1. Phenomenology of a Noncommutative Spacetime
Xavier Calmet
University of Brussels (ULB)

2. Outline Why do we believe in a minimal length Motivations and goals
Local gauge symmetries on noncommutative spaces
Bounds on space-time noncommutativity
Space-Time symmetries of noncommutative spaces
Gravity on noncommutative spaces
Conclusions

3. Why do we believe in a minimal length?

4. A minimal length from QM and GR
Assumptions:
Hoop Conjecture (GR): if an amount of energy E is confined to a ball of size R, where R < E, then that region will eventually evolve into a black hole.
Quantum Mechanics: uncertainty relation.
Claim: GR and QM imply that no operational procedure exists which can measure a distance less than the Planck length.
Minimal Ball of uncertainty:
Consider a particle of Energy E which is not already a Black hole.
Its size r must satisfy:
where 1/E is the Compton wavelength and E comes from the
Hoop Conjecture. We find:

5. Could an interferometer do better?
Our concrete model:
We assume that the position operator
has discrete eigenvalues separated by a
distance lP or smaller.
This lattice model covers quantum foam, strings etc.

6. Let us start from the standard inequality:
Suppose that the position of a test mass is measured at time t=0 and again at a later time. The position operator at a later time t is:
The commutator between the position operators at t=0 and t is
so using the standard inequality we have:
Quantization of position doesn’t imply quantization of momentum. Thus displacement operator does not necessarily have discrete eigenvalues. Since the time evolution operator is unitary the eigenvalues of x(t) are the same as x(0). However, the spectrum of x(0) (or x(t)) is completely unrelated to the spectrum of the p(0).
A measurement of arbitrarily small displacement does not exclude our model of minimum length. To exclude it requires measurements of a position eigenvalue x and a nearby eigenvalue x', with |x - x'| << lP.

7. At least one of the uncertainties x(0) or x(t) must be larger than:
A measurement of the discreteness of x(0) requires two position measurements, so it is limited by the greater of
x(0) or x(t):
This is the bound we obtain from Quantum Mechanics.

8. However, by causality R cannot exceed t. GR and causality imply:
To avoid gravitational collapse, the size R of our measuring device must also grow such that R > M.
However, by causality R cannot exceed t.
GR and causality imply:
Combined with the QM bound, they require x > 1 in Planck units or
This derivation was not specific to an interferometer - the result is device independent: no device subject to quantum mechanics, gravity and causality can exclude the quantization of position on distances less than the Planck length.
However, by causality R cannot exceed t. Any component of the device a distance greater than t away cannot affect the measurement, hence we should not consider it part of the device.
These considerations are summarized in the inequalities:
Combined with the QM bound, they require x > 1 in Planck units or
This derivation was not specific to an interferometer - the result is device independent: no device subject to quantum mechanics, gravity and causality can exclude the quantization of position on distances less than the Planck length.

9. Motivations
Space-time noncommutativity is an extension of quantum mechanics:
Heisenberg algebra:
is extended with new noncommutative (NC) relations:
that lead to new uncertainty relations:

10. This is a nice analogy to the Heisenberg uncertainty relations.
Quantum mechanics and general relativity considered together imply the existence of a minimal length in Nature: Gauge theories with a fundamental length are thus very interesting.
A class of models with a fundamental length are gauge theories on noncommutative spaces (length ~ ).
Noncommutative coordinates appear in nature: e.g. electron in a strong B field (first Landau level can be described in terms of NC coordinates). Tools which are developed can prove useful for solid states physics.

11. Idea of a noncommutative space-time is not new
Idea of a noncommutative space-time is not new! It can be traced back to Snyder, Heisenberg, Pauli etc. At that time the motivation was that a cutoff could provide a solution to the infinities appearing in quantum field theory.
Nowadays, we know that renormalization does the job for infinites of the Standard Model, but modifying space-time at short distances will help for quantum gravity.

12. Furthermore, the Standard Model needs to be extended if it is
coupled to gravity since it is then inconsistent: noncommutative
gauge theories are a natural candidate to solve this problem.
Another motivation is string theory where these noncommutative
relations appear. But the situation is in that case very different!

13. Gauge Symmetries on Noncommutative Spaces

14. Goals
How does the Standard Model of particle physics which is a gauge theory based on the group SU(3)SU(2)U(1), emerge as a low energy action of a noncommutative gauge theory?
The main difficulty is to implement symmetries on NC spaces.
We need to understand how to implement SU(N) gauge symmetries on NC spaces.
Are there space-time symmetries (Lorentz invariance) for noncommutative spaces?

15. Symmetries and Particle Physics commutative space-time case
Impose invariance of the action under certain transformations.
Two symmetries are crucial in order to formulate the Standard Model of particle physics:
Space-time: Lorentz invariance, and combinations of C, P and T e.g.:
Local gauge symmetries
Problem on NC spaces
Problem on NC spaces

16. Enveloping algebra approach to NC
Goal: derive low energy effective actions for NC actions which are too difficult to handle.
Strategy: map NC actions to an effective action on a commutative space-time such that higher order operators describe this special property of space-time.
There is an alternative to taking fields in the Lie algebra: consider fields in the enveloping algebra

17. Definitions and Gauge Transf.
def. 1: consider the algebra
algebra of noncommutative functions
def. 2: generators of the algebra: ``coordinates´´
def. 3: : elements of the algebra
infinitesimal gauge transformation:
note that the coordinates do not transform under a
gauge transformation:

18. one has:
that’s not covariant! Introduce a covariant coordinate
such that
i.e let’s set
this implies:
that’s the central result: relation between coord. gauge fields and Yang-Mills fields!
That’s not trivial: problem with direct product!

19. Star product & Weyl quantization
def: commutative algebra of functions:
aim: construct a vector space isomorphism W. Choose a way to “decompose” elements of :
(basis):

20. we need to def. the product (noncommutative multiplication) in :
Weyl quantization procedure:
Let us use the Campbell-Baker-Hausdorff formula:
We then have:

21. to leading order:
We now have the first map:
we know how to replace the argument of the functions, i.e. the NC coordinates by usual coordinates: price to pay is the star product. This is done using the isomorphism
The second map will map the function , this second map (Seiberg-Witten map) is linked to gauge invariance, more later.

22. Let us start from the relations:
Field Theory
Let us start from the relations:
the Yang-Mills gauge potential is defined as
has the usual transformation property:
The covariant coordinate leads to the Yang-Mills potential!

23. Local gauge theories on NC spaces
Let be Lie-algebra valued gauge transformations, the commutator:
is a gauge transformation only for U(N) gauge transformations in the (anti)fundamental or adjoint representation.
Problem: Standard Model requires SU(N)!
BUT, it can close for all groups if we take the fields and gauge transformations to be in the enveloping algebra:
Is there an infinite number of degrees of freedom?
No! They can be reduced using Seiberg-Witten maps!

24. Consistency condition and Seiberg-Witten map
Replace the noncommutative variable by a commutative one. Price to pay is the introduction of the star product:
Let us consider the commutator once again:

25. Let us now assume that are in the enveloping algebra:
one finds
in 0th order in  and
in the leading order in .

26. Previous partial differential equation is solved by:
Expanding the star product and the fields via the SW maps in the leading order in theta, one finds:
Action is SU(N)
invariant!

27. SM on NC Space-Time Problems: Solutions: a) direct product of groups
b) charge quantization
c) Yukawa couplings
d) “Trace” in the enveloping algebra
Solutions:
One can’t introduce 3 NC gauge potentials:
must remain covariant!
solution: introduce a master field:
SW map for
Note that

28. b) Charge quantization problem:
solution to charge quantization: introduce n NC photons:
Too many degrees of freedom?
No  Seiberg-Witten map!
there is only one classical photon!

29. Complication for Yukawa couplings:
c) Yukawa couplings: left/right makes a difference!
Complication for Yukawa couplings:
is not NC gauge invariant if transforms only on the r.h.s. or l.h.s.
Solution:
Hybrid SW map:
with

30. d) trace for the gauge part of the action:
is a huge matrix. There is not a unique way to fix the trace, gauge inv. only requires:
Minimal model:

31. Other choice

33. How to bound these models?
Tree level
Tree level + test of Lorentz inv.
Quantum level
Quantum level + test of Lorentz inv.
Rigorous but low scale
not a direct test: Warning!
High energy scale accessible but
It’s maybe not yet clear how to build
a quantum theory: Warning!
High energy accessible but
not yet clear how to regularize this theory and not a direct test: Warning!

34. What is θ?
So in principle we have 6 scales!

35. Bounds on NC scale From colliders:
Lots of corrections to SM processes, but large background:
search for rare decays.
Smoking gun for NC: Z-->   or Z--> g g.
Limit on NC from LEP is around 143 GeV.

36. Bounds on NC scale From colliders: From low energy experiments:
Lots of corrections to SM processes, but large background:
search for rare decays.
Smoking gun for NC: Z-->   or Z--> g g.
Limit on NC from LEP is around 250 GeV
From low energy experiments:
Bounds on
imply NC TeV from atomic clock comparison (Be9).
Note that the bound comes from Lorentz violation, and is thus not a
“direct” test of the noncommutative nature of space-time.

37. “Quantum Level” Bound One loop operator generated in NCQCD:
(Carlson et al. hep-ph/ ), but there is a problem with that paper: operator giving the bound
is actually vanishing.
They considered the one loop correction to the quark mass and wavefunction renormalization and performed their calculation using Pauli-Villars regularization:

38. They considered 3 operators separately
Bound from first operator: this is wrong!!! Let us look again at the matrix element:
Using the Dirac equation it is obviously vanishing. Quarks are onshell at this order in perturbation theory.

39. Quantum Mechanics and EDM
There are claims in the literature that EDMs can put very tight bounds on the scale for spacetime noncommutativity.
A formulation of Quantum Mechanics on a NC spacetime is needed to address this question.
Let us start from the QED action on a NC spacetime:

40. Two maps lead to the following action:

41. And the Dirac eq. easily follows:
Let us now prepare the non-relativistic expansion:
And we

42. From this it is easy to obtain the low energy Hamiltonian:
3 operators are CP violating:

43. Let us look at one of them:
However it is not of the shape: i.e. there is no spin flip!
Experiments searching for an EDM are not sensitive to this operator: there is no bound!
These experiments measure the energy difference between a two-levels system. Here the effect cancels out.

44. Space-time symmetries of NC spaces
Consider NC:
Furthermore, one has the Heisenberg algebra:
Let us now do a variable transformation:
It leads to the following algebra:
Breaks Lorentz
invariance!

45. Let us consider transformations of the commuting coordinates:
one also has
The invariant length is given by:
It is invariant if
We can now implement this transformation for the NC coordinates:

46. The invariant length is given by:
the derivative is given by:
it transforms as
under a noncommutative Lorentz transformation.

47. The NC Yang-Mills potential transforms as:
and the covariant derivative as:
The field strength transforms as:
and a spinor as:

48. This represents an extension of special relativity
This represents an extension of special relativity. The limit   0 is well defined: one recovers the usual Lorentz invariance. Note: we do not deform the Poincaré algebra!
It is easy to verify that the actions discussed previously are indeed invariant under these transformations.
This symmetry is important because bounds on space-time noncommutativity come from bounds on Lorentz violation (atomic clocks). The bounds will be affected.
Any operator derived from loop calculations must be invariant under this symmetry: beware of artifacts of regularization procedure.

49. Is microcausality violated?
Let us look at the light cone of a photon on a NC spacetime:
which is not !
Let us now compute (at equal time) the expectation value of the commutator between and as done by Greenberg. It is proportional to
He concludes that microcausality is violated. However this precisely corresponds to our light cone: microcausality is not violated!

50. Quantization of Noncommutative QED
Misuse of the term effective theory: mapped theory?
Seiberg-Witten expansion is an expansion in .
If one expands in and then quantize the theory (expansion in terms of one can miss important resummation effects.
This is indeed the case because of the vertices phases as we shall see.
Let us start from the unexpanded action:
Fields are representation of the Lorentz group: quantize the fields which are in the enveloping algebra.

51. Add Faddeev-Popov terms
Feynman rules are then given by:

52. It is then straightforward to compute the beta function
And the renormalized vertex :
The quantized and renormalized action can then be mapped on a commutative spacetime.
The vertex correction is given by

53. Gravity on Noncommutative Spaces

54. Gravity on noncommutative spaces
Hypothesis:  is a constant of nature and it has the same value in every coordinate frame.
Well if that is the situation, what are the coordinate transformations allowed by the NC algebra:
Let us consider the transformations:
and study the NC algebra:
It is invariant iff
The solutions are:
They form a subgroup of 4-Volume preserving coord. transformations.

55. Consider the enveloping algebra:
We now want to implement this symmetry for a NC gravity action. We consider iso(3,1) but restrict ourselves to the coord. transformations that preserve 
Consider the enveloping algebra:
Consistency condition:

56. Differential equations: 0th order in  :
1st order in  :
Solution:
Now for the spin connection:
One thus has:

57. For the field strength one has:
classically one has:
Note that our covariant derivative is torsion free.
Field strength for the local Lorentz symmetry:
Noncommutative Riemann tensor:

58. we can then define a noncommutative Ricci tensor:
and a noncommutative Ricci scalar:
It is easy to see that the leading order correction is vanishing:
with which vanishes!
Second order corrections have to be calculated.
However the result is quite complicated.
It implies a that we need to know solutions to the consistency conditions to second order in theta.
OK get ready for the result:

60. Action:
Equations of motion
Can be massaged into:
Remarkable: on a canonical NC spacetime: the cosmological constant is an integration constant uncorrelated to parameters of the action!

61. Some open questions
Understand how to formulate gravity on NC spaces (non constant ).
CMB signatures of NC physics?
Do NC black holes have singularities? What about horizon? Hawking radiation? Information loss?
Is quantum gravity formulated on NC spaces renormalizable?
Fine tuning issues and NC physics. Certain short distance modifications of space-time can modify the high energy behavior of loops.
New ideas to break gauge symmetries: after all lots of ideas come from solid state physics and we have quite a few models in solid state physics that are described by NC gauge theories. This will lead to new phenomenology for the LHC.

62. Conclusions
Noncommutative gauge theories are examples of non-local theories with a minimal length.
SU(N) gauge symmetries, which are crucial for the Standard Model, can be implemented on noncommutative spaces.
Noncommutative Lorentz transformations can be introduced.
Constraints are not very severe! Bounds of the order of few TeV only.
Lots of open issues e.g. Loops, general relativity: work in progress!
Applications to solid state physics, cosmology etc.
Even if there is a grand desert, Planck scale physics might be accessible with low energy experiments.
Exciting field in development.