By Silvan S. Schweber
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Additional resources for An Introduction to Relativistic Quantum Field Theory
Exercise Set 5 1. Compute the transformation rules for each of the following, and hence decide whether or not they are tensors. Sub-and superscripted quantities (other than coordinates) are understood to be tensors. 37 dXij ∂2˙ ∂xi ∂Xi (c) (d) dt ∂xi∂xj ∂xj ∂xj 2. (Rund, p. 4) Show that if Aj is a type (0, 1) tensor, then ∂Aj ∂Ah - j ∂xh ∂x is a type (0, 2) tensor. 3. Show that, if M and N are tensors of type (1, 1), then: (a) M ij Npq is a tensor of type (2, 2) (a) (b) (e) ∂2xl ∂xi∂xj (b) M ij Njq is a tensor of type (1, 1) (c) M ij Nji is a tensor of type (0, 0) (that is, a scalar field) 4.
Similarly, by sending light beams in the other directions, we see that the other columns of D are orthogonal to the fourth column. If, instead of subtracting, we now add (*) and (**), and divide by 2, we get c2[D11D11 + D12 D12 + D13 D13 - c2D14 D14] + [D41D41 + D42 D42 + D43 D43 - c2D44 D44] = 0, showing that c2“column 1, column 1‘ = -“column 4, column 4‘. So, if we write “column 1, column 1‘ = k, 54 then “column 4, column 4‘ = -c2k … (***) Similarly (by choosing other photons) we can replace column 1 by either column 2 or column 3, showing that if we take “column 1, column 1‘ = k, we have k “column i, column i‘ = -kc2 if 1 ≤ i ≤ 3 if i = 4 .
2. Give and example of a contravariant vector field that is not covariant. Justify your claim. 3. Verify the following claim If V and W are contravariant (or covariant) vector fields on M, and if å is a real number, then V+W and åV are again contravariant (or covariant) vector fields on M. 4. 7: If Ci is covariant and Vj is contravariant, then Ck Vk is a scalar. 5. Let ˙: Sn’E1 be the scalar field defined by ˙(p1, p2, . . , pn+1) = pn+1. (a) Express ˙ as a function of the xi and as a function of the x–j (the charts for stereographic projection).