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Snyder spacetime is generally considered one of the first models of non-commutative and quantum spacetime. There is no time operator in Quantum Mechanics. Is time in quantized Snyder space a true operator?

Qmechanic
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riemannium
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1 Answers1

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  1. One can in principle rewrite a $\color{red}{\text{non-commutative}}$ spacetime $$ \{ x^{\mu},x^{\nu}\}_{PB} ~=~\color{red}{\theta^{\mu\nu}}, \qquad \{x^{\mu},p_{\nu}\}_{PB}~=~\delta^{\mu}_{\nu}, \qquad \{p_{\mu},p_{\nu}\}_{PB}~=~0 , \qquad \mu,\nu~\in~\{0,1,2,3\}, $$ as a usual commutative spacetime $$ \{ X^{\mu},X^{\nu}\}_{PB} ~=~0, \qquad \{X^{\mu},P_{\nu}\}_{PB}~=~\delta^{\mu}_{\nu}, \qquad \{P_{\mu},P_{\nu}\}_{PB}~=~0 , \qquad \mu,\nu~\in~\{0,1,2,3\},$$ by going to Darboux coordinates. For constant $\theta^{\mu\nu}$, a possible transformation is $$ X^{\mu}~=~x^{\mu}+\frac{1}{2}\theta^{\mu\nu}p_{\nu}, \qquad P_{\mu}~=~p_{\mu}, \qquad \mu~\in~\{0,1,2,3\}. $$

  2. Because of point 1 the question

    Is time an operator/observable $\hat{X}^0$ or merely a parameter $X^0$ in quantum theories?

    remains more or less the same for commutative and $\color{red}{\text{non-commutative}}$ spacetimes, cf. Pauli's theorem and e.g. this Phys.SE post and links therein.

Qmechanic
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