Quantum mechanics can combine with special relativity {gauge theory}.
boson
Forces have force fields and exchange bosons. Bosons are quanta. Field quanta are bosons. Gauge transformations are boson exchanges. Boson exchange carries energy and momentum quanta between fermions. Field is for relativity, and quanta are for quantum mechanics.
Higgs particles are bosons that generate masses for particles. Hadrons are bosons in multiplets for charge and isotopic spin.
groups
Conservation laws determine symmetries and gauge transformations, which form mathematical groups. Quantum electrodynamics is lepton gauge theory and uses symmetry group U(1). Quantum chromodynamics is hadron gauge theory and uses symmetry group SU(3). Electroweak theory [1973] is gauge theory for weak interactions and electromagnetism and uses symmetry group SU(2) x U(1).
Symmetry {gauge symmetry}| requires that only quantum differences are important, not absolute values.
Continuous point sets are manifolds {base space}. Manifold points can have internal spaces {fiber space}, with internal dimensions {fiber, mathematics}. Fiber spaces are manifolds. Fibers do not intersect. Fibers project to points {canonical projection}.
fiber bundles
Combined base and fiber space {fiber bundle}| {bundle} has dimension number equal to sum of fiber-space and base-space dimensions. Base space can be curve. Curve points have line tangents to curve. Tangents are fiber spaces.
Curved-surface points have planes tangent to surface. Tangent planes are fiber spaces.
vector bundle
Fiber spaces can be vector spaces {vector bundle}.
twisting
If fiber spaces are the same for all base-space points, base space and fiber space can make product space {untwisted bundle}. If fiber spaces are not all the same, base space and fiber space can make a symmetrical locally untwisted product space {twisted bundle} with a mathematical group. For example, particle spins can be fiber bundles. Base-space spins go to fiber-space phase relations.
curvature
Curvature can be connections between fibers in fiber bundles, with rule {path-lifting rule} for getting to fiber-space point from base-space point.
gauge fields
Gauge fields can be connections between fiber-bundle fibers. Bundles can have locally constant values {bundle connection}, which are like gauge connections. Connections represent field phase shifts {path lift}.
tangent bundle
Base spaces can have tangent vectors as fiber spaces {tangent bundle} or covectors as fiber spaces {cotangent bundle}. Base spaces can be two-dimensional spheres. Fiber spaces can be circles. Bundles {Hopf fibration} {Clifford bundle} can be three-dimensional spheres.
Quantum mechanics can combine with general relativity by gauge-theory extension {relativistic gauge theory}. Base field or space represents physical space-time events. Total field or space represents quantum wavefunctions or symmetry transformations. Base-space points project to total-space points to make fibers.
5-Physics-Quantum Mechanics-Theory-Quantum Relativity-Theories
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Date Modified: 2022.0225