Matter is composed of particles with different versions.
Particles have properties like mass and electric charge.
1922 Stern-Gerlach Experiment: Identified a new property of particles by sending a beam of silver atoms through an inhomogeneous magnetic field, which split into two beams.
Half deflected upwards, half downwards.
Initially thought electrons behaved like tiny marbles spinning on themselves (electromagnets).
Calculation showed electrons would need to rotate faster than light, which is impossible.
Current understanding: Particles are dimensionless points with an intrinsic property called spin.
Spin and Quantum Field Theory
Spin is an intrinsic property similar to mass or charge.
Origin of spin explained using geometry of space, rotations, and group theory.
Group Theory
Group: Set of operations that can be combined or reversed (e.g., quarter-turn rotations - 0°, 90°, 180°, 270° forms a group Z4).
Symmetry of Objects: An object remains unchanged under certain operations.
Z4: Group with four rotations.
Other examples: Z12 (hours on a clock), Z^2 (checkerboard displacements), SO(2) (2D rotations, 0° to 360°), SO(3,1) (SpaceTime rotations).
Representation of Particles using Groups
Gemstones Analogy: Different gemstones transform differently under rotation groups.
Blue stone: No change (spin 0).
Red stone: Changes and has four unique states (spin 1).
Green stone: Changes after first rotation but returns to initial state after the second (spin 2).
State Space: Abstract space where each particle state is plotted.
Rotation in physical space corresponds to rotation in state space.
Spin Number
Describes how rotations in space affect particle states in state space.
Spin 0: State does not change (e.g., Higgs boson).
Spin 1: State changes per rotation (e.g., photon - vector field).
Spin 2: Returns to state after 180° rotation (e.g., hypothetical graviton - rank 2 tensor field).
Spin 1/2: Requires two full turns to return to initial state (e.g., electrons).
Quantum Mechanics: Allows particles with half-integer spins (superposition principle).
Quantum Mechanics and Superposition
Superposition Principle: A particle can be in a superposition state (e.g., a coin in heads and tails simultaneously until observed).
State Space for Quantum Particles: Heads and tails act as axes; superpositions are sums (e.g., heads + tails).
Equivalence of Opposite States: Physically indistinguishable but mathematically different.
Spin 1/2 Particles (Fermions)
Allowed in quantum mechanics due to superposition and equivalence of opposite states in measurements.
Spinners: Mathematical tool for modeling spin 1/2 particles.
Mathematical Modeling of Spin
Spin 0: Modeled as a point/number.
Spin 1: Modeled by a vector.
Spin 2: Modeled by a rank 2 tensor.
Spin 1/2: Modeled by a spinner.
Electron Spin and Experiments
Electrons (spin 1/2) modeled by a field of spinners.
Spin states (up/down) cause deflection in experiments (e.g., Stern-Gerlach).
Spin number characterizes the angular momentum-like behavior.
Quantum Mechanics: Uses complex numbers for quantum states.
Summary
Spin Number: Intrinsic property describing particle behavior under rotation.
Integer spins: 0, 1, 2.
Half-integer spins: 1/2 (e.g., electrons).
Measurements: Collapse particle state to one of the axes in abstract space.
Dirac Equation (1928): Added time dimension, predicting particles and antiparticles.
Antimatter: Mathematically described as opposite types of spinners.
Key Takeaways
Spin is a fundamental property described using quantum mechanics and group theory.
Quantum particles exhibit unique behaviors like superposition and spin 1/2.
Mathematical tools (spinners, vectors, tensors) help model these properties and predict experimental outcomes.