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MIT OpenCourseWare http://ocw.mit.edu 8.323 Relativistic Quantum Field Theory I Spring 2008 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms.� � MASSACHUSETTS INSTITUTE OF TECHNOLOGY Physics Department 8.323: Relativistic Quantum Field Theory I Prof. Alan Guth February 16, 2008 PROBLEM SET 2 REFERENCES: Peskin and Schroeder, Section 2.3 and part of 2.4, through p. 29. Also Lecture Notes 1 (or Lecture Slides 1): Quantization of the Free Scalar Field. Problem 1: Complex scalar fields (10 points) Peskin and Schroeder, Problem 2.2. The problem as stated in the original printing asked you to find 4 conserved currents for the theory with two complex scalar fields. There are actually 6 conserved currents, as is indicated on the Peskin and Schroeder corrections web page, http://www.slac.stanford.edu/∼mpeskin/QFT.html and in the newer printings of the book. You will get full credit for finding the same four that Peskin and Schroeder found, and their generalization for n fields. If you can find all six currents for two fields and their generalization for n fields, you will get extra credit, for a maximum of 12 points. Problem 2: Lorentz transformations and Noether’s theorem for scalar fields (continued) (10 points) In Problem 4 of Problem Set 1, you showed that the Lorentz invariance of the theory of a single scalar field leads to a conserved current (∂µ jµλσ = 0)thatcan be written as jµλσ = x λ Tµσ − x σ Tµλ , where Tµν = ∂µφ∂ν φ − ηµν � . The conserved quantities are then Mλσ ≡ d3xj0λσ . Express these conserved quantities in terms of creation and annihilation operators. [Hint: For the case where λ and σ are both spacelike, the conserved quantity can be written as Mij ≡ ijkJk , where ijk is the fully antisymmetric Levi-Civita tensor. One then finds that Ji = −iijk (2d3πp )3 pj a †(p) ∂a∂p(pk ) . In the course of the derivation you will find ill-defined c-number contributions. You should verify the above expression for Ji, arguing that the c-number contributions vanish due to a symmetry argument. You should also calculate the answer for the case where λ =0, σ = i.]8.323 PROBLEM SET 2, SPRING 2008 p. 2 Problem 3: Lorentz transformations and Noether’s theorem for the electro-magetic potential Aµ(x) (10 points) Consider again the electromagnetic potential Aµ(x), as discussed in problem 2.1 of Peskin and Schroeder. The Lagrangian is 1 � = − Fµν F µν , (1)4 where Fµν = ∂µAν − ∂ν Aµ . (2) In problem 2.1 we learned that translation invariance and Noether’s theorem lead to a nonsymmetric energy-momentum tensor, which can be made symmetric by adding a piece that has the form of a total derivative that is automatically conserved regardless of the equations of motion. We discussed in lecture, however, how the conservation of angular momentum forces one to use a symmetric energy-momentum tensor, so that the cross product of r and the momentum density T 0i gives a conserved angular momentum density � i = ijk xj T 0k . (3) If T µν is both conserved and symmetric, then this angular momentum density can be written as the 0th component of the divergenceless current Kµλσ = x λ T µσ − x σ T µλ , (4) where ∂µKµλσ =0 and � i = ijk K0jk . (5) One might hope, therefore, that if one derived the conservation of angular momentum by using rotational symmetry and Noether’s theorem, then one would be led directly to a symmetric energy-momentum tensor. This hope, however, is not realized, as will be shown in this problem. We are interested mainly in rotations, but for the sake of generality we will consider arbitrary Lorentz transformations, which include rotations as a special case. Since Aµ(x) is a Lorentz vector, under a Lorentz transformation xµ =Λµν xν it transforms as Aµ(x )=Λµν Aν (x) . (6) For an infinitesimal Lorentz transformation Λµν = δνµ −Σµν ,where Σµν is antisymmetric as discussed in the previous problem, the symmetry tranformation becomes Aµ(x)= Aµ(x)+Σλσ � x σ ∂λAµ(x) − ηµλ Aσ (x) � . (7)8.323 PROBLEM SET 2, SPRING 2008 p. 3 (a) Show that the above symmetry leads via Noether’s theorem to the conserved current jµλσ = x σ � Fµκ ∂λAκ + ηµλ � � − Fµλ Aσ − (λ ↔ σ) . (8) Here “−(λ ↔ σ)” means to subtract an expression identical to everything previously written on the right-hand side, except that the subscripts λ and σ are interchanged. (b) Show directly from the equations of motion that the above current is conserved (∂µjµλσ = 0). Thus Noether’s theorem leads to a conserved current, as it must, but since Eq. (8) does not match the form of Eq. (4), the conserved angular momentum current can be constructed without using a symmetric energy-momentum tensor. (c) As in Peskin and Schroeder’s problem 2.1, we can construct a modified form of the conserved current by adding a derivative term: ˆµλσ = jµλσ + ∂κNκµλσ , (9) where Nκµλσ is antisymmetric in its first two indices and in its last two indices. Show that if Nκµλσ = x λ Fµκ Aσ − x σ Fµκ Aλ , (10) then ˆµλσ can be written in the form of Eq. (4), with a symmetric energy-momentum tensor. Problem 4: N on-uniqueness of the harmonic oscillator quantization (10 points) On p. 20 of Peskin and Schroeder, the authors show that the Fourier expansion function φ(p,t) obeys the equation of motion of a harmonic oscillator, and they then reflexively invoke the standard quantization procedure. While the results they obtain are certainly correct, we show in this problem that their logic is insufficient. The classical equations of motion do not imply a unique quantization. Rather, one needs a classical canonical formulation to determine the corresponding quantum theory. Consider for example a classical quantity x(t) that obeys the simple harmonic equa-tion of motion d2x = −ω2 x. (1)dt2 We will show in this problem that there are nonstandard ways in which such a quantity can appear in a quantum theory. We will start by formulating a standard harmonic oscillator, described by the La-grangian 1 1 L = q˙2 − ω02 q 2 . (2)2 2� � � � � � � 8.323 PROBLEM SET 2, SPRING 2008 p. 4


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