A New Model for Physics Based on Mathematics of Symmetric Domains. : A New Model for Physics Based on Mathematics of Symmetric Domains. Yaakov Friedman Jerusalem College of Technology Jerusalem, Israel
2 Sivan 5762 בס”ד
Interaction between Mathematics and Physics : Interaction between Mathematics and Physics Geometry, trigonometry
Land measure, astronomy
Calculus, differential equations
Mechanics,
aero-hydro-electrodynamics
Linear algebra, operator theory, group theory
Quantum mechanics, special relativity
Differential geometry
General relativity
Algebraic topology,
algebraic geometry Quantum fields, physics
of elementary particles
Objective : Objective Defining a mathematical model that can unify different areas of physics
Symmetry (invariance) principle : Symmetry (invariance) principle Symmetry of a law :
A law of science should not change if we change the point from which we observe the natural phenomena.
What Is Symmetry? : What Is Symmetry? Fundamental organizing principle in nature and art
Preserves distances, angles, sizes and shapes
Reflection : Reflection Produce an object’s mirror image
A reflection must have a mirror plane
Symmetry operator : Symmetry operator S: S2 =I Eigenvalue of S {-1,1}
Slide8 : system1 Space-time transformations between inertial systems Principle of Relativity
Symmetry in Special Relativity : Symmetry in Special Relativity
Black-box model for space-time transformation : Black-box model for space-time transformation Linearity of the transformation
Space-time symmetry : Space-time symmetry
Lorentz transformation : Lorentz transformation
Eigenspaces of the symmetry : Eigenspaces of the symmetry
Interval conservation : Interval conservation Invariant velocity When is Sv self-adjoint? isometric? Galilean transformation ?
Slide15 : Ball of Possible Velocities Lorentz
Transformation Einstein's velocity addition Ball of possible velocities-
Bounded Symmetric Domain
Bounded Symmetric Domain D (BSD) : Bounded Symmetric Domain D (BSD) x0 D S: D D smooth (Aut D)
Fix S ={ x0 }
Smooth: projective, conformal, analytic BSD – homogeneous : x,y D
Aut D (x)=y
Relativistic Dynamic Equation : Relativistic Dynamic Equation Dynamic equation - proper time m0du/d =F-u F-u : Generator of boosts on velocity ball