1. Complex TransformationsG. t Hooft and the Gerard t Hooft Spinoza Institute Utrecht Utrecht University

2. Complex TransformationsG. t Hooft The Cosmological Constant Problem stretchability very small stiffness very large

3. Complex TransformationsG. t Hooft The inflating Universe

4. Complex TransformationsG. t Hooft During inflation, one can try to follow the effects o f on the cosmological cnstant term ++... but the subtraction term is crucial: X

5. Complex TransformationsG. t Hooft Various theories claim such an effect, but the physical forces for that remain obscure. Can thist happen ? Does tend to zero during inflation ???

6. Complex TransformationsG. t Hooft With S. Nobbenhuis, gr-qc/ 0602076 De Sitter Anti-De Sitter

7. Complex TransformationsG. t Hooft 1. Classical scalar field: Gravity:

8. Complex TransformationsG. t Hooft 2. Non-relativistic quantum particle: Example:

9. Complex TransformationsG. t Hooft There is a condition that must be obeyed: translation invariance: real or complex Note: change of hermiticity properties There can exist only one state,, obeying boundary conditions both at and

10. Complex TransformationsG. t Hooft 3. Harmonic oscillator: Change hermiticity but not the algebra: We get all negative energy states, but the vacuum is not invariant ! E 0

11. Complex TransformationsG. t Hooft 4. Second quantization: Definition of delta function: divergent !

12. Complex TransformationsG. t Hooft x -space hermiticity conditions: (Now, just 1 space-dimension)

13. Complex TransformationsG. t Hooft Then, the Hamiltonian becomes: Express the new in terms of = 0 by contour integration

14. Complex TransformationsG. t Hooft In that case: and then the vacuum is invariant ! The vacuum energy is forced to vanish because of the x ix symmetry

15. Complex TransformationsG. t Hooft 5. Difficulties Particle masses and interactions appear to violate the symmetry: Higgs potential: Perhaps there is a Higgs pair ? x x

16. Complex TransformationsG. t Hooft The pure Maxwell case can be done, but gauge fields ?? The image of the gauge group then becomes non-compact...

17. Complex TransformationsG. t Hooft The x ix symmetry might work, but only if, at some scale of physics, Yang-Mills fields become emergent... q uestions: relation with supersymmetry... ? interactions... renormalization group....