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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.
Contents
- Bicrossproduct model spacetime
- q Deformed spacetime
- Fuzzy or spin model spacetime
- Heisenberg model spacetimes
- Noncommutative extensions to spacetime
- References
Physical reasons have been given to believe that physical spacetime is a quantum spacetime. In quantum mechanics position and momentum variables
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:
This would permit wave equations for particles and fields and facilitate predictions for experimental deviations from classical spacetime physics that can then be tested experimentally.
Several models were found in the 1990s more or less meeting most of the above criteria.
Bicrossproduct model spacetime
The bicrossproduct model spacetime was introduced by Shahn Majid and Henri Ruegg and has Lie algebra relations
for the spatial variables
The momentum generators
in simplified units. The upshot is that Lorentz-boosting a momentum will never increase it above the Planck momentum. The existence of a highest momentum scale or lowest distance scale fits the physical picture. This squashing comes from the non-linearity of the Lorentz boost and is an endemic feature of bicrossproduct quantum groups known since their introduction in 1988. Some physicists dub the bicrossproduct model doubly special relativity, since it sets an upper limit to both speed and momentum.
Another consequence of the squashing is that the propagation of particles is deformed, even of light, leading to a variable speed of light. This prediction requires the particular
in other words a form which is sufficiently close to classical that one might plausibly believe the interpretation. At the moment such wave analysis represents the best hope to obtain physically testable predictions from the model.
Prior to this work there were a number of unsupported claims to make predictions from the model based solely on the form of the Poincaré quantum group. There were also claims based on an earlier
q-Deformed spacetime
This model was introduced independently by a team working under Julius Wess in 1990 and by Majid and coworkers in a series of papers on braided matrices starting a year later. The point of view in the second approach is that usual Minkowski spacetime has a nice description via Pauli matrices as the space of 2 x 2 hermitian matrices. In quantum group theory and using braided monoidal category methods one has a natural q-version of this defined here for real values of
These relations say that the generators commute as
is given by a natural braided trace of the matrix and commutes with the other generators (so this model has a very different flavour from the bicrossproduct one). The braided-matrix picture also leads naturally to a quantity
which as
A full understanding of this model requires (and was concurrent with the development of) a full theory of `braided linear algebra' for such spaces. The momentum space for the theory is another copy of the same algebra and there is a certain `braided addition' of momentum on it expressed as the structure of a braided Hopf algebra or quantum group in a certain braided monoidal category). This theory by 1993 had provided the corresponding
In the process it was discovered that the Poincaré group not only had to be deformed but had to be extended to include dilations of the quantum spacetime. For such a theory to be exact we would need all particles in the theory to be massless, which is consistent with experiment as masses of elementary particles are indeed vanishingly small compared to the Planck mass. If current thinking in cosmology is correct then this model is more appropriate, but it is significantly more complicated and for this reason its physical predictions have yet to be worked out.
Fuzzy or spin model spacetime
This refers in modern usage to the angular momentum algebra
familiar from quantum mechanics but interpreted in this context as coordinates of a quantum space or spacetime. These relations were proposed by Roger Penrose in his earliest spin network theory of space. It is a toy model of quantum gravity in 3 spacetime dimensions (not the physical 4) with a Euclidean (not the physical Minkowskian) signature. It was again proposed in this context by Gerardus 't Hooft. A further development including a quantum differential calculus and an action of a certain `quantum double' quantum group as deformed Euclidean group of motions was given by Majid and E. Batista
A striking feature of the noncommutative geometry here is that the smallest covariant quantum differential calculus has one dimension higher than expected, namely 4, suggesting that the above can also be viewed as the spatial part of a 4-dimensional quantum spacetime. The model should not be confused with fuzzy spheres which are finite-dimensional matrix algebras which one can think of as spheres in the spin model spacetime of fixed radius.
Heisenberg model spacetimes
The quantum spacetime of Hartland Snyder proposes that
where the
The idea was revived in a modern context by Sergio Doplicher, Claus Fredenhagen and John Roberts in 1995 by letting
An even simpler variant of this model is to let
Noncommutative extensions to spacetime
Although not quantum spacetime in the sense above, another use of noncommutative geometry is to tack on `noncommutative extra dimensions' at each point of ordinary spacetime. Instead of invisible curled up extra dimensions as in string theory, Alain Connes and coworkers have argued that the coordinate algebra of this extra part should be replaced by a finite-dimensional noncommutative algebra. For a certain reasonable choice of this algebra, its representation and extended Dirac operator, one is able to recover the Standard Model of elementary particles. In this point of view the different kinds of matter particles are manifestations of geometry in these extra noncommutative directions. Connes's first works here date from 1989 but has been developed considerably since then. Such an approach can theoretically be combined with quantum spacetime as above.