13. Symmetry of laws of physics is not the
same as symmetry of solutions!!
– The solution of the two-body gravitation
problem is not symmetric w.r.t. rotations…
…but we are saying that physics laws are.
– Where is the snag?
14. This is what we mean when we say that
gravitation law has rotation symmetry:
Initial state Time
evolution
Rotation
Same final
state
21. If the Gravitational Constant
depended on time, pumped-
storage hydroelectric plants could
generate energy from scratch!!
22. If we privilege a position in
space, momentum
conservation could seem
violated!!
23. Gauge Theories
• Particle physics theories
–Symmetrical against
local transformations
•Specially w.r.t. groups
SU(n), U(N)
–Called “gauge symmetry”
–Extraordinarily successful
24. SU(n)
U(n) Unitary Group of degree n
Continuous group
Complex matrices Unxn
Unitary U*
U=UU*
=1
SU(n) Special Unitary Group
Special: det(U)=1
n2-1 generators
28. –Symmetrical w.r.t. local U(1) transf.
–Acting on the phase of the complex
field of the electron.
–Conserved magnitude: electric charge!!
–Prediction capability: 10-8
• Relation between the electron
anomalous magnetic dipole and the
Rydberg constant
Quantum Electrodynamics (QED)
34. Spin and Rotations
• Spin values as generalization of rotations.
• Space vector of a given dimension (1, 2, 3…)
• Transformations on this space vector with the
same rules as rotations in 3 dimensions
Spin 0: Scalars (dimension 1).
Spin 1/2: Vectors of dimension 2.
Spin 1: Vectors of dimension 3
Spin 3/2: Vectors of dimension 4
35. Identity of Particles
• Fundamental particles of a given kind
are identical!!
– Can be distinguished only by their state
(including spin value)
• Status of a system of quantum particles:
– Wave function
– Depending on the particles and their status
36. Spin-statistics Theorem
• Bosons: Wave function is symmetrical w.r.t.
swapping of two identical bosons
– Bosons can have the same quantum numbers.
– Bose-Einstein statistics
• Fermions: Wave function is anti-symmetrical w.r.t.
swapping of two identical fermions
– Fermions cannot have the same quantum state
(Pauli’s exclusion principle)
– Fermi-Dirac statistics
