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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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slide1

Utrecht University

Complex Transformations

and the

Cosmological Constant

Problem

Gerard ’t Hooft

Spinoza Institute

Utrecht

slide3

= density of matter at

= spacelike curvature, usually assumed to vanish

= cosmological constant

The expanding Universe

slide5

The Cosmological Constant Problem

stretchability

very small

stiffness

very large

slide7

During inflation, one can try to follow the effects of

quantum corrections

on the cosmological cnstant term

+

+ ...

but the subtraction term is crucial:

X

slide8

Does tend to zero

during inflation ???

Various theories claim such an effect,

but the physical forces for that

remain obscure.

Can this happen ?

slide9

Anti-De Sitter

With S. Nobbenhuis,

gr-qc/ 0602076

De Sitter

slide12

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:

slide13

E

0

3. Harmonic oscillator:

Change hermiticity but not the algebra:

We get all negative energy states, but the vacuum is not invariant !

slide14

4. Second quantization:

Definition of delta function:

divergent !

divergent !

divergent !

divergent !

slide15

(Now, just 1

space-dimension)

x -space hermiticity conditions:

slide16

Express the new in terms of

Then, the Hamiltonian becomes:

“ = 0 by contour integration ”

slide17

In that case:

and then the vacuum is invariant !

The vacuum energy is forced to vanish because of the

x↔ ix symmetry

slide18

5. Difficulties

Particle masses and interactions appear to violate the symmetry:

Higgs potential:

Perhaps there is

a Higgs pair ?

slide19

The pure Maxwell case can be done,

but gauge fields ??

The image of the gauge group then becomes non-compact ...

slide20

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