Lecture 13: Field-theoretic formulation of Langevin models Outline: Functional (path) integral formulation Stratonovich and Ito, again the Martin-Siggia-Rose formalism free fields perturbation theory Stratonovich and supersymmetry

generating functionals In the equilibrium case, the partition function

generating functionals In the equilibrium case, the partition function

generating functionals In the equilibrium case, the partition function

generating functionals In the equilibrium case, the partition function is a generating functional:

generating functionals In the equilibrium case, the partition function is a generating functional:

generating functionals In the equilibrium case, the partition function is a generating functional:

generating functionals In the equilibrium case, the partition function is a generating functional:

generating functionals In the equilibrium case, the partition function is a generating functional:

generating functionals In the equilibrium case, the partition function is a generating functional: Here we will construct a generating functional for time-dependent correlation functions in the Langevin-Landau-Ginzburg model

Dynamics (single variable) Start from the equation of motion

Dynamics (single variable) Start from the equation of motion or, more generally,

Dynamics (single variable) Start from the equation of motion or, more generally, Discretize time:

Dynamics (single variable) Start from the equation of motion or, more generally, Discretize time: Gaussian noise:

equations of motion Ito:

equations of motion Ito:

equations of motion Ito: Stratonovich:

equations of motion Ito: Stratonovich:

equations of motion Ito: Stratonovich: Change variables from η (0 < n < M-1) to ϕ (1 < n < M):

equations of motion Ito: Stratonovich: Change variables from η (0 < n < M-1) to ϕ (1 < n < M):

equations of motion Ito: Stratonovich: Change variables from η (0 < n < M-1) to ϕ (1 < n < M): (Ito)

equations of motion Ito: Stratonovich: Change variables from η (0 < n < M-1) to ϕ (1 < n < M): (Ito) Jacobian:

equations of motion Ito: Stratonovich: Change variables from η (0 < n < M-1) to ϕ (1 < n < M): (Ito) Jacobian: ______ diagonal

equations of motion Ito: Stratonovich: Change variables from η (0 < n < M-1) to ϕ (1 < n < M): (Ito) Jacobian: ______ diagonal ______________ 1 below diagonal

equations of motion Ito: Stratonovich: Change variables from η (0 < n < M-1) to ϕ (1 < n < M): (Ito) Jacobian: (Ito) ______ diagonal ______________ 1 below diagonal

equations of motion Ito: Stratonovich: Change variables from η (0 < n < M-1) to ϕ (1 < n < M): (Ito) Jacobian: (Ito) ______ diagonal ______________ 1 below diagonal (all elements above the diagonal vanish, so det = product of diagonal elements)

Stratonovich

Stratonovich

Stratonovich Jacobian:

Stratonovich Jacobian:

Stratonovich Jacobian:

Stratonovich Jacobian:

Stratonovich Jacobian:

Stratonovich Jacobian: (back to this later)

Martin-Siggia-Rose back to Ito:

Martin-Siggia-Rose back to Ito:

Martin-Siggia-Rose back to Ito: use

Martin-Siggia-Rose back to Ito: use

Martin-Siggia-Rose back to Ito: use generating function (multivariate characteristic function)

Martin-Siggia-Rose back to Ito: use generating function (multivariate characteristic function)

Martin-Siggia-Rose back to Ito: use generating function (multivariate characteristic function)

a field theory: Δ -> 0: “action” putting space back in, using ______________________ quadratic: L0

a field theory: Δ -> 0: “action”

a field theory: Δ -> 0: “action” putting space back in, using

a field theory: Δ -> 0: “action” putting space back in, using

a field theory: Δ -> 0: “action” putting space back in, using ______________________ quadratic: L0

a field theory: Δ -> 0: “action” putting space back in, using ______________________ quadratic: L0 _____ inter- action term L1

a field theory: Δ -> 0: “action” putting space back in, using ______________________ quadratic: L0 _____ inter- action term L1 ________ “source” terms

a field theory: Δ -> 0: “action” putting space back in, using ______________________ quadratic: L0 _____ inter- action term L1 ________ “source” terms note:

a field theory: Δ -> 0: “action” putting space back in, using ______________________ quadratic: L0 _____ inter- action term L1 ________ “source” terms note: (normalization of P(ϕ|h))

correlation functions

correlation functions magnetization

correlation functions magnetization

correlation functions magnetization correlation functions:

correlation functions magnetization correlation functions:

correlation functions magnetization correlation functions:

correlation functions magnetization correlation functions:

correlation functions magnetization correlation functions: = susceptibility / response function

correlation functions magnetization correlation functions: = susceptibility / response function

correlation functions magnetization correlation functions: = susceptibility / response function

free action

free action free action:

free action free action:

free action free action: generating functional:

free action free action: generating functional: in Fourier components:

free fields (invert the matrix in the exponent in S0)

free fields (invert the matrix in the exponent in S0)

free fields (invert the matrix in the exponent in S0) back to time domain:

free fields (invert the matrix in the exponent in S0) back to time domain:

free fields (invert the matrix in the exponent in S0) back to time domain: in agreement with what we found using the direct approach in Lect. 11

perturbation theory

perturbation theory want to evaluate quantities like

perturbation theory want to evaluate quantities like expand

perturbation theory want to evaluate quantities like expand

1st order:

1st order:

1st order: average of product of 6 Gaussian variables: Use Wick’s theorem sum of products of all pairwise averages)

1st order: average of product of 6 Gaussian variables: Use Wick’s theorem sum of products of all pairwise averages) (graphs on blackboard)

1st order: average of product of 6 Gaussian variables: Use Wick’s theorem sum of products of all pairwise averages) (graphs on blackboard) but most of these vanish:

1st order: average of product of 6 Gaussian variables: Use Wick’s theorem sum of products of all pairwise averages) (graphs on blackboard) but most of these vanish:

1st order: average of product of 6 Gaussian variables: Use Wick’s theorem sum of products of all pairwise averages) (graphs on blackboard) but most of these vanish: (Ito)

Feynman graphs The surviving term:

Feynman graphs The surviving term: C(3,3) G(1,3) 1 3 G(3,2) 2

Feynman graphs The surviving term: C(3,3) G(1,3) 1 3 G(3,2) … 2

Feynman graphs The surviving term: C(3,3) G(1,3) 1 3 G(3,2) … 2 Can generate a diagrammatic expansion like that in Lect 7

Feynman graphs The surviving term: C(3,3) G(1,3) 1 3 G(3,2) … 2 Can generate a diagrammatic expansion like that in Lect 7 In fact, it is exactly the same diagrammatic expansion

Feynman graphs The surviving term: C(3,3) G(1,3) 1 3 G(3,2) … 2 Can generate a diagrammatic expansion like that in Lect 7 In fact, it is exactly the same diagrammatic expansion (except that ϕ and the correlation and response functions now depend on space as well as time)

Feynman graphs The surviving term: C(3,3) G(1,3) 1 3 G(3,2) … 2 Can generate a diagrammatic expansion like that in Lect 7 In fact, it is exactly the same diagrammatic expansion (except that ϕ and the correlation and response functions now depend on space as well as time) all closed loops of response functions (including all disconnected diagrams) vanish because for Ito G(t=0) = 0.

Stratonovich, again

Stratonovich, again Can take the continuous-time limit of the exponent as in Ito, but now J ≠ 1.

Stratonovich, again Can take the continuous-time limit of the exponent as in Ito, but now J ≠ 1. Use Grassman variables

Stratonovich, again Can take the continuous-time limit of the exponent as in Ito, but now J ≠ 1. Use Grassman variables

Stratonovich, again Can take the continuous-time limit of the exponent as in Ito, but now J ≠ 1. Use Grassman variables Taylor series expansions terminate:

Stratonovich, again Can take the continuous-time limit of the exponent as in Ito, but now J ≠ 1. Use Grassman variables Taylor series expansions terminate:

Stratonovich, again Can take the continuous-time limit of the exponent as in Ito, but now J ≠ 1. Use Grassman variables Taylor series expansions terminate: “integrals”:

Stratonovich, again Can take the continuous-time limit of the exponent as in Ito, but now J ≠ 1. Use Grassman variables Taylor series expansions terminate: “integrals”:

determinants

determinants cf for real x

determinants cf for real x and for complex z

determinants cf for real x and for complex z so represent J as

determinants cf for real x and for complex z so represent J as “ghost” variables

Stratonovich generating functional (one variable)

Stratonovich generating functional (one variable)

Stratonovich generating functional (one variable) (field)

Stratonovich generating functional (one variable) (field) free action:

Stratonovich generating functional (one variable) (field) free action: interactions:

ghost correlations:

ghost correlations: Now when we expand

ghost correlations: Now when we expand we get

ghost correlations: Now when we expand we get new terms

cancellation of closed loops Because of the -1 in the ghost correlation function, these just cancel the terms

cancellation of closed loops Because of the -1 in the ghost correlation function, these just cancel the terms

cancellation of closed loops Because of the -1 in the ghost correlation function, these just cancel the terms that were zero with Ito convention but not Stratonovich

cancellation of closed loops Because of the -1 in the ghost correlation function, these just cancel the terms that were zero with Ito convention but not Stratonovich This theory has a supersymmetry

the superfield Define a combination of the real and Grassman fields

the superfield Define a combination of the real and Grassman fields Grassman numbers

the superfield Define a combination of the real and Grassman fields Grassman numbers Then if

the superfield Define a combination of the real and Grassman fields Grassman numbers Then if the generating functional can be written

How does this happen? Expand the potential term:

How does this happen? Expand the potential term:

How does this happen? Expand the potential term: Integrate over the “Grassman time”

How does this happen? Expand the potential term: Integrate over the “Grassman time”

How does this happen? Expand the potential term: Integrate over the “Grassman time” which are the terms in the action involving f.