<Physics From Symmetry>
by Jakob Schwichtenberg
http://physicsfromsymmetry.com/
2.6 Invariance, Symmetry and Covariance
Covariance means something similar, but may not be confused with invariance. An equation is called covariant, if it takes the same #form# when the objects in it are transformed. For instance, if we have an equation
E1 = aA^2 + bBA + cC^4
and after the transformation this equation reads
E1 = aA'^2 + bB' A' + cC'^4
这里A，B，C表示一个坐标系下的一组变量；A'，B'，C' 表示另一个坐标系下的一组变量。
O(2) group: the group of all orthogonal 2 × 2 matrices. 二维旋转，保持向量长度不变。
The transformations of the group with det ( O ) = 1 are rotations. SO(2)
The transformations of the group with det ( O ) = -1 are reflections.
SO(2): O^T O = I, det(O) = 1
The unit complex numbers form a group, called U(1) under ordinary complex number multiplication:
U* U = 1 (U*表示U的共轭)
用矩阵表示二维旋转(SO(2))：
用复数表示二维旋转(U(1))：R^θ = e^iθ = cos(θ) + i sin (θ)
所以，如果定义i为矩阵：
则可将两者联系起来：
所以，认为SO(2)≈U(1)
三维情况：
用矩阵表示三维旋转(SO(3))：
用四元数表示三维旋转：
（用单位四元数可以表示绕三维单位向量u旋转θ: q = cos(θ/2) + sin(θ/2) u）
所以，如果定义矩阵i， j，k (SU(2), 注意不是U(2) )：
则可将两者联系起来：
(其中，a=cos(θ/2), b=sin(θ/2) ux, c=sin(θ/2) uy, d=sin(θ/2) uz, 且ux^2 +uy^2 +uz^2=1)
所以，认为SO(3)≈SU(2)
其他：
- det(q)= a^2 + b^2 + c^2 + d^2. 满足U† U=1 and det(U)=1.
定义，绕u的旋转被表示为：q^−1 uq （q是单位四元数）
- q†q = 1
The symbol † is called "dagger"，it means: a† = (a*)^T .(为了让复数能够变成实数，发明了这个操作。给定复数q，q†q 就得到了实数)
- SU(2) is called the double-cover of SO(3)
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3.4 Lie Algebras
Hermitian matrices：
A matrix fulfilling the condition Ji† = Ji is called Hermitian。
every Hermitian traceless 2 × 2 matrix can be written as a linear combination of Pauli matrices。
Pauli matrices：
注意，与SU(2)里的i, j, k相比,有: σ1 = i j , σ2= i i , σ3= i k
为了去掉系数2，定义：
所以：
把SU(1)对应单位环S1，SU(2)对应S3
参考书：
E. Taylor and J. Wheeler - Spacetime Physics: Introduction to Special Relativity
• D. Fleisch - A Student’s Guide to Vectors and Tensors（SR，张量）
• N. Jeevanjee - An Introduction to Tensors and Group Theory for Physicists (SR)
• A. Zee - Einstein Gravity in a nutshell（SR，GR）
Herbert Goldstein - Classical Mechanics
See, for example, page 90 in Matthew Robinson. Symmetry and the Standard Model. Springer, 1st edition, August 2011. ISBN 978-1-4419-8267-4
See, for example, page 189 in Nadir Jeevanjee. An Introduction to Tensors and Group Theory for Physicists. Birkhaeuser, 1st edition, August 2011. ISBN 978- 0817647148
