Fundamental theories of physics

Symmetries

Gerard ’t Hooft

Published Jan 1, 2016• 0 citations

Abstract

In classical, deterministic systems, we may have classical symmetries, transforming ontological states into other ontological states. In line with our general philosophy, we now extend the group of symmetries using quantum symmetries, where ontological states are compared with superimposed states. The groups of symmetry transformations obtained this way may become much larger than that of the classical symmetries. Some of the mathematical procedures, borrowed from ‘ordinary’ quantum mechanics, are explained. These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

DOI: 10.1007/978-3-319-41285-6_18

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Publisher

Abstract

DOI: 10.1007/978-3-319-41285-6_18

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u : H \rightarrow H

satisfying one and hence all of the following equivalent conditions:

uu^*=u^{*}u=1_H

; u is invertible with

u^{-1}=u^{*}

; u is a surjective isometry (or, if

\mathrm{dim}(H)<\infty

, just an isometry); u is invertible and preserves the inner product, i.e.,

\langle u \varphi , u \psi \rangle = \langle \varphi , \psi \rangle (\varphi , \psi \in H).

These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

\mathit{SU}(2)

for this model, and we explain how to take the ‘continuum limit’. These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.