Golden ratios in quantum mechanics?

Loren Booda
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Do golden ratios appear in quantum mechanics - such as with the standard model, string theory or quantum gravity?
 
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I don't think so, Loren--at least, not yet. However, I was very startled to see the question! Your natural tendency to think far out of the box is your talent.

In a world dominated by two fold symmetry, 1.414... to 1 is a common ratio we see occurring often. Nature seems to prefer 2:1 over any other.

0.866... is the square root of 3. We don't see it as often. Quarks seem to the the first known, fundamental 3-fold symmetry.

The golden ratio is about 5-fold symmetries. To my knowledge, there is yet to be discovered any elemental 5-fold symmetries in nature.
 
It appears in relation to the quantum dimension of the (nontrivial) anyon in the Fibonacci anyon model. Whether or not you consider this a physical example is another matter...
 
Is it true that golden ratios do not play as significant a part in the microverse as they play in the macroverse?
 

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Thread 'Lesser Green's function'

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The lesser Green's function is defined as: $$G^{<}(t,t')=i\langle C_{\nu}^{\dagger}(t')C_{\nu}(t)\rangle=i\bra{n}C_{\nu}^{\dagger}(t')C_{\nu}(t)\ket{n}$$ where ##\ket{n}## is the many particle ground state. $$G^{<}(t,t')=i\bra{n}e^{iHt'}C_{\nu}^{\dagger}(0)e^{-iHt'}e^{iHt}C_{\nu}(0)e^{-iHt}\ket{n}$$ First consider the case t <t' Define, $$\ket{\alpha}=e^{-iH(t'-t)}C_{\nu}(0)e^{-iHt}\ket{n}$$ $$\ket{\beta}=C_{\nu}(0)e^{-iHt'}\ket{n}$$ $$G^{<}(t,t')=i\bra{\beta}\ket{\alpha}$$ ##\ket{\alpha}##...
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