theories
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| - | Next, when we are dealing with tiny objects that move at speeds close to the speed of light, both, quantum mechanics and special relativity, fail. If we combine the lessons learned in quantum mechanics and special relativity properly we end up with [[theories:quantum_field_theory|quantum field theory]]. Here the fields obey a non-commutative algebra and it is conventional to describe quantum fields either in Hilbert space or configuration space((see the discussion at [[theories:quantum_field_theory|quantum field theory]])). The Hilbert or configuration space in //interacting// quantum field theory is more complicated since the particles described by quantum fields can have an internal structure and this internal structure becomes important when fields interact((Examples, for internal structure are [[advanced_tools:gauge_symmetry|gauge symmetry]] and [[basic_notions:spin|spin]]. For example, for spin $1/2$ particles the internal space is $\mathbb{C}^2$. For [[advanced_tools:group_theory:u1|$U(1)$]] gauge symmetry we get a copy of the unit circle $S^1$ (since $U(1)\simeq S^1$) above each spacetime point.)). Hence our physics not only happens in spacetime but also in internal spaces. The theory that deals with physics in internal space is known as [[theories:gauge_theory|gauge theory]]. The appropriate geometrical tool to describe physics in spacetime and internal spaces at the same time are [[advanced_tools:fiber_bundles|fiber bundles]]. | + | Next, when we are dealing with tiny objects that move at speeds close to the speed of light, both, quantum mechanics and special relativity, fail. If we combine the lessons learned in quantum mechanics and special relativity properly we end up with [[theories:quantum_field_theory|quantum field theory]]. Here the fields obey a non-commutative algebra and it is conventional to describe quantum fields either in Hilbert space or configuration space((see the discussion at [[theories:quantum_field_theory|quantum field theory]])). The Hilbert or configuration space in //interacting// quantum field theory is more complicated since the particles described by quantum fields can have an internal structure and this internal structure becomes important when fields interact((Examples, for internal structure are [[advanced_tools:gauge_symmetry|gauge symmetry]] and [[basic_notions:spin|spin]]. For example, for spin $1/2$ particles the internal space is $\mathbb{C}^2$. For [[advanced_tools:group_theory:u1|$U(1)$]] gauge symmetry we get a copy of the unit circle $S^1$ (since $U(1)\simeq S^1$) above each spacetime point.)). Hence our physics not only happens in spacetime but also in internal spaces. The theory that deals with physics in internal space is known as [[models:gauge_theory|gauge theory]]. The appropriate geometrical tool to describe physics in spacetime and internal spaces at the same time are [[advanced_tools:fiber_bundles|fiber bundles]]. |
| A third instance where classical mechanics fails is in systems where gravity is strong((For example, in the neighborhood of large stars or [[advanced_notions:black_hole|black holes]])). In such systems, the correct theory is Einstein's [[models:general_relativity|general relativity]]. Here the Minkowski space of special relativity becomes replaced with a more general Lorentzian manifold. Hence spacetime only looks locally like Minkowksi space and is otherwise curved. | A third instance where classical mechanics fails is in systems where gravity is strong((For example, in the neighborhood of large stars or [[advanced_notions:black_hole|black holes]])). In such systems, the correct theory is Einstein's [[models:general_relativity|general relativity]]. Here the Minkowski space of special relativity becomes replaced with a more general Lorentzian manifold. Hence spacetime only looks locally like Minkowksi space and is otherwise curved. | ||
theories.txt · Last modified: 2020/04/09 20:35 by 68.142.63.195
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