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Phenomenology of a Noncommutative Spacetime

Phenomenology of a Noncommutative Spacetime. Xavier Calmet University of Brussels (ULB). Outline. Why do we believe in a minimal length Motivations and goals Local gauge symmetries on noncommutative spaces Bounds on space-time noncommutativity

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Phenomenology of a Noncommutative Spacetime

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  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 Claim:GR and QM imply that no operational procedure exists which can measure a distance less than the Planck length. 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. 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.

  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:

  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. 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.

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

  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. 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: 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. 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

  32. 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! How to bound these models?

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

  34. 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NCfrom LEP is around 143 GeV.

  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NCfrom LEP is around 250 GeV • From low energy experiments: • Bounds on • implyNCTeVfrom 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.

  36. “Quantum Level” Bound • One loop operator generated in NCQCD: (Carlson et al. hep-ph/0107291), 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:

  37. 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.

  38. 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:

  39. Two maps lead to the following action:

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

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

  42. 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.

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

  44. 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:

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

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

  47. 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.

  48. 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!

  49. 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.

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