An Introduction To Quantum Field Theory - M. Peskin, D. Schroeder (Perseus, 1995) · Problem 3.3

The Dirac Field — M. Peskin, D. Schroeder (Perseus, 1995)

Physics — QFT · massless-spinor-products

Problem

Spinor products. (This problem, together with Problems 5.3 and 5.6, introduces an efficient computational method for processes involving massless particles.) Let k0μk_0^\mu, k1μk_1^\mu be fixed 4-vectors satisfying k02=0k_0^2 = 0, k12=−1k_1^2 = -1, k0⋅k1=0k_0 \cdot k_1 = 0. Define basic spinors in the following way: Let uL0u_{L0} be the left-handed spinor for a fermion with momentum k0k_0. Let uR0≠k1uL0u_{R0} = \not{k}_1 u_{L0}. Then, for any pp such that pp is lightlike (p2=0p^2 = 0), define uL(p)=12p⋅k0p̸uR0anduR(p)=12p⋅k0p̸uL0.u_L(p) = \frac{1}{\sqrt{2p\cdot k_0}}\not{p}u_{R0} \qquad \text{and} \qquad u_R(p) = \frac{1}{\sqrt{2p\cdot k_0}}\not{p}u_{L0}. This set of conventions defines the phases of spinors unambiguously (except when pp is parallel to k0k_0). (a) Show that ̸k0uR0=0\not{k}_0 u_{R0} = 0. Show that, for any lightlike pp, p̸uL(p)=p̸uR(p)=0\not{p}u_L(p) = \not{p}u_R(p) = 0. (b) For the choices k0=(E,0,0,−E)k_0 = (E, 0, 0, -E), k1=(0,1,0,0)k_1 = (0, 1, 0, 0), construct uL0u_{L0}, uR0u_{R0}, uL(p)u_L(p), and uR(p)u_R(p) explicitly. (c) Define the *spinor products* s(p1,p2)s(p_1, p_2) and t(p1,p2)t(p_1, p_2), for p1p_1, p2p_2 lightlike, by s(p1,p2)=uˉR(p1)uL(p2),t(p1,p2)=uˉL(p1)uR(p2).s(p_1, p_2) = \bar{u}_R(p_1)u_L(p_2), \qquad t(p_1, p_2) = \bar{u}_L(p_1)u_R(p_2). Using the explicit forms for the uλu_\lambda given in part (b), compute the spinor products explicitly and show that t(p1,p2)=(s(p2,p1))∗t(p_1, p_2) = (s(p_2, p_1))^* and s(p1,p2)=−s(p2,p1)s(p_1, p_2) = -s(p_2, p_1). In addition, show that ∣s(p1,p2)∣2=2p1⋅p2.|s(p_1, p_2)|^2 = 2p_1 \cdot p_2. Thus the spinor products are the square roots of 4-vector dot products.

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