Quantum spacetime

Last updated

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. [1] Snyder's Lie algebra was made simple by C. N. Yang in the same year.


Physical spacetime is a quantum spacetime when in quantum mechanics position and momentum variables 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:

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 [3] and has Lie algebra relations

for the spatial variables and the time variable . Here has dimensions of time and is therefore expected to be something like the Planck time. The Poincaré group here is correspondingly deformed, now to a certain bicrossproduct quantum group with the following characteristic features.

Orbits for the action of the Lorentz group on momentum space in the construction of the bicrossproduct model in units of
{\displaystyle \lambda ^{-1}}
. Mass-shell hyperboloids are 'squashed' into a cylinder. 2dhyp.png
Orbits for the action of the Lorentz group on momentum space in the construction of the bicrossproduct model in units of . Mass-shell hyperboloids are 'squashed' into a cylinder.

The momentum generators commute among themselves but addition of momenta, reflected in the quantum group structure, is deformed (momentum space becomes a non-abelian group). Meanwhile, the Lorentz group generators enjoy their usual relations among themselves but act non-linearly on the momentum space. The orbits for this action are depicted in the figure as a cross-section of against one of the . The on-shell region describing particles in the upper center of the image would normally be hyperboloids but these are now 'squashed' into the cylinder

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. [4] 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 to be the physical energy and spatial momentum (as opposed to some other function of them). Arguments for this identification were provided in 1999 by Giovanni Amelino-Camelia and Majid [5] through a study of plane waves for a quantum differential calculus in the model. They take the form

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 -Poincaré quantum group introduced by Jurek Lukierski and co-workers [6] which should be viewed as an important precursor to the bicrossproduct one, albeit without the actual quantum spacetime and with different proposed generators for which the above picture does not apply. The bicrossproduct model spacetime has also been called -deformed spacetime with .

q-Deformed spacetime

This model was introduced independently by a team [7] working under Julius Wess in 1990 and by Shahn Majid and coworkers in a series of papers on braided matrices starting a year later. [8] 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 as a 'braided hermitian matrix' of generators and relations

These relations say that the generators commute as thereby recovering usual Minkowski space. One can work with more familiar variables as linear combinations of these. In particular, time

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 returns us the usual Minkowski distance (this translates to a metric in the quantum differential geometry). The parameter or is dimensionless and is thought to be a ratio of the Planck scale and the cosmological length. That is, there are indications that this model relates to quantum gravity with non-zero cosmological constant, the choice of depending on whether this is positive or negative. We have described the mathematically better understood but perhaps less physically justified positive case here.

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 -deformed Poincaré group as generated by such translations and -Lorentz transformations, completing the interpretation as a quantum spacetime. [9]

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.[ how? ]

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 [10] 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 [11]

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 generate the Lorentz group. This quantum spacetime and that of C. N. Yang entail a radical unification of spacetime, energy-momentum, and angular momentum.

The idea was revived in a modern context by Sergio Doplicher, Klaus Fredenhagen and John Roberts in 1995 [12] by letting simply be viewed as some function of as defined by the above relation, and any relations involving it viewed as higher order relations among the . The Lorentz symmetry is arranged so as to transform the indices as usual and without being deformed.

An even simpler variant of this model is to let here be a numerical antisymmetric tensor, in which context it is usually denoted , so the relations are . In even dimensions , any nondegenerate such theta can be transformed to a normal form in which this really is just the Heisenberg algebra but the difference that the variables are being proposed as those of spacetime. This proposal was for a time quite popular because of its familiar form of relations and because it has been argued [13] that it emerges from the theory of open strings landing on D-branes, see noncommutative quantum field theory and Moyal plane. However, this D-brane lives in some of the higher spacetime dimensions in the theory and hence it is not our physical spacetime that string theory suggests to be effectively quantum in this way. You also have to subscribe to D-branes as an approach to quantum gravity in the first place. Even when posited as quantum spacetime it is hard to obtain physical predictions and one reason for this is that if is a tensor then by dimensional analysis it should have dimensions of length, and if this length is speculated to be the Planck length then the effects would be even harder to ever detect than for other models.

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 [14] but has been developed considerably since then. Such an approach can theoretically be combined with quantum spacetime as above.

See also

Related Research Articles

Kaluza–Klein theory Unified field theory

In physics, Kaluza–Klein theory is a classical unified field theory of gravitation and electromagnetism built around the idea of a fifth dimension beyond the common 4D of space and time and considered an important precursor to string theory. Gunnar Nordström had an earlier, similar idea. But in that case, a fifth component was added to the electromagnetic vector potential, representing the Newtonian gravitational potential, and writing the Maxwell equations in five dimensions.

Quantum field theory Theoretical framework combining classical field theory, special relativity, and quantum mechanics

In theoretical physics, quantum field theory (QFT) is a theoretical framework that combines classical field theory, special relativity and quantum mechanics. QFT is used in particle physics to construct physical models of subatomic particles and in condensed matter physics to construct models of quasiparticles.

Special relativity Theory of interwoven space and time by Albert Einstein

In physics, the special theory of relativity, or special relativity for short, is a scientific theory regarding the relationship between space and time. In Albert Einstein's original treatment, the theory is based on two postulates:

  1. The laws of physics are invariant in all inertial frames of reference.
  2. The speed of light in vacuum is the same for all observers, regardless of the motion of the light source or observer.
Dirac equation Relativistic quantum mechanical wave equation

In particle physics, the Dirac equation is a relativistic wave equation derived by British physicist Paul Dirac in 1928. In its free form, or including electromagnetic interactions, it describes all spin-12 massive particles such as electrons and quarks for which parity is a symmetry. It is consistent with both the principles of quantum mechanics and the theory of special relativity, and was the first theory to account fully for special relativity in the context of quantum mechanics. It was validated by accounting for the fine details of the hydrogen spectrum in a completely rigorous way.

Loop quantum gravity Theory of quantum gravity, merging quantum mechanics and general relativity

Loop quantum gravity (LQG) is a theory of quantum gravity, which aims to merge quantum mechanics and general relativity, incorporating matter of the Standard Model into the framework established for the pure quantum gravity case. It is an attempt to develop a quantum theory of gravity based directly on Einstein's geometric formulation rather than the treatment of gravity as a force.

Special unitary group

In mathematics, the special unitary group of degree n, denoted SU(n), is the Lie group of n × n unitary matrices with determinant 1.

Poincaré group

The Poincaré group, named after Henri Poincaré (1906), was first defined by Hermann Minkowski (1908) as the group of Minkowski spacetime isometries. It is a ten-dimensional non-abelian Lie group, which is of importance as a model in our understanding the most basic fundamentals of physics.

Lorentz group Lie group of Lorentz transformations

In physics and mathematics, the Lorentz group is the group of all Lorentz transformations of Minkowski spacetime, the classical and quantum setting for all (non-gravitational) physical phenomena. The Lorentz group is named for the Dutch physicist Hendrik Lorentz.

In mathematics and physics, n-dimensional anti-de Sitter space (AdSn) is a maximally symmetric Lorentzian manifold with constant negative scalar curvature. Anti-de Sitter space and de Sitter space are named after Willem de Sitter (1872–1934), professor of astronomy at Leiden University and director of the Leiden Observatory. Willem de Sitter and Albert Einstein worked together closely in Leiden in the 1920s on the spacetime structure of the universe.

Quantum group Algebraic construct of interest in theoretical physics

In mathematics and theoretical physics, the term quantum group denotes one of a few different kinds of noncommutative algebras with additional structure. These include Drinfeld–Jimbo type quantum groups, compact matrix quantum groups, and bicrossproduct quantum groups.

Einstein field equations Field equations in general relativity

In the general theory of relativity the Einstein field equations relate the geometry of spacetime to the distribution of matter within it.

In mathematical physics, noncommutative quantum field theory is an application of noncommutative mathematics to the spacetime of quantum field theory that is an outgrowth of noncommutative geometry and index theory in which the coordinate functions are noncommutative. One commonly studied version of such theories has the "canonical" commutation relation:

Representation theory of the Lorentz group Representation of the symmetry group of spacetime in special relativity

The Lorentz group is a Lie group of symmetries of the spacetime of special relativity. This group can be realized as a collection of matrices, linear transformations, or unitary operators on some Hilbert space; it has a variety of representations. This group is significant because special relativity together with quantum mechanics are the two physical theories that are most thoroughly established, and the conjunction of these two theories is the study of the infinite-dimensional unitary representations of the Lorentz group. These have both historical importance in mainstream physics, as well as connections to more speculative present-day theories.

In relativity, rapidity is commonly used as a measure for relativistic velocity. Mathematically, rapidity can be defined as the hyperbolic angle that differentiates two frames of reference in relative motion, each frame being associated with distance and time coordinates.

In quantum geometry or noncommutative geometry a quantum differential calculus or noncommutative differential structure on an algebra over a field means the specification of a space of differential forms over the algebra. The algebra here is regarded as a coordinate ring but it is important that it may be noncommutative and hence not an actual algebra of coordinate functions on any actual space, so this represents a point of view replacing the specification of a differentiable structure for an actual space. In ordinary differential geometry one can multiply differential 1-forms by functions from the left and from the right, and there exists an exterior derivative. Correspondingly, a first order quantum differential calculus means at least the following:

Shahn Majid

Shahn Majid is an English pure mathematician and theoretical physicist, trained at Cambridge University and Harvard University and, since 2001, a Professor of Mathematics at the School of Mathematical Sciences, Queen Mary, University of London.

In physics, Liouville field theory is a two-dimensional conformal field theory whose classical equation of motion is a generalization of Liouville's equation.

Bargmann–Wigner equations

In relativistic quantum mechanics and quantum field theory, the Bargmann–Wigner equations describe free particles of arbitrary spin j, an integer for bosons or half-integer for fermions. The solutions to the equations are wavefunctions, mathematically in the form of multi-component spinor fields.

Symmetry in quantum mechanics Properties underlying modern physics

Symmetries in quantum mechanics describe features of spacetime and particles which are unchanged under some transformation, in the context of quantum mechanics, relativistic quantum mechanics and quantum field theory, and with applications in the mathematical formulation of the standard model and condensed matter physics. In general, symmetry in physics, invariance, and conservation laws, are fundamentally important constraints for formulating physical theories and models. In practice, they are powerful methods for solving problems and predicting what can happen. While conservation laws do not always give the answer to the problem directly, they form the correct constraints and the first steps to solving a multitude of problems.

Dual graviton

In theoretical physics, the dual graviton is a hypothetical elementary particle that is a dual of the graviton under electric-magnetic duality, as an S-duality, predicted by some formulations of supergravity in eleven dimensions.


  1. Snyder, H. (1947), "Quantized space-time", Physical Review, 67 (1): 38–41, Bibcode:1947PhRv...71...38S, doi:10.1103/PhysRev.71.38
  2. Yang, I. E. Segal 1947
  3. Majid, S.; Ruegg, H. (1994), "Bicrossproduct structure of the -Poincaré group and noncommutative geometry", Physics Letters B, 334 (3–4): 348–354, arXiv: hep-th/9405107 , Bibcode:1994PhLB..334..348M, doi:10.1016/0370-2693(94)90699-8
  4. Majid, Shahn (1988), "Hopf algebras for physics at the Planck scale", Classical and Quantum Gravity, 5 (12): 1587–1607, Bibcode:1988CQGra...5.1587M, CiteSeerX , doi:10.1088/0264-9381/5/12/010
  5. Amelino-Camelia, G.; Majid, S. (2000), "Waves on noncommutative spacetime and gamma-ray bursts", International Journal of Modern Physics A, 15 (27): 4301–4323, arXiv: hep-th/9907110 , Bibcode:2000IJMPA..15.4301A, doi:10.1142/s0217751x00002779
  6. Lukierski, J; Nowicki, A; Ruegg, H; Tolstoy, V.N. (1991), "-Deformation of Poincaré algebras", Physics Letters B, 264 (3–4): 331–338, Bibcode:1991PhLB..264..331L, doi:10.1016/0370-2693(91)90358-w
  7. Carow-Watamura, U.; Schlieker, M.; Scholl, M.; Watamura, S. (1990), "Tensor representation of the quantum group and quantum Minkowski space", Zeitschrift für Physik C, 48 (1): 159, doi:10.1007/BF01565619
  8. Majid, S. (1991), "Examples of braided groups and braided matrices", Journal of Mathematical Physics, 32 (12): 3246–3253, Bibcode:1991JMP....32.3246M, doi:10.1063/1.529485
  9. Majid, S. (1993), "Braided momentum in the q-Poincaré group", Journal of Mathematical Physics, 34 (5): 2045–2058, arXiv: hep-th/9210141 , Bibcode:1993JMP....34.2045M, doi:10.1063/1.530154
  10. 't Hooft, G. (1996), "Quantization of point particles in (2 + 1)-dimensional gravity and spacetime discreteness", Classical and Quantum Gravity, 13 (5): 1023–1039, arXiv: gr-qc/9601014 , Bibcode:1996CQGra..13.1023T, doi:10.1088/0264-9381/13/5/018
  11. Batista, E.; Majid, S. (2003), "Noncommutative geometry of angular momentum space U(su_2)", Journal of Mathematical Physics, 44 (1): 107–137, arXiv: hep-th/0205128 , Bibcode:2003JMP....44..107B, doi:10.1063/1.1517395
  12. Doplicher, S.; Fredenhagen, K.; Roberts, J.E. (1995), "The quantum structure of spacetime at the Planck scale and quantum fields", Communications in Mathematical Physics, 172 (1): 187–220, arXiv: hep-th/0303037 , Bibcode:1995CMaPh.172..187D, doi:10.1007/BF02104515
  13. Seiberg, N.; Witten, E. (1999), "String theory and noncommutative geometry", Journal of High Energy Physics, 1999 (9): 9909, 032, arXiv: hep-th/9908142 , Bibcode:1999JHEP...09..032S, doi:10.1088/1126-6708/1999/09/032
  14. Connes, A.; Lott, J. (1989), "Particle models and noncommutative geometry" (PDF), Nuclear Physics B: Proceedings Supplements, 18 (2): 29, Bibcode:1991NuPhS..18...29C, doi:10.1016/0920-5632(91)90120-4, hdl: 2027.42/29524

Further reading