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group_theory

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group_theory [2017/04/09 08:32] jakobadmin [Notions] |
group_theory [2017/06/22 08:32] jakobadmin [Quotes] |
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</blockquote> | </blockquote> | ||

+ | <blockquote> | ||

+ | Physicists are mostly agreed that the ultimate laws of Nature enjoy a high degree of | ||

+ | symmetry. By this I mean that the formulation of these laws, be it in mathematical | ||

+ | terms or perhaps in other accurate descriptions, is unchanged when various transformations are performed. Presence of symmetry implies absence of complicated and | ||

+ | irrelevant structure, and our conviction that this is fundamentally true reflects an | ||

+ | ancient aesthetic prejudice - physicists are happy in the belief that Nature in its | ||

+ | fundamental workings is essentially simple. Moreover, there are practical consequences of the simplicity entailed by symmetry: it is easier to understand the predictions of physical laws. For example, working out the details of very-many-body motion is beyond the reach of actual calculations, even with the help of computers. | ||

+ | But taking into account the symmetries that are present allows one to understand | ||

+ | at least some aspects of the motion, and to chart regularities within it. | ||

+ | <cite>[[https://arxiv.org/abs/hep-th/9602122|The Unreasonable Effectiveness of Quantum Field Theory]] by R. Jackiw</cite> | ||

+ | </blockquote> | ||

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</blockquote> | </blockquote> | ||

- | + | <blockquote> | |

+ | Rovelli: [...] . I | ||

+ | learned from your book that the world in which we happen to live is not Poincare | ||

+ | invariant, and is not described by Poincare invariant theory. There is no sense in | ||

+ | which general relativity is Poincare invariant. (If it were, what would be the Poincare transform of the closed Friedman-Robertson-Walker solution of the Einstein | ||

+ | equation?) Thus, Poincare invariance is neither a symmetry of our universe, nor a | ||

+ | symmetry of the laws governing our universe. Don't you find it a bit disturbing | ||

+ | basing the foundation of our understanding of the world on a symmetry which is | ||

+ | not a symmetry of the world nor of its laws? | ||

+ | | ||

+ | Weinberg: Well, I think there's always been a distinction that we have to make | ||

+ | between the symmetries of laws and the symmetries of things. You look at a | ||

+ | chair; it's not rotationally invariant. Do you conclude that there's something | ||

+ | wrong with rotation invariance? Actually, it's fairly subtle why the chair breaks | ||

+ | rotational invariance: it's because the chair is big. In fact an isolated chair in its | ||

+ | ground state in empty space, without any external perturbations, will not violate | ||

+ | rotational invariance. It will be spinning in a state with zero rotational quantum | ||

+ | numbers, and be rotationally invariant. But because it's big, the states of different | ||

+ | angular momentum of the chair are incredibly close together (since the rotational | ||

+ | energy differences go inversely with the moment of inertia), so that any tiny pertur- | ||

+ | bation will make the chair line up in a certain direction. That's why chairs break | ||

+ | rotational invariance. That's why the universe breaks symmetries like chiral invar- | ||

+ | iance; it is very big, even bigger than a chair. This doesn't seem to me to be relevant | ||

+ | to what we take as our fundamental principles. You can still talk about Lorentz | ||

+ | invariance as a fundamental law of nature and live in a Lorentz non-invariant | ||

+ | universe, and in fact sit on a Lorentz non-invariant chair, as you are doing. | ||

+ | [Added note: Lorentz invariance is incorporated in general relativity, as the | ||

+ | holonomy group, or in other words, the symmetry group in locally inertial frames.] | ||

+ | | ||

+ | <cite>Conceptual Foundations of Quantum Field Theory edited by Cao</cite> | ||

+ | </blockquote> | ||

==== Matrix Groups and Matrix Algebras ==== | ==== Matrix Groups and Matrix Algebras ==== | ||

group_theory.txt · Last modified: 2017/12/06 09:33 (external edit)