Holographic Brownian Motion in 1+1 Dimensions

We study the motion of a stochastic string in the background of a BTZ black hole. In the 1+1 dimensional boundary theory this corresponds to a very heavy external particle (e.g, a quark), interacting with the fields of a CFT at finite temperature, and describing Brownian motion. The equations of motion for a string in the BTZ background can be solved exactly. Thus we can use holographic techniques to obtain the Schwinger-Keldysh Green function for the boundary theory for the force acting on the quark. We write down the generalized Langevin equation describing the motion of the external particle and calculate the drag and the thermal mass shift. Interestingly we obtain dissipation even at zero temperature for this 1+1 system. Even so, this does not violate boost (Lorentz) invariance because the drag force on a constant velocity quark continues to be zero. Furthermore since the Green function is exact, it is possible to write down an effective membrane action, and thus a Langevin equation, located at a"stretched horizon"at an arbitrary finite distance from the horizon.


AdS/CFT : A theorist's tool
Strongly coupled field theory ⇔ Weakly coupled gravity.
Real time correlators for (scalar) field theory can be obtained by choosing appropriate boundary conditions.

Then the Langevin equation reads
Fluctuation-Dissipation relation The same metric in dimensionless coordinate, r ≡ br where, b = 1 2πT L 2 , f (r ) = 1 − 1 r 2 and T is Hawking temperature.

Generalized Langevin Equation from Holography
The background metric AdS3-BTZ is defined as The same metric in dimensionless coordinate, r ≡ br where, b = 1 2πT L 2 , f (r ) = 1 − 1 r 2 and T is Hawking temperature.
The Nambu-Goto action is The string world sheet action becomes where, m ≡ (2πT )L 2 2πl 2 The EOM of the string − m f ∂ 2 t x + ∂r (T0(r )∂r x) = 0 (11) The solution to this EOM is given by The solution to this EOM is given by Modes satisfying boundary conditions The solution to this EOM is given by Modes satisfying boundary conditions The retarded correlator G R (ω) is defined as

Generalized Langevin Equation from Holography
Zero temperature mass of the particle Retarded propagator Retarded propagator Expanding G R in small frequencies Retarded propagator Expanding G R in small frequencies Generically when G R (ω) is expanded in small ω it takes the form Thermal mass shift Pinaki Banerjee (IMSc)

Generalized Langevin Equation from Holography
Viscous drag Thermal mass shift Higher order "dissipation coefficient"

Generalized Langevin Equation from Holography
Viscous drag Thermal mass shift Higher order "dissipation coefficient" In the "large frequency" limit It is finite and therefore no need to renormalize by adding counterterms.
It cannot be renormalized away in the boundary theory by Hermitian counter terms.
Quark moving at constant velocity doesn't feel any drag at T= 0.
Some 1+1 condensed matter systems exhibit such dissipation (or decoherence) at absolute zero due to zero-point fluctuations.
A possible explanation : Energy can cascade from high frequencies to low frequencies in a nonlinear system. One can expect a large Poincare recurrence time and the energy is effectively lost for good. This would then show up as a dissipation! t=−∞ t=+∞ t=+∞ − iσ t=−∞ − iβ field "1" field "2" field "2" field "1"

Brownian Motion at Stretched Horizon
Softening of delta function

Results
Natural softening of delta function in Langevin equation.
Temperature dependent mass correction is zero (in the extreme UV limit).
A temperature independent dissipation at all frequencies.
The "stretched horizon" can be placed at an arbitrary radius and an effective action obtained.

Results
Natural softening of delta function in Langevin equation.
Temperature dependent mass correction is zero (in the extreme UV limit).
A temperature independent dissipation at all frequencies.
The "stretched horizon" can be placed at an arbitrary radius and an effective action obtained.

Can be done
Study the holographic RG interpretation in this case.
Same problem using a charged BTZ , thereby introducing a chemical potential.

Results
Natural softening of delta function in Langevin equation.
Temperature dependent mass correction is zero (in the extreme UV limit).
A temperature independent dissipation at all frequencies.
The "stretched horizon" can be placed at an arbitrary radius and an effective action obtained.

Can be done
Study the holographic RG interpretation in this case.
Same problem using a charged BTZ , thereby introducing a chemical potential.