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group_theory [2017/04/09 08:32]
jakobadmin [Notions]
group_theory [2017/12/06 09:33] (current)
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 </​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://​​abs/​hep-th/​9602122|The Unreasonable Effectiveness of Quantum Field Theory]] by R. Jackiw</​cite>​
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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>​ 
 ==== Matrix Groups and Matrix Algebras ==== ==== Matrix Groups and Matrix Algebras ====
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