SYMMETRIES AND QUANTUM FIELD THEORY Master 2 ICFP, Program ‘Quantum Physics’, 2014-2015 Instructor: Giuliano Orso, Lab. MPQ. Webpage: https://sites.google.com/site/giulianoorso/ Sheet 2 0 Useful formulas σ1 = 0 1 1 0 σ2 = 0 i −i 0 σ3 = 1 0 ~·K ~ + iθ~ · J) ~ General transformation of L↑+ : Λ = exp(iφ Infinitesimal transformations :  0  −φ ~·K ~ + iθ~ · J) ~ µν =  1 ω µν = (iφ −φ2 −φ3 0 −1 −φ1 0 −θ3 θ2 σi σj = δij 1l + iijk σk −φ2 θ3 0 −θ1  −φ3 −θ2   θ1  0 The Lie algebra of the restricted Lorentz group L↑+ consists in two independent su(2) algebras with ~ ± = (J~ ± iK)/2. ~ generators N 1 Representations of the restricted Lorentz group In this exercise one wants to study the irreducible representations of L↑+ of dimension 2. 1. Express Ji and Ki in terms of Pauli matrices for the representation ( 21 , 0) and for the representation (0, 12 ). ~ φ) ~ associated with a general restricted Lorentz trans2. Write down (explicitly) the matrix ΛL (θ, 1 formation in the representation ( 2 , 0) (you have to use the exponential of an infinitesimal transfor~ φ) ~ for the representation (0, 1 ). Remark : the mation). In a similar way, write down the matrix ΛR (θ, 2 indices R and L stand for left and right. One feels that parity should play a role ... 3. Are those two representations equivalent ? (Hint : it is sufficient to consider transformations close to the identity) 4. Show that ΛR = (Λ†L )−1 . 5. Let ψL and χR be two spinors (i.e. column vectors with two complex components) transforming 0 0 according to ψL = ΛL ψL and ψR = ΛR ψL , respectively. What is the transformation rule for the † composite object ψL · χR ? ? 6. What is the transformation rule for ψL ? Demonstrate the identity σ2 σi σ2 = −σi? for all ? i = 1, 2, 3. By using this identity, show that σ2 ψL transforms as a right spinor. What about σ2 χ?R ? Henceforth, we can define an operation (called charge conjugation 1 ) that allows to build a right spinor C ? starting from a left spinor : ψR = ψL = iσ2 ψL , and, viceversa, it transforms a right spinor into a left C ? spinor : χL = χR = −iσ2 χR 1. for reasons that shall become clear later in the course. 1 7. What is the connexion between this last result with the following decomposition (that you will justify) of the tensor product of two spinors (each term in this decomposition is an irreducible representation of the restricted Lorentz group) : 2 ( 12 , 0) ⊗ (0, 12 ) = ( 12 , 21 ) ( 12 , 0) ⊗ ( 12 , 0) = (0, 0) ⊕ (1, 0) (1) How to build quadrivectors from spinors Let us consider the left spinors ψL and χL and the right spinors ψR and χR . † 1. How does ψL transform ? † µ † µ 2. Let σ ¯ µ = (1l, −~σ ) and σ µ = (1l, ~σ ). How does ψL σ ¯ χL transform ? What about ψR σ χR with σ = (1l, σi ) ? (Hint : You can restrict yourself to transformations close to the identity). µ 3 Parity In the previous exercise we focused on the restricted Lorentz group. Here we investigate how the parity operation P (change of sign of spatial coordinates, P : ~x → −~x) behaves under a Lorentz tranformation. 1. Calculate P ΛP for a transformation in L↑+ close to the identity. 2. By taking the exponential of this formula, find P ΛP for an arbitrary transformation. 3. In the light of the preceding question, how can we interpret the difference between ΛL and ΛR of exercise 1 ? 4. In order to represent the parity action on spinors, one associates to each left spinor ψL a right spinor ψR . The action of parity is to exchange those spinors. By using ψL , ψR , χL and χR , build a scalar and a pseudo-scalar under the orthochrone Lorentz group (the latter one changes sign under parity, the former one is invariant). 5. Build a quadrivector and a pseudo-quadrivectorr using the spinors. 4 Higher (larger) representations of the restricted Lorentz group One wishes to study the Lorentz group irreducible representations with higher spin. We shall first consider (j+ , j− ) = (1, 0) and (0, 1). To this end, given a rank-2 antisymmetric tensor with complex entries Aµν , one defines the duality operation D(A)µν = 2i µνρσ Aρσ . Such an antisymmetric tensor will be said ”self-dual” whenever D(A) = A and ”anti-self-dual” if D(A) = −A. 1. Show that the properties of (i) antisymmetry and of (ii) being self-dual (or anti-self-dual) are preserved by Lorentz transformations (one should remember that the tensor is Lorentz invariant, i.e. 0µνρσ = det(Λ)µνρσ = µνρσ . Hence the linear space of self-dual tensors (A+ ) and anti-self-dual tensors (A− ) must both admit a decomposition as direct sum of irreducible Lorentz representations. 2. Show that A+ and A− have dimension 3 (on the complex field). 3. Is is possible to find a Lorentz scalar in A± ? Show that A+ is the (1, 0) representation and A is the (0, 1) representation. Note that those two representations are exchanged under the parity operation. − 2 0 −E 1 −E 2  E1 0 −B 3 =   E2 B3 0 E3 B2 B1 A2± A3± A1± 0 ∓iA3± ±iA2± 3 ±iA± 0 ∓iA1± 2 1 ∓iA± ±iA± 0  4. The electromagnetic field tensor is F µν = ∂ µ Aν − ∂ ν Aµ  Check that one can write F µν = F+µν + F−µν with F±µν 0  −A1± =  −A2 ± −A3±  −E 3 B2  . −B 1  0   , where  ~ ± = 1 −E ~ ∓ iB ~ . Show that F µν are respectively self-dual et anti-self-dual. In what representation A ± 2 does F µν transform ? Notes that this is in agreement with the spin content of the representations (1,0) and (0,1). 5. We shall now characterize the (1, 1) representation. Let us consider the space S of all rank-2 symmetric tensors Rµν with real entries. Show that S is stable under the restricted Lorentz group, and also under parity. 6. Show that the trace Rµµ is a Lorentz invariant, and that it is thus possible to decompose S = S0 ⊕ T (where T is the space of matrices proportional to ηµν and S0 is the sub-space of S made of matrices with vanishing trace Rµµ = 0) with the property that those two subspaces are stable under Lorentz trsnaformations. 7. From the results of the preceding question, one can thus decompose M α α S0 = (j+ , j− ) (2) α into a sum of irreducible representations of L↑+ . Remembering that 3 × 3 real symmetric matrices with vanishing trace form the spin j = 2 representation of SO(3), show that there exists at least a pair in 0 0 0 0 = 1. = j− = 2. By invoking parity, show that one has necessarily j+ + j− the sum (2) such that j+ Conclude. The representation (1, 1) is that in which the putative graviton transforms. 3