schrödinger equation A. A. Deriglazov and B. F. Rizzuti Depto. de Matemática, ICE, Universidade Federal de Juiz de Fora, MG, Brazil

More information

Path Integral for Spin

Path Integral for Spin Path Integral for Spin Altland-Simons have a good discussion in 3.3 Applications of the Feynman Path Integral to the quantization of spin, which is defined by the commutation relations [Ŝj, Ŝk = iɛ jk

More information

Quantum Field Theory. Badis Ydri Department of Physics, Faculty of Sciences, Annaba University, Annaba, Algeria. December 10, 2012

Quantum Field Theory. Badis Ydri Department of Physics, Faculty of Sciences, Annaba University, Annaba, Algeria. December 10, 2012 Quantum Field Theory Badis Ydri Department of Physics, Faculty of Sciences, Annaba University, Annaba, Algeria. December 10, 2012 1 Concours Doctorat Physique Théorique Théorie Des Champs Quantique v1

More information

Quantization of the E-M field

Quantization of the E-M field Quantization of the E-M field 0.1 Classical E&M First we will wor in the transverse gauge where there are no sources. Then A = 0, nabla A = B, and E = 1 A and Maxwell s equations are B = 1 E E = 1 B E

More information

The Dirac Equation. Topic 3 Spinors, Fermion Fields, Dirac Fields Lecture 13

The Dirac Equation. Topic 3 Spinors, Fermion Fields, Dirac Fields Lecture 13 The Dirac Equation Dirac s discovery of a relativistic wave equation for the electron was published in 1928 soon after the concept of intrisic spin angular momentum was proposed by Goudsmit and Uhlenbeck

More information

Lagrangian. µ = 0 0 E x E y E z 1 E x 0 B z B y 2 E y B z 0 B x 3 E z B y B x 0. field tensor. ν =

Lagrangian. µ = 0 0 E x E y E z 1 E x 0 B z B y 2 E y B z 0 B x 3 E z B y B x 0. field tensor. ν = Lagrangian L = 1 4 F µνf µν j µ A µ where F µν = µ A ν ν A µ = F νµ. F µν = ν = 0 1 2 3 µ = 0 0 E x E y E z 1 E x 0 B z B y 2 E y B z 0 B x 3 E z B y B x 0 field tensor. Note that F µν = g µρ F ρσ g σν

More information

Biquaternion formulation of relativistic tensor dynamics

Biquaternion formulation of relativistic tensor dynamics Juli, 8, 2008 To be submitted... 1 Biquaternion formulation of relativistic tensor dynamics E.P.J. de Haas High school teacher of physics Nijmegen, The Netherlands Email: epjhaas@telfort.nl In this paper

More information

arxiv: v1 [gr-qc] 15 Jul 2011

arxiv: v1 [gr-qc] 15 Jul 2011 Comment on Hamiltonian formulation for the theory of gravity and canonical transformations in extended phase space by T P Shestakova N Kiriushcheva, P G Komorowski, and S V Kuzmin The Department of Applied

More information

Lecture 5: Sept. 19, 2013 First Applications of Noether s Theorem. 1 Translation Invariance. Last Latexed: September 18, 2013 at 14:24 1

Lecture 5: Sept. 19, 2013 First Applications of Noether s Theorem. 1 Translation Invariance. Last Latexed: September 18, 2013 at 14:24 1 Last Latexed: September 18, 2013 at 14:24 1 Lecture 5: Sept. 19, 2013 First Applications of Noether s Theorem Copyright c 2005 by Joel A. Shapiro Now it is time to use the very powerful though abstract

More information

Chapter 4: Quantum Electrodynamics

Chapter 4: Quantum Electrodynamics Chapter 4: Quantum Electrodynamics This chapter provides a survey of quantum electrodynamics, the quantum theory of the electromagnetic field and its interaction with electrically charged particles, such

More information

Quantization of scalar fields

Quantization of scalar fields Quantization of scalar fields March 8, 06 We have introduced several distinct types of fields, with actions that give their field equations. These include scalar fields, S α ϕ α ϕ m ϕ d 4 x and complex

More information