Upload
rene-garcia
View
78
Download
3
Embed Size (px)
Citation preview
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.
A relativistic example
See: counting-paths-in-spacetime, random-walks-in-a-lattice, andcorners-distribuition
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.
Euclidean Gravity
This is Wick’s rotation:
−dt2 + dx2 → d(i t)2 + dx2
It makes gravity euclidean.
And turns amplitudes into probabilities!
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.