40 CHAPTER 2. FUNDAMENTAL PRINCIPLES OF PHYSICS
which are applicable to different physical fields. Here are the five currently known important
symmetries and their corresponding ingredients:
1) Galileo Invariance:- Space: Euclidean spaceR^3 ,
- Transformation group: Galileo group andSO( 3 ),
- Tensor types: Cartesian tensors,
- Fields: classical mechanics, fluid dynamics, and astrophysics.
2) Lorentz Invariance:- Space: Minkowski space,
- Transformation group: Lorentz groups,
- Tensor types: 4-dimensional tensors and spinors,
- Fields: Quantum physics and Interactions.
3) Einstein General Relativity:- Space: 4-dimensional Riemann manifolds,
- Transformation group:GL( 4 ),
- Tensor types: General tensors and Riemann metric,
- Fields: Gravitation and Astrophysics.
4) Gauge Invariance:- Space: complex vector bundleM⊗pCn,
- Transformation groups:U( 1 )andSU(n),
- Tensor types: Wave functions and gauge fields,
- Fields: Quantum physics and Interactions.
5) Representation Invariance:- Space: Tangent space ofSU(n),
- Transformation:GL(Cn),
- Tensor types:SU(n)tensors,
- Fields: Quantum physics and Interactions.
We now use examples in classical mechanics to illustrate themain characteristics of these
symmetries and to clarify how tensors describe invariances.
Principle 2.7(Galilean Invariance).Mechanical laws are invariant under the following trans-
formations: