Quantum spacetime | Wikipedia audio article
Channel: wikipedia tts
Date: 24/12/2018

This is an audio version of the Wikipedia Article:
https://en.wikipedia.org/wiki/Quantum_spacetime


00:04:21 1 Bicrossproduct model spacetime
00:09:14 2 iq/i-Deformed spacetime
00:14:11 3 Fuzzy or spin model spacetime
00:16:26 4 Heisenberg model spacetimes
00:19:52 5 Noncommutative extensions to spacetime
00:20:53 6 See also



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SUMMARY
=======
In mathematical physics, the concept of quantum spacetime is a generalization of the usual concept of spacetime in which some variables that ordinarily commute are assumed not to commute and form a different Lie algebra.  The choice of that algebra still varies from theory to theory.
As a result of this change some variables that are usually continuous may become discrete.
Often only such discrete variables are called "quantized"; usage varies.
The idea of quantum spacetime was proposed in the early days of quantum theory by Heisenberg and Ivanenko
as a way to eliminate infinities from quantum field theory.
The germ of the idea passed from Heisenberg to Rudolf Peierls, who noted that electrons in a magnetic field
can be regarded as moving in a quantum space-time, and to Robert Oppenheimer, who carried it
to Hartland Snyder,
who published the first concrete example.
Snyder's Lie algebra was made simple by C. N. Yang in the same year.
Physical reasons have been given to believe that physical spacetime is a quantum spacetime.
In quantum mechanics position and momentum variables 
  
    
      
        x
        ,
        p
      
    
    {\displaystyle x,p}
   are already noncommutative, obey the Heisenberg uncertainty principle, and are continuous.
Because of the Heisenberg uncertainty relations, greater energy is needed to probe smaller distances.
Ultimately, according to gravity theory, the probing particles form black holes that destroy what was to be measured. The process cannot be repeated, so it cannot be counted as a measurement.
This limited measurability led many to expect that our usual picture of continuous commutative spacetime breaks down at Planck scale distances, if not sooner.
Again, physical spacetime is expected to be quantum because physical coordinates are already slightly noncommutative.
The astronomical coordinates of a star are modified by gravitational fields between us and the star, as in the deflection of light by the sun, one of the classic tests of general relativity.
Therefore, the coordinates actually depend on gravitational field variables.
According to quantum theories of gravity these field variables do not commute;
therefore coordinates that depend on them likely do not commute.
Both arguments are based on pure gravity and quantum theory, and they limit the measurement of time
by the only time constant in pure quantum gravity, the Planck time.
Our instruments, however, are not purely gravitational but are made of particles. They may set a more severe, larger, limit than the Planck time.
Quantum spacetimes are often described mathematically using the noncommutative geometry of Connes,
quantum geometry, or quantum groups.
Any noncommutative algebra with at least four generators could be interpreted as a quantum spacetime, but
the following desiderata have been suggested:

Local Lorentz group and Poincaré group symmetries should be retained, possibly in a generalised form. Their generalisation often takes the form of a quantum group acting on the quantum spacetime algebra.
The  algebra might plausibly arise in an effective description of quantum gravity effects in some regime of that theory. For example, a physical parameter 
  
    
      
        λ
      
    
    {\displaystyle \lambda }
  , perhaps the Planck length, might control the deviation from commutative classical spacetime, so that ordinary Lorentzian spacetime arises as 
  
    
      
        λ
        →
        0
      
    
    {\displaystyle \lambda \to 0}
  .
There might be a notion of quantum differentia ...