Tham khảo tài liệu 'mathematical method in science and engineering episode 8', kỹ thuật - công nghệ, cơ khí - chế tạo máy phục vụ nhu cầu học tập, nghiên cứu và làm việc hiệu quả | GROUP INVARIANTS 233 We can write this as X2 y2 X y 0 y 1L55 For a linear transformation between x y and x y we write X _ a b X ỹ c d y Invariance of jC2 y2 can now be expressed as s2-ỹ2 xy 1 0 0 -1 x y a c b d 1 0 0 -1 X y X y. X2 - y2 . a2 c2 ab cd ab cd b2 d2 10 X 0 -1 J y From above we see that for x2 y2 to remain invariant under the transformation Eq. components of the transformation matrix must satisfy a2 c2 1 b2 d2 -1 ab cd 0 . This means that only one of a b c d can be independent. Defining a new parameter X 88 a cosh X we see that the transformation matrix in Equation can be written as cosh X sinh X sinh X cosh X Introducing cosh X 7 sinh X tanh X X y a b c d X y 234 CONTINUOUS GROUPS AND REPRESENTATIONS where y 7 and 3 v c along with the identification X ct y x we obtain CN r X f3y nct 1. X 17 7 X 1 This is nothing but the Lorentz transformation Eqs. cĩ . . ct vx c a 1- 2 v X . . . x vt . which leaves distances in spacetime that is c2t2 a 2 invariant. UNITARY GROUP IN TWO DIMENSIONS 17 2 Quantum mechanics is formulated in complex space. Hence the components of the transformation matrix are in general complex numbers. The scalar or inner product of two vectors in n-dimensional complex space is defined as x y X I xt y2 d-----1- x nyn where X means the complex conjugate of X. Unitary transformations are linear transformations which leaves x x x 2 X Xi X2X2 d------ -X Xn invariant. All such transformations form the unitary group v ri . An element of t 2 can be written as . _ A B u c D UNI TAR Y GROUP IN TWO DIMENSIONS u 2 235 where and D are in general complex numbers. Invariance of x x gives the unitarity condition as u u Ulf I where I Ũ is called the Hermitian conjugate of u. Using the unitarity condition we can write A c B D AB c D M 2 C 2 AB D C A B C D B 2 D 2 . 1 0 0 1 which gives K C 2 1 B 2 D 2 1
