theories
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theories [2018/05/06 13:07] jakobadmin |
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| | | ,| -| AA |-|.|AA=[[theories:speculative_theories:quantum_gravity|Quantum Gravity]] | | | ,| -| AA |-|.|AA=[[theories:speculative_theories:quantum_gravity|Quantum Gravity]] | ||
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| - | | BB | ||| CC | | |BB=[[theories:quantum_field_theory|Quantum Field Theory]]|CC=[[theories:general_relativity|General Relativity]] | + | | BB | ||| CC | | |BB=[[theories:quantum_field_theory|Quantum Field Theory]]|CC=[[models:general_relativity|General Relativity]] |
| | |!@4| | | ||| !@4 | | | | |!@4| | | ||| !@4 | | | ||
| - | | DD| ||| EE | | |DD=[[theories:quantum_mechanics|Quantum Mechanics]]|EE=[[theories:special_relativity|Special Relativity]] | + | | DD| ||| EE | | |DD=[[theories:quantum_mechanics|Quantum Mechanics]]|EE=[[models:special_relativity|Special Relativity]] |
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| | | || ||| FF | | |FF=[[theories:classical_mechanics|Classical Mechanics]] | | | || ||| FF | | |FF=[[theories:classical_mechanics|Classical Mechanics]] | ||
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| [[theories:classical_mechanics|Classical mechanics]] describes everyday physics and can be described by using everyday space, configuration space, phase space or Hilbert space((See the discussion at [[theories:classical_mechanics|classical mechanics]])). Classical mechanics works perfectly for phenomena like billiard-ball collisions. However, we can ask what happens if we make the billiard balls smaller and smaller? What if we scale the whole system down by a factor 10, a factor 1000, or even a factor 10000? Will the laws of classical mechanics still hold? | [[theories:classical_mechanics|Classical mechanics]] describes everyday physics and can be described by using everyday space, configuration space, phase space or Hilbert space((See the discussion at [[theories:classical_mechanics|classical mechanics]])). Classical mechanics works perfectly for phenomena like billiard-ball collisions. However, we can ask what happens if we make the billiard balls smaller and smaller? What if we scale the whole system down by a factor 10, a factor 1000, or even a factor 10000? Will the laws of classical mechanics still hold? | ||
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| + | {{ :theoriesoverviewbyscale.png?nolink&400|}} | ||
| In principle this could be since the $m$ in [[equations:newtons_second_law|Newton's second law]] does not come with a predefined allowed mass range, say 0.1 kg up to 10000 kg. But the correct answer is no. If we make things smaller and smaller we eventually end up in a regime where the laws of classical mechanics no longer apply. The scale where this happens is encoded in Planck's constant $\hbar$. If we are dealing with systems where the [[formalisms:lagrangian_formalism|action]] is much larger than $\hbar$ we can treat it using classical mechanics. Effectively this means that we set $\hbar$ to zero. If the action of the system is close to $\hbar$ we need to take quantum effects into account. | In principle this could be since the $m$ in [[equations:newtons_second_law|Newton's second law]] does not come with a predefined allowed mass range, say 0.1 kg up to 10000 kg. But the correct answer is no. If we make things smaller and smaller we eventually end up in a regime where the laws of classical mechanics no longer apply. The scale where this happens is encoded in Planck's constant $\hbar$. If we are dealing with systems where the [[formalisms:lagrangian_formalism|action]] is much larger than $\hbar$ we can treat it using classical mechanics. Effectively this means that we set $\hbar$ to zero. If the action of the system is close to $\hbar$ we need to take quantum effects into account. | ||
| - | {{ :paper.journal.6.png?nolink&500|}} | + | |
| The crucial difference between classical and quantum mechanics is that the algebra for observables is non-commutative, as encoded in [[advanced_notions:uncertainty_principle|Heisenberg's uncertainty principle]]((In words the non-commutativity, it makes a tiny difference ($sim \hbar$) whether we first measure the momentum or the location of a particle)). Apart from this difference, we can again use everyday space, configuration space, phase space or Hilbert space((See the discussion at [[theories:quantum_mechanics|quantum mechanics]])). The question why we need to change our "naive" algebra to a non-commutative one is the what discussions about the [[https://physicstravelguide.com/theories/quantum_mechanics#tab__interpretations|interpretation of quantum mechanics are all about]]. The procedure where we start with a classical theory an construct the corresponding quantum theory is known as [[advanced_tools:quantization|quantization]]. | The crucial difference between classical and quantum mechanics is that the algebra for observables is non-commutative, as encoded in [[advanced_notions:uncertainty_principle|Heisenberg's uncertainty principle]]((In words the non-commutativity, it makes a tiny difference ($sim \hbar$) whether we first measure the momentum or the location of a particle)). Apart from this difference, we can again use everyday space, configuration space, phase space or Hilbert space((See the discussion at [[theories:quantum_mechanics|quantum mechanics]])). The question why we need to change our "naive" algebra to a non-commutative one is the what discussions about the [[https://physicstravelguide.com/theories/quantum_mechanics#tab__interpretations|interpretation of quantum mechanics are all about]]. The procedure where we start with a classical theory an construct the corresponding quantum theory is known as [[advanced_tools:quantization|quantization]]. | ||
| - | Another instance where classical mechanics becomes invalid is when our objects move at speeds close to the speed of light $c$. In such systems, the correct theory is [[theories:special_relativity|special relativity]]. The difference between special relativity and classical mechanics is that we now use Minkowski space instead of our everyday Euclidean space. Again we can equivalently use the corresponding configuration space, phase space or a Hilbert space. When the objects in our system move so slowly that we can treat effectively $c$ as infinity, we can use classical mechanics. | ||
| - | 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]]. | ||
| - | 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 [[theories: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. | + | Another instance where classical mechanics becomes invalid is when our objects move at speeds close to the speed of light $c$. In such systems, the correct theory is [[models:special_relativity|special relativity]]. The difference between special relativity and classical mechanics is that we now use Minkowski space instead of our everyday Euclidean space. Again we can equivalently use the corresponding configuration space, phase space or a Hilbert space. When the objects in our system move so slowly that we can treat effectively $c$ as infinity, we can use classical mechanics. |
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| + | {{:paper.journal.6.png?nolink&400 |}} | ||
| + | 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]]. | ||
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| + | 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. | ||
| So far, there is no theory that works for systems with strong gravity and tiny objects that move close to the speed of light. The //hypothetical// theory that combines the lessons of general relativity and quantum field theory is known as [[theories:speculative_theories:quantum_gravity|quantum gravity]]. | So far, there is no theory that works for systems with strong gravity and tiny objects that move close to the speed of light. The //hypothetical// theory that combines the lessons of general relativity and quantum field theory is known as [[theories:speculative_theories:quantum_gravity|quantum gravity]]. | ||
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| - | + | <tabbox Field Theories vs. Particle Theories> | |
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| - | <tabbox Field Theories vs. Particle Theories# | + | |
| In a particle theory, the solutions describe __particle trajectories__: | In a particle theory, the solutions describe __particle trajectories__: | ||
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