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Complex Transformations

Utrecht University. Complex Transformations. and the. Cosmological Constant Problem. Gerard ’t Hooft Spinoza Institute Utrecht. One sees here an accelerated expansion . = density of matter at = spacelike curvature, usually assumed to vanish = cosmological constant.

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Complex Transformations

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  1. Utrecht University Complex Transformations and the Cosmological Constant Problem Gerard ’t Hooft Spinoza Institute Utrecht

  2. One sees here an accelerated expansion ...

  3. = density of matter at = spacelike curvature, usually assumed to vanish = cosmological constant The expanding Universe

  4. The Cosmological Constant Problem stretchability very small stiffness very large

  5. The inflating Universe

  6. During inflation, one can try to follow the effects of quantum corrections on the cosmological cnstant term + + ... but the subtraction term is crucial: X

  7. Does tend to zero during inflation ??? Various theories claim such an effect, but the physical forces for that remain obscure. Can this happen ?

  8. Anti-De Sitter With S. Nobbenhuis, gr-qc/ 0602076 De Sitter

  9. 1. Classical scalar field: Gravity:

  10. 2. Non-relativistic quantum particle: Example:

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

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

  13. 4. Second quantization: Definition of delta function: divergent ! divergent ! divergent ! divergent !

  14. (Now, just 1 space-dimension) x -space hermiticity conditions:

  15. Express the new in terms of Then, the Hamiltonian becomes: “ = 0 by contour integration ”

  16. In that case: and then the vacuum is invariant ! The vacuum energy is forced to vanish because of the x↔ ix symmetry

  17. 5. Difficulties Particle masses and interactions appear to violate the symmetry: Higgs potential: Perhaps there is a Higgs pair ?

  18. The pure Maxwell case can be done, but gauge fields ?? The image of the gauge group then becomes non-compact ...

  19. The x↔ ix symmetry might work, but only if, at some scale of physics, Yang-Mills fields become emergent ... questions: relation with supersymmetry ... ? interactions ... renormalization group .... The END

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