2. Complex TransformationsG. ’t Hooft and the Gerard ’t Hooft Spinoza Institute Utrecht Utrecht University

3. Complex TransformationsG. ’t Hooft One sees here an accelerated expansion...

4. Complex TransformationsG. ’t Hooft The expanding Universe = density of matter at = spacelike curvature, usually assumed to vanish = cosmological constant

5. Complex TransformationsG. ’t Hooft

6. Complex TransformationsG. ’t Hooft The Cosmological Constant Problem stretchability very small stiffness very large

7. Complex TransformationsG. ’t Hooft The inflating Universe

8. 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

9. Complex TransformationsG. ’t Hooft Various theories claim such an effect, but the physical forces for that remain obscure. Can this happen ? Does tend to zero during inflation ???

10. Complex TransformationsG. ’t Hooft With S. Nobbenhuis, gr-qc/ 0602076 De Sitter Anti-De Sitter

11. Complex TransformationsG. ’t Hooft 1. Classical scalar field: Gravity:

12. Complex TransformationsG. ’t Hooft 2. Non-relativistic quantum particle: Example:

13. 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

14. 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

15. Complex TransformationsG. ’t Hooft 4. Second quantization: Definition of delta function: divergent !

16. Complex TransformationsG. ’t Hooft x -space hermiticity conditions: (Now, just 1 space-dimension)

17. Complex TransformationsG. ’t Hooft Then, the Hamiltonian becomes: Express the new in terms of “ = 0 by contour integration ”

18. 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

19. Complex TransformationsG. ’t Hooft 5. Difficulties Particle masses and interactions appear to violate the symmetry: Higgs potential: Perhaps there is a Higgs pair ?

20. Complex TransformationsG. ’t Hooft The pure Maxwell case can be done, but gauge fields ?? The image of the gauge group then becomes non-compact...

21. 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....