Causal Dynamical Triangulations

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Causal Dynamical Triangulations Israel Garc´ ıa @renegarxia 20 de mayo de 2015

Transcript of Causal Dynamical Triangulations

Causal Dynamical Triangulations

Israel Garcıa@renegarxia

20 de mayo de 2015

Table of contents

The classical action

Quantum mechanics

Path Integrals in relativity

Path Integrals

Some waves

More waves (click to watch them)

Path Integrals

With xa, xb fixed, how do we minimize S?

S =

∫ tb

ta

L(x , x , t)dt (action)

L =m

2x2 − V (x , t) (Lagrangian)

Path Integrals

Set x 7→ x + δx , then

L 7→ m

2(x + δx)2 − V (x + δx , t),

=m

2x2 + mxδx +

m

2δx2 − V (x , t)− ∂xV (x , t)δx − O(δx2),

Therefore,

L 7→ L + mxδx +m

2δx2 − ∂xV (x , t)δx − O(δx2),

What happens to the action? (S)

The Action

S 7→∫ tb

ta

L + mxδx +m

2δx2 − ∂xV (x , t)δx − O(δx2) dt,

= S +

∫ tb

ta

mxδx − ∂xV (x , t)δx dt (up to first order)

With fixed extremes, δxa = δxb = 0. Integration by parts follows:

S 7→ S −∫ tb

ta

(mx + ∂xV (x , t)) δx dt

Least action

The true x is such that, if

x 7→ x + δx ,

then S is the same, to first order. The true trajectory obeys thisequation:

mx = −∂xV (x , t),

which is the second law of mechanics in disguise.

Quantum mechanics a la Feynman

The path integral:

K (a, b) =

∫ b

ae i/~SDx(t)

Path integrals in relativity

There’s an action principle for general relativity:

SEH =1

G

∫d4x

√det g (R − 2Λ) (Einstein-Hilbert action)

Problem: Make sense of the path integral:∫g∈GDg e iSEH

So, you want to quantize gravity?

I String theory.

I Loop quantum gravity.

I Euclidean quantum gravity.

I Causal dynamical triangulations.

What is curvature?

Discrete curvature

Euclidean Gravity

This is Wick’s rotation:

−dt2 + dx2 → d(i t)2 + dx2

It makes gravity euclidean.

And turns amplitudes into probabilities!

First attempt: failed!

Causality (you can’t kill your parents...)

...before you are born...

Global hyperbolicity

This is acceptable: This is not:

Wormhole, baby universes.

Causal triangulations

Quantum gravity in your desktop

References I

J. Ambjorn, A. Goerlich, J. Jurkiewicz, and R. Loll, QuantumGravity via Causal Dynamical Triangulations.

Jan Ambjorn, J. Jurkiewicz, and R. Loll, Dynamicallytriangulating Lorentzian quantum gravity, Nucl.Phys. B610(2001), 347–382.

J. Ambjorn, J. Jurkiewicz, and R. Loll, The Universe fromscratch, Contemp.Phys. 47 (2006), 103–117.

R. Loll, The Emergence of spacetime or quantum gravity onyour desktop, Class.Quant.Grav. 25 (2008), 114006.