Kunze Linear Algebra · Problem 15

Inner Product Spaces

Mathematics — Methods · lorentz-transformations

Problem

A unitary operator on R4R^4 (with the standard inner product) is simply a linear operator which preserves the quadratic form ∣∣(x,y,z,t)∣∣2=x2+y2+z2+t2||(x,y,z,t)||^2 = x^2 + y^2 + z^2 + t^2 that is, a linear operator UU such that ∣∣Uα∣∣2=∣∣α∣∣2||U\alpha||^2 = ||\alpha||^2 for all α\alpha in R4R^4. In a certain part of the theory of relativity, it is of interest to find the linear operators TT which preserve the form ∣∣(x,y,z,t)∣∣L2=t2−x2−y2−z2.||(x,y,z,t)||_L^2 = t^2 - x^2 - y^2 - z^2. Now ∣∣  ∣∣L2||\;||_L^2 does not come from an inner product, but from something called the 'Lorentz metric' (which we shall not go into). For that reason, a linear operator TT on R4R^4 such that ∣∣Tα∣∣L2=∣∣α∣∣L2||T\alpha||_L^2 = ||\alpha||_L^2, for every α\alpha in R4R^4, is called a Lorentz transformation. (a) Show that the function UU defined by U(x,y,z,t)=[t+xy+izy−izt−x]U(x,y,z,t) = \begin{bmatrix} t+x & y+iz \\ y-iz & t-x \end{bmatrix} is an isomorphism of R4R^4 onto the real vector space HH of all self-adjoint 2×22 \times 2 complex matrices. (b) Show that ∣∣α∣∣L2=det⁡(Uα)||\alpha||_L^2 = \det(U\alpha). (c) Suppose TT is a (real) linear operator on the space HH of 2×22 \times 2 self-adjoint matrices. Show that L=U−1TUL = U^{-1}TU is a linear operator on R4R^4. (d) Let MM be any 2×22 \times 2 complex matrix. Show that TM(A)=M∗AMT_M(A) = M^*AM defines a linear operator TMT_M on HH. (Be sure you check that TMT_M maps HH into HH.) (e) If MM is a 2×22 \times 2 matrix such that ∣det⁡M∣=1|\det M| = 1, show that LM=U−1TMUL_M = U^{-1}T_MU is a Lorentz transformation on R4R^4. (f) Find a Lorentz transformation which is not an LML_M.

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