quantum_gravity:concepts:quantized_spacetime

It is commonly assumed that quantum gravity sets a fundamental length scale, the Planck scale [137], which can not be resolved by any physical experiment. Different approaches to quantum gravity, such as string theory or loop quantum gravity, incorporate such a scale. This leads to the idea that some kind of “space discreteness” should be apparent even in a low-energy “effective” theory. The idea of putting quantum mechanics on a discrete lattice seems to have been first considered by Heisenberg in the spring of 1930 [130], in an attempt to remove the divergence in the electron self-energy. Because the absence of continuous spacetime symmetries leads to violations of energy and momentum conservation, this approach was not pursued further, but later in the same year Heisenberg considered modifying the commutation relations involving position operators instead [130].

See: Atoms of space and time by Lee Smolin

the proposal by Snyder [9]

$$\{x_µ,x_ν\} = -a^2 (x_µp_ν - x_νp_µ), \quad \{p_µ,p_ν\} = 0, \quad \{x_µ,p_ν\} = \eta_{\mu \nu} - a^2 p_µp_ν,$$ where a is a newly introduced length scale usually associated with the Planck scale. In Snyder’s description these quantities should be thought of as Hermitian operators acting on some Hilbert space. The physical motivation behind this is that space-time should be fundamentally “discrete”. It was shown in [9] that this can be done by identifying xµ with generators of SO(4, 1), such that all brackets involving Lorentz transformations are unchanged and Lorentz invariance is maintained.

- Quantized Space-Time by Hartland S. Snyder

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