Lecture 13: Field-theoretic formulation of Langevin models Outline: Functional (path) integral...

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



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Z[β,h] = Dφexp∫ −βE[φ]( )



€


€

= Dφexp − 12 dd x∫ r0φ

2(x) + 12 u0φ

4 (x) + (∇φ(x))2 − 2h(x)φ(x)[ ]{ }∫



€


€


2(x) + 12 u0φ

4 (x) + (∇φ(x))2 − 2h(x)φ(x)[ ]{ }∫is a generating functional:



€


€


2(x) + 12 u0φ


€

∂ logZ

∂β= E



€


€


2(x) + 12 u0φ


€

∂ logZ

∂β= E

€

∂2 logZ

∂β 2= E 2 − E

2



€


€


2(x) + 12 u0φ


€

δ logZ

δh(x)= φ(x)

€

∂ logZ

∂β= E

€

∂2 logZ

∂β 2= E 2 − E

2



€


€


2(x) + 12 u0φ


€

δ logZ

δh(x)= φ(x)

€

δ 2 logZ

δh(x)δh( ′ x )= φ(x)φ( ′ x ) − φ(x) φ( ′ x )

€

∂ logZ

∂β= E

€

∂2 logZ

∂β 2= E 2 − E

2



€


€


2(x) + 12 u0φ


€

δ logZ

δh(x)= φ(x)

€

δ 2 logZ

δh(x)δh( ′ x )= φ(x)φ( ′ x ) − φ(x) φ( ′ x )

€

∂ logZ

∂β= E

€

∂2 logZ

∂β 2= E 2 − E

2

Here we will construct a generating functional for time-dependentcorrelation functions in the Langevin-Landau-Ginzburg model

Dynamics (single variable)

Start from the equation of motion

€

η(t)η ( ′ t ) = 2Tδ(t − ′ t )

€

dφ

dt= −γφ + h(t) + η (t)



€

dφ

dt= f (φ) + h(t) + η (t)

€

η(t)η ( ′ t ) = 2Tδ(t − ′ t )

€

dφ

dt= −γφ + h(t) + η (t)

or, more generally,



€

dφ

dt= f (φ) + h(t) + η (t)

€

η(t)η ( ′ t ) = 2Tδ(t − ′ t )

€

dφ

dt= −γφ + h(t) + η (t)

or, more generally,

Discretize time:

€

t = Δ, 2Δ,L MΔ; η m = η (t)dtmΔ

(m +1)Δ

∫



€

dφ

dt= f (φ) + h(t) + η (t)

€

η(t)η ( ′ t ) = 2Tδ(t − ′ t )

€

dφ

dt= −γφ + h(t) + η (t)

or, more generally,

Discretize time:

€

t = Δ, 2Δ,L MΔ; η m = η (t)dtmΔ

(m +1)Δ

∫

€

P[η ] =1

(4πTΔ)M / 2exp −

1

4TΔη n

2

n= 0

M −1

∑ ⎡

⎣ ⎢

⎤

⎦ ⎥

η mη n = 2TΔδmn

Gaussian noise:

equations of motion

Ito:

equations of motion

€

φn +1 − φn = Δ f (φn ) + hn( ) + η nIto:

equations of motion

€


Stratonovich:

equations of motion

€


€

φn +1 − φn = 12 Δ f (φn ) + hn( ) + 1

2 Δ f (φn +1) + hn +1( ) + η n

Stratonovich:

equations of motion

€


€

φn +1 − φn = 12 Δ f (φn ) + hn( ) + 1

2 Δ f (φn +1) + hn +1( ) + η n

Stratonovich:

Change variables from η (0 < n < M-1) to ϕ (1 < n < M):

equations of motion

€


€

φn +1 − φn = 12 Δ f (φn ) + hn( ) + 1

2 Δ f (φn +1) + hn +1( ) + η n

Stratonovich:


€

P[φ] =J[φ]


1

4TΔφn +1 − φn − f (φn ) + hn[ ]Δ( )

2

n=1

M

∑ ⎡

⎣ ⎢

⎤

⎦ ⎥

equations of motion

€


€

φn +1 − φn = 12 Δ f (φn ) + hn( ) + 1

2 Δ f (φn +1) + hn +1( ) + η n

Stratonovich:


€

P[φ] =J[φ]


1

4TΔφn +1 − φn − f (φn ) + hn[ ]Δ( )

2

n=1

M

∑ ⎡

⎣ ⎢

⎤

⎦ ⎥ (Ito)

equations of motion

€


€

φn +1 − φn = 12 Δ f (φn ) + hn( ) + 1

2 Δ f (φn +1) + hn +1( ) + η n

Stratonovich:


€

P[φ] =J[φ]


1

4TΔφn +1 − φn − f (φn ) + hn[ ]Δ( )

2

n=1

M

∑ ⎡

⎣ ⎢

⎤

⎦ ⎥ (Ito)

€

J[φ] = det∂η

∂φ

⎛

⎝ ⎜

⎞

⎠ ⎟= det δm,n +1 − 1+ Δ ′ f (φn )( )δm,n[ ] =1Jacobian:

equations of motion

€


€

φn +1 − φn = 12 Δ f (φn ) + hn( ) + 1

2 Δ f (φn +1) + hn +1( ) + η n

Stratonovich:


€

P[φ] =J[φ]


1

4TΔφn +1 − φn − f (φn ) + hn[ ]Δ( )

2

n=1

M

∑ ⎡

⎣ ⎢

⎤

⎦ ⎥ (Ito)

€

J[φ] = det∂η

∂φ

⎛

⎝ ⎜

⎞


______diagonal

equations of motion

€


€

φn +1 − φn = 12 Δ f (φn ) + hn( ) + 1

2 Δ f (φn +1) + hn +1( ) + η n

Stratonovich:


€

P[φ] =J[φ]


1

4TΔφn +1 − φn − f (φn ) + hn[ ]Δ( )

2

n=1

M

∑ ⎡

⎣ ⎢

⎤

⎦ ⎥ (Ito)

€

J[φ] = det∂η

∂φ

⎛

⎝ ⎜

⎞


______diagonal

______________1 belowdiagonal

equations of motion

€


€

φn +1 − φn = 12 Δ f (φn ) + hn( ) + 1

2 Δ f (φn +1) + hn +1( ) + η n

Stratonovich:


€

P[φ] =J[φ]


1

4TΔφn +1 − φn − f (φn ) + hn[ ]Δ( )

2

n=1

M

∑ ⎡

⎣ ⎢

⎤

⎦ ⎥ (Ito)

€

J[φ] = det∂η

∂φ

⎛

⎝ ⎜

⎞

⎠ ⎟= det δm,n +1 − 1+ Δ ′ f (φn )( )δm,n[ ] =1Jacobian: (Ito)

______diagonal


equations of motion

€


€

φn +1 − φn = 12 Δ f (φn ) + hn( ) + 1

2 Δ f (φn +1) + hn +1( ) + η n

Stratonovich:


€

P[φ] =J[φ]


1

4TΔφn +1 − φn − f (φn ) + hn[ ]Δ( )

2

n=1

M

∑ ⎡

⎣ ⎢

⎤

⎦ ⎥ (Ito)

€

J[φ] = det∂η

∂φ

⎛

⎝ ⎜

⎞

⎠ ⎟= det δm,n +1 − 1+ Δ ′ f (φn )( )δm,n[ ] =1Jacobian: (Ito)

______diagonal


(all elements above the diagonal vanish, so det = product of diagonal elements)

Stratonovich

€

φn +1 − φn = 12 Δ f (φn ) + hn( ) + 1

2 Δ f (φn +1) + hn +1( ) + η n ⇒

Stratonovich

€

P[φ] =J[φ]


1

4TΔφn +1 − φn − 1

2 Δ fn + hn( ) − 12 Δ fn +1 + hn +1( )( )

2

n=1

M

∑ ⎡

⎣ ⎢

⎤

⎦ ⎥

€

φn +1 − φn = 12 Δ f (φn ) + hn( ) + 1

2 Δ f (φn +1) + hn +1( ) + η n ⇒

Stratonovich

€

P[φ] =J[φ]


1

4TΔφn +1 − φn − 1

2 Δ fn + hn( ) − 12 Δ fn +1 + hn +1( )( )

2

n=1

M

∑ ⎡

⎣ ⎢

⎤

⎦ ⎥

€

φn +1 − φn = 12 Δ f (φn ) + hn( ) + 1

2 Δ f (φn +1) + hn +1( ) + η n ⇒

Jacobian:

Stratonovich

€

P[φ] =J[φ]


1

4TΔφn +1 − φn − 1

2 Δ fn + hn( ) − 12 Δ fn +1 + hn +1( )( )

2

n=1

M

∑ ⎡

⎣ ⎢

⎤

⎦ ⎥

€

φn +1 − φn = 12 Δ f (φn ) + hn( ) + 1

2 Δ f (φn +1) + hn +1( ) + η n ⇒

€

J[φ] = det (1− 12 Δ ′ f n +1)δm,n +1 − 1+ 1

2 Δ ′ f n( )δm,n[ ]Jacobian:

Stratonovich

€

P[φ] =J[φ]


1

4TΔφn +1 − φn − 1

2 Δ fn + hn( ) − 12 Δ fn +1 + hn +1( )( )

2

n=1

M

∑ ⎡

⎣ ⎢

⎤

⎦ ⎥

€

φn +1 − φn = 12 Δ f (φn ) + hn( ) + 1

2 Δ f (φn +1) + hn +1( ) + η n ⇒

€

J[φ] = det (1− 12 Δ ′ f n +1)δm,n +1 − 1+ 1

2 Δ ′ f n( )δm,n[ ]

= (1− 12 Δ ′ f n )

n=1

M

∏

Jacobian:

Stratonovich

€

P[φ] =J[φ]


1

4TΔφn +1 − φn − 1

2 Δ fn + hn( ) − 12 Δ fn +1 + hn +1( )( )

2

n=1

M

∑ ⎡

⎣ ⎢

⎤

⎦ ⎥

€

φn +1 − φn = 12 Δ f (φn ) + hn( ) + 1

2 Δ f (φn +1) + hn +1( ) + η n ⇒

€

J[φ] = det (1− 12 Δ ′ f n +1)δm,n +1 − 1+ 1

2 Δ ′ f n( )δm,n[ ]

= (1− 12 Δ ′ f n ) = exp log(1− 1

2 Δ ′ f n )n=1

M

∑ ⎡

⎣ ⎢

⎤

⎦ ⎥

n=1

M

∏

Jacobian:

Stratonovich

€

P[φ] =J[φ]


1

4TΔφn +1 − φn − 1

2 Δ fn + hn( ) − 12 Δ fn +1 + hn +1( )( )

2

n=1

M

∑ ⎡

⎣ ⎢

⎤

⎦ ⎥

€

φn +1 − φn = 12 Δ f (φn ) + hn( ) + 1

2 Δ f (φn +1) + hn +1( ) + η n ⇒

€

J[φ] = det (1− 12 Δ ′ f n +1)δm,n +1 − 1+ 1

2 Δ ′ f n( )δm,n[ ]

= (1− 12 Δ ′ f n ) = exp log(1− 1

2 Δ ′ f n )n=1

M

∑ ⎡

⎣ ⎢

⎤

⎦ ⎥

n=1

M

∏ = exp − 12 Δ ′ f n

n=1

M

∑ ⎡

⎣ ⎢

⎤

⎦ ⎥

Jacobian:

Stratonovich

€

P[φ] =J[φ]


1

4TΔφn +1 − φn − 1

2 Δ fn + hn( ) − 12 Δ fn +1 + hn +1( )( )

2

n=1

M

∑ ⎡

⎣ ⎢

⎤

⎦ ⎥

€

φn +1 − φn = 12 Δ f (φn ) + hn( ) + 1

2 Δ f (φn +1) + hn +1( ) + η n ⇒

€

J[φ] = det (1− 12 Δ ′ f n +1)δm,n +1 − 1+ 1

2 Δ ′ f n( )δm,n[ ]

= (1− 12 Δ ′ f n ) = exp log(1− 1

2 Δ ′ f n )n=1

M

∑ ⎡

⎣ ⎢

⎤

⎦ ⎥

n=1

M

∏ = exp − 12 Δ ′ f n

n=1

M

∑ ⎡

⎣ ⎢

⎤

⎦ ⎥

Jacobian:

(back to this later)

Martin-Siggia-Rose

back to Ito:

Martin-Siggia-Rose

€

P[φ | h] =1


1

4TΔφn +1 − φn − f (φn ) + hn[ ]Δ( )

2

n=1

M

∑ ⎡

⎣ ⎢

⎤

⎦ ⎥back to Ito:

Martin-Siggia-Rose

€

P[φ | h] =1


1

4TΔφn +1 − φn − f (φn ) + hn[ ]Δ( )

2

n=1

M

∑ ⎡

⎣ ⎢

⎤

⎦ ⎥back to Ito:

€

exp − 14 a2

( ) =dy

2πexp −y 2 + iya( )

−∞

∞

∫ ⇒use

Martin-Siggia-Rose

€

P[φ | h] =1


1

4TΔφn +1 − φn − f (φn ) + hn[ ]Δ( )

2

n=1

M

∑ ⎡

⎣ ⎢

⎤

⎦ ⎥back to Ito:

€

exp − 14 a2

( ) =dy


−∞

∞

∫ ⇒use

€

P[φ | h] =1

(2π )Md ˆ φ n

n= 0

M −1

∏ exp∫ −TΔ ˆ φ n2 + i ˆ φ n −φn +1 + φn + Δ( fn + hn )[ ]( )

n= 0

M −1

∑ ⎡

⎣ ⎢

⎤

⎦ ⎥

Martin-Siggia-Rose

€

P[φ | h] =1


1

4TΔφn +1 − φn − f (φn ) + hn[ ]Δ( )

2

n=1

M

∑ ⎡

⎣ ⎢

⎤

⎦ ⎥back to Ito:

€

exp − 14 a2

( ) =dy


−∞

∞

∫ ⇒use

€

P[φ | h] =1

(2π )Md ˆ φ n

n= 0

M −1


n= 0

M −1

∑ ⎡

⎣ ⎢

⎤

⎦ ⎥

generating function (multivariate characteristic function)

Martin-Siggia-Rose

€

P[φ | h] =1


1

4TΔφn +1 − φn − f (φn ) + hn[ ]Δ( )

2

n=1

M

∑ ⎡

⎣ ⎢

⎤

⎦ ⎥back to Ito:

€

exp − 14 a2

( ) =dy


−∞

∞

∫ ⇒use

€

P[φ | h] =1

(2π )Md ˆ φ n

n= 0

M −1


n= 0

M −1

∑ ⎡

⎣ ⎢

⎤

⎦ ⎥


€

Z[h,θ] = dφn

n=1

M

∏∫ e iθ nφn

n=1

M

∏ ⎛

⎝ ⎜

⎞

⎠ ⎟P[φ]

Martin-Siggia-Rose

€

P[φ | h] =1


1

4TΔφn +1 − φn − f (φn ) + hn[ ]Δ( )

2

n=1

M

∑ ⎡

⎣ ⎢

⎤

⎦ ⎥back to Ito:

€

exp − 14 a2

( ) =dy


−∞

∞

∫ ⇒use

€

P[φ | h] =1

(2π )Md ˆ φ n

n= 0

M −1


n= 0

M −1

∑ ⎡

⎣ ⎢

⎤

⎦ ⎥


€

Z[h,θ] = dφn

n=1

M

∏∫ e iθ nφn

n=1

M

∏ ⎛

⎝ ⎜

⎞

⎠ ⎟P[φ]

=1

(2π )Mdφn d ˆ φ n

n= 0

M −1

∏n=1

M

∏ exp∫ i θnφn

n=1

M

∑ + −TΔ ˆ φ n2 + i ˆ φ n −φn +1 + φn + Δ( fn + hn )[ ]( )

n= 0

M −1

∑ ⎡

⎣ ⎢

⎤

⎦ ⎥

a field theory:

Δ -> 0:

€

Z[h,θ] = DφD ˆ φ exp −S[φ, ˆ φ ,h,θ]( )∫−S[φ, ˆ φ ,h,θ] = d∫ t −T ˆ φ 2(t) + i ˆ φ (t) − ˙ φ (t) + f (φ) + h(t)( ) + iθ(t)φ(t)[ ]“action”

putting space back in, using

€

S[φ, ˆ φ ,h,θ] = d∫ t ddx L φ(x, t), ˙ φ (x, t), ˆ φ (x, t),h(x, t),θ(x, t)( )

L(φ, ˙ φ , ˆ φ ,h,θ) = T ˆ φ 2 + i ˆ φ ∂φ

∂t+ ir0

ˆ φ φ − i ˆ φ ∇ 2φ + iu0ˆ φ φ3 − i ˆ φ h − iθφ€

f (x, t) = − r0 −∇ 2( )φ(x, t) − u0φ

3(x, t)

______________________ quadratic: L0

a field theory:

Δ -> 0:

€


a field theory:

Δ -> 0:

€



€

f (x, t) = − r0 −∇ 2( )φ(x, t) − u0φ

3(x, t)

a field theory:

Δ -> 0:

€



€


L(φ, ˙ φ , ˆ φ ,h,θ) = T ˆ φ 2 + i ˆ φ ∂φ

∂t+ ir0


f (x, t) = − r0 −∇ 2( )φ(x, t) − u0φ

3(x, t)

a field theory:

Δ -> 0:

€



€


L(φ, ˙ φ , ˆ φ ,h,θ) = T ˆ φ 2 + i ˆ φ ∂φ

∂t+ ir0


f (x, t) = − r0 −∇ 2( )φ(x, t) − u0φ

3(x, t)

______________________ quadratic: L0

a field theory:

Δ -> 0:

€



€


L(φ, ˙ φ , ˆ φ ,h,θ) = T ˆ φ 2 + i ˆ φ ∂φ

∂t+ ir0


f (x, t) = − r0 −∇ 2( )φ(x, t) − u0φ

3(x, t)

______________________ quadratic: L0

_____inter-actionterm L1

a field theory:

Δ -> 0:

€



€


L(φ, ˙ φ , ˆ φ ,h,θ) = T ˆ φ 2 + i ˆ φ ∂φ

∂t+ ir0


f (x, t) = − r0 −∇ 2( )φ(x, t) − u0φ

3(x, t)

______________________ quadratic: L0


________“source” terms

a field theory:

Δ -> 0:

€



€


L(φ, ˙ φ , ˆ φ ,h,θ) = T ˆ φ 2 + i ˆ φ ∂φ

∂t+ ir0


f (x, t) = − r0 −∇ 2( )φ(x, t) − u0φ

3(x, t)

______________________ quadratic: L0



note:

€

Z[h,0] = DφD ˆ φ exp −S[φ, ˆ φ ,h,θ]( )∫ =1

a field theory:

Δ -> 0:

€



€


L(φ, ˙ φ , ˆ φ ,h,θ) = T ˆ φ 2 + i ˆ φ ∂φ

∂t+ ir0


f (x, t) = − r0 −∇ 2( )φ(x, t) − u0φ

3(x, t)

______________________ quadratic: L0



note:

€

Z[h,0] = DφD ˆ φ exp −S[φ, ˆ φ ,h,θ]( )∫ =1 (normalization of P(ϕ|h))

correlation functions


magnetization


€

M(x, t) = φ(x, t) = −i limθ →0

δZ

δθ(x, t)magnetization


€

M(x, t) = φ(x, t) = −i limθ →0

δZ


correlation functions:


€

C(x, t; ′ x , ′ t ) = φ(x, t)φ( ′ x , ′ t ) − φ(x, t) φ( ′ x , ′ t ) = −limθ →0

δ 2Z

δθ(x, t)δθ( ′ x , ′ t )

€

M(x, t) = φ(x, t) = −i limθ →0

δZ




€


δ 2Z

δθ(x, t)δθ( ′ x , ′ t )

€

M(x, t) = φ(x, t) = −i limθ →0

δZ

δθ(x, t)

€

G(x, t; ′ x , ′ t ) = i ˆ φ (x, t)φ( ′ x , ′ t ) = −i limθ →0

δ 2Z

δθ(x, t)δh( ′ x , ′ t )

magnetization



€


δ 2Z

δθ(x, t)δθ( ′ x , ′ t )

€

M(x, t) = φ(x, t) = −i limθ →0

δZ

δθ(x, t)

€


δ 2Z

δθ(x, t)δh( ′ x , ′ t )

magnetization


€

=δM(x, t)

δh( ′ x , ′ t )


€


δ 2Z

δθ(x, t)δθ( ′ x , ′ t )

€

M(x, t) = φ(x, t) = −i limθ →0

δZ

δθ(x, t)

€


δ 2Z

δθ(x, t)δh( ′ x , ′ t )

magnetization


€

=δM(x, t)

δh( ′ x , ′ t )= susceptibility / response function


€


δ 2Z

δθ(x, t)δθ( ′ x , ′ t )

€

M(x, t) = φ(x, t) = −i limθ →0

δZ

δθ(x, t)

€


δ 2Z

δθ(x, t)δh( ′ x , ′ t )

magnetization


€

=δM(x, t)


€

ˆ φ (x, t) = −limθ →0

δZ

δh(x, t)= 0


€


δ 2Z

δθ(x, t)δθ( ′ x , ′ t )

€

M(x, t) = φ(x, t) = −i limθ →0

δZ

δθ(x, t)

€


δ 2Z

δθ(x, t)δh( ′ x , ′ t )

magnetization


€

=δM(x, t)


€

ˆ φ (x, t) = −limθ →0

δZ

δh(x, t)= 0

€

ˆ φ (x, t) ˆ φ ( ′ x , ′ t ) = −limθ →0

δ 2Z

δh(x, t)δh( ′ x , ′ t )= 0

free action

€

L0 = T ˆ φ 2 + i ˆ φ ∂φ

∂t+ ir0

ˆ φ φ − i ˆ φ ∇ 2φ

free action

€

L0 = T ˆ φ 2 + i ˆ φ ∂φ

∂t+ ir0

ˆ φ φ − i ˆ φ ∇ 2φ

free action:

free action

€

L0 = T ˆ φ 2 + i ˆ φ ∂φ

∂t+ ir0

ˆ φ φ − i ˆ φ ∇ 2φ

€

S0 = L0dt =∫ 12 dt∫ φ ˆ φ ( )

0 ∂t + r0 −∇ 2

−∂t + r0 −∇ 2 2T

⎛

⎝ ⎜

⎞

⎠ ⎟φˆ φ

⎛

⎝ ⎜

⎞

⎠ ⎟free action:

free action

€

L0 = T ˆ φ 2 + i ˆ φ ∂φ

∂t+ ir0

ˆ φ φ − i ˆ φ ∇ 2φ

€

S0 = L0dt =∫ 12 dt∫ φ ˆ φ ( )

0 ∂t + r0 −∇ 2

−∂t + r0 −∇ 2 2T

⎛

⎝ ⎜

⎞

⎠ ⎟φˆ φ

⎛

⎝ ⎜

⎞

⎠ ⎟free action:

€

Z0(h,θ) = DφD ˆ φ exp − 12 dt∫ φ ˆ φ ( )

0 ∂t + r0 −∇ 2

−∂t + r0 −∇ 2 2T

⎛

⎝ ⎜

⎞

⎠ ⎟φˆ φ

⎛

⎝ ⎜

⎞

⎠ ⎟+ ih ˆ φ + iθφ

⎡

⎣ ⎢

⎤

⎦ ⎥∫

generating functional:

free action

€

L0 = T ˆ φ 2 + i ˆ φ ∂φ

∂t+ ir0

ˆ φ φ − i ˆ φ ∇ 2φ

€

S0 = L0dt =∫ 12 dt∫ φ ˆ φ ( )

0 ∂t + r0 −∇ 2

−∂t + r0 −∇ 2 2T

⎛

⎝ ⎜

⎞

⎠ ⎟φˆ φ

⎛

⎝ ⎜

⎞

⎠ ⎟free action:

€

Z0(h,θ) = DφD ˆ φ exp − 12 dt∫ φ ˆ φ ( )

0 ∂t + r0 −∇ 2

−∂t + r0 −∇ 2 2T

⎛

⎝ ⎜

⎞

⎠ ⎟φˆ φ

⎛

⎝ ⎜

⎞

⎠ ⎟+ ih ˆ φ + iθφ

⎡

⎣ ⎢

⎤

⎦ ⎥∫

generating functional:

in Fourier components:

€

Z0(h,θ) = DφD ˆ φ exp − 12 φ− p,−ω

ˆ φ − p,−ω( )0 −iω + r0 + p2

iω + r0 + p2 2T

⎛

⎝ ⎜

⎞

⎠ ⎟φpω

ˆ φ pω

⎛

⎝ ⎜

⎞

⎠ ⎟

p,ω

∫ ⎡

⎣ ⎢ ⎢

⎤

⎦ ⎥ ⎥

∫

⋅exp i h− p,−ωˆ φ pω + i θ− p,−ωφpω

p,ω

∫p,ω

∫ ⎡

⎣ ⎢ ⎢

⎤

⎦ ⎥ ⎥

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

free fields

€

C0( p,ω) =φ− p,−ω

ˆ φ − p,−ω

⎛

⎝ ⎜

⎞

⎠ ⎟ φp,ω

ˆ φ p,ω( ) =

2T

ω2 + (r0 + p2)2

1

−iω + r0 + p2

1

+iω + r0 + p20

⎛

⎝

⎜ ⎜ ⎜ ⎜

⎞

⎠

⎟ ⎟ ⎟ ⎟

(invert the matrix in the exponent in S0)

free fields

€

C0( p,ω) =φ− p,−ω

ˆ φ − p,−ω

⎛

⎝ ⎜

⎞

⎠ ⎟ φp,ω

ˆ φ p,ω( ) =

2T

ω2 + (r0 + p2)2

1

−iω + r0 + p2

1

+iω + r0 + p20

⎛

⎝

⎜ ⎜ ⎜ ⎜

⎞

⎠

⎟ ⎟ ⎟ ⎟


back to time domain:

free fields

€

C0( p,ω) =φ− p,−ω

ˆ φ − p,−ω

⎛

⎝ ⎜

⎞

⎠ ⎟ φp,ω

ˆ φ p,ω( ) =

2T

ω2 + (r0 + p2)2

1

−iω + r0 + p2

1

+iω + r0 + p20

⎛

⎝

⎜ ⎜ ⎜ ⎜

⎞

⎠

⎟ ⎟ ⎟ ⎟



€

C0( p, t − ′ t ) =φ− p (t)ˆ φ − p (t)

⎛

⎝ ⎜

⎞

⎠ ⎟ φp ( ′ t ) ˆ φ p ( ′ t )( )

=T exp −(r0 + p2) t − ′ t [ ] Θ(t − ′ t )exp −(r0 + p2)(t − ′ t )[ ]

Θ( ′ t − t)exp −(r0 + p2)( ′ t − t)[ ] 0

⎛

⎝ ⎜ ⎜

⎞

⎠ ⎟ ⎟

free fields

€

C0( p,ω) =φ− p,−ω

ˆ φ − p,−ω

⎛

⎝ ⎜

⎞

⎠ ⎟ φp,ω

ˆ φ p,ω( ) =

2T

ω2 + (r0 + p2)2

1

−iω + r0 + p2

1

+iω + r0 + p20

⎛

⎝

⎜ ⎜ ⎜ ⎜

⎞

⎠

⎟ ⎟ ⎟ ⎟



€

C0( p, t − ′ t ) =φ− p (t)ˆ φ − p (t)

⎛

⎝ ⎜

⎞

⎠ ⎟ φp ( ′ t ) ˆ φ p ( ′ t )( )

=T exp −(r0 + p2) t − ′ t [ ] Θ(t − ′ t )exp −(r0 + p2)(t − ′ t )[ ]

Θ( ′ t − t)exp −(r0 + p2)( ′ t − t)[ ] 0

⎛

⎝ ⎜ ⎜

⎞

⎠ ⎟ ⎟

in agreement with what we found using the direct approach in Lect. 11

perturbation theory

perturbation theory

want to evaluate quantities like

€

φ(1) ˆ φ (2) = DφD ˆ φ ∫ φ(1) ˆ φ (2)exp −S( )

= DφD ˆ φ ∫ φ(1) ˆ φ (2)exp −S1( )exp −S0( )

= φ(1) ˆ φ (2)exp −S1( )0

(1) = (x1, t1), etc.

perturbation theory

expand

€

exp −S1( ) = exp − dt dd x L1(x, t)∫( )

= exp −iu0 d∫ 1 ˆ φ (1)φ3(1)( )


€



= φ(1) ˆ φ (2)exp −S1( )0

(1) = (x1, t1), etc.

perturbation theory

expand

€

exp −S1( ) = exp − dt dd x L1(x, t)∫( )

= exp −iu0 d∫ 1 ˆ φ (1)φ3(1)( )

€

=1− iu0 d∫ 1 ˆ φ (1)φ3(1) − 12 u0

2 d∫ 1d2 ˆ φ (1)φ3(1) ˆ φ (2)φ3(2) +L


€



= φ(1) ˆ φ (2)exp −S1( )0

(1) = (x1, t1), etc.

1st order:

1st order:

€

φ(1) ˆ φ (2) = φ(1) ˆ φ (2) 1− S1 +L( )0

= C0(1,2) + iu0 d∫ 3 φ(1) ˆ φ (2) ˆ φ (3)φ3(3)0

+L

1st order:

€

φ(1) ˆ φ (2) = φ(1) ˆ φ (2) 1− S1 +L( )0

= C0(1,2) + iu0 d∫ 3 φ(1) ˆ φ (2) ˆ φ (3)φ3(3)0

+L

average of product of 6 Gaussian variables: Use Wick’s theoremsum of products of all pairwise averages)

1st order:

€

φ(1) ˆ φ (2) = φ(1) ˆ φ (2) 1− S1 +L( )0

= C0(1,2) + iu0 d∫ 3 φ(1) ˆ φ (2) ˆ φ (3)φ3(3)0

+L


€

3 φ(1) ˆ φ (2)0

d3∫ ˆ φ (3)φ(3)0

φ2(3)0

+3 d3∫ φ(1) ˆ φ (3)0

φ2(3)0

φ(3) ˆ φ (2)0

+6 d3 φ(1)φ(3)0

φ(3) ˆ φ (3)∫0

φ(3) ˆ φ (2)0

(graphs onblackboard)

1st order:

€

φ(1) ˆ φ (2) = φ(1) ˆ φ (2) 1− S1 +L( )0

= C0(1,2) + iu0 d∫ 3 φ(1) ˆ φ (2) ˆ φ (3)φ3(3)0

+L


€

3 φ(1) ˆ φ (2)0

d3∫ ˆ φ (3)φ(3)0

φ2(3)0

+3 d3∫ φ(1) ˆ φ (3)0

φ2(3)0

φ(3) ˆ φ (2)0

+6 d3 φ(1)φ(3)0

φ(3) ˆ φ (3)∫0

φ(3) ˆ φ (2)0


but most of these vanish:

1st order:

€

φ(1) ˆ φ (2) = φ(1) ˆ φ (2) 1− S1 +L( )0

= C0(1,2) + iu0 d∫ 3 φ(1) ˆ φ (2) ˆ φ (3)φ3(3)0

+L


€

3 φ(1) ˆ φ (2)0

d3∫ ˆ φ (3)φ(3)0

φ2(3)0

+3 d3∫ φ(1) ˆ φ (3)0

φ2(3)0

φ(3) ˆ φ (2)0

+6 d3 φ(1)φ(3)0

φ(3) ˆ φ (3)∫0

φ(3) ˆ φ (2)0



€

φ(3) ˆ φ (3) = G( p, t = 0)p

∫ = 0

1st order:

€

φ(1) ˆ φ (2) = φ(1) ˆ φ (2) 1− S1 +L( )0

= C0(1,2) + iu0 d∫ 3 φ(1) ˆ φ (2) ˆ φ (3)φ3(3)0

+L


€

3 φ(1) ˆ φ (2)0

d3∫ ˆ φ (3)φ(3)0

φ2(3)0

+3 d3∫ φ(1) ˆ φ (3)0

φ2(3)0

φ(3) ˆ φ (2)0

+6 d3 φ(1)φ(3)0

φ(3) ˆ φ (3)∫0

φ(3) ˆ φ (2)0



€

φ(3) ˆ φ (3) = G( p, t = 0)p

∫ = 0 (Ito)

Feynman graphs

The surviving term:

€

3 d3∫ φ(1) ˆ φ (3)0

φ2(3)0

φ(3) ˆ φ (2)0

Feynman graphs

1

2

3

G(1,3)

G(3,2)

C(3,3)The surviving term:

€

3 d3∫ φ(1) ˆ φ (3)0

φ2(3)0

φ(3) ˆ φ (2)0

Feynman graphs

1

2

3

G(1,3)

G(3,2)


…

€

3 d3∫ φ(1) ˆ φ (3)0

φ2(3)0

φ(3) ˆ φ (2)0

Feynman graphs

Can generate a diagrammatic expansion like that in Lect 7

1

2

3

G(1,3)

G(3,2)


…

€

3 d3∫ φ(1) ˆ φ (3)0

φ2(3)0

φ(3) ˆ φ (2)0

Feynman graphs

Can generate a diagrammatic expansion like that in Lect 7In fact, it is exactly the same diagrammatic expansion

1

2

3

G(1,3)

G(3,2)


…

€

3 d3∫ φ(1) ˆ φ (3)0

φ2(3)0

φ(3) ˆ φ (2)0

Feynman graphs

Can generate a diagrammatic expansion like that in Lect 7In fact, it is exactly the same diagrammatic expansion(except that ϕ and the correlation and response functionsnow depend on space as well as time)

1

2

3

G(1,3)

G(3,2)


…

€

3 d3∫ φ(1) ˆ φ (3)0

φ2(3)0

φ(3) ˆ φ (2)0

Feynman graphs

Can generate a diagrammatic expansion like that in Lect 7In fact, it is exactly the same diagrammatic expansion(except that ϕ and the correlation and response functionsnow depend on space as well as time)

1

2

3

G(1,3)

G(3,2)


…

€

3 d3∫ φ(1) ˆ φ (3)0

φ2(3)0

φ(3) ˆ φ (2)0

all closed loops of response functions (including alldisconnected diagrams) vanish because for Ito G(t=0) = 0.

Stratonovich, again

€

P[φ] =J[φ]


1

4TΔφn +1 − φn − 1

2 Δ fn + hn( ) − 12 Δ fn +1 + hn +1( )( )

2

n=1

M

∑ ⎡

⎣ ⎢

⎤

⎦ ⎥

Stratonovich, again

€

P[φ] =J[φ]


1

4TΔφn +1 − φn − 1

2 Δ fn + hn( ) − 12 Δ fn +1 + hn +1( )( )

2

n=1

M

∑ ⎡

⎣ ⎢

⎤

⎦ ⎥

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

Stratonovich, again

€

P[φ] =J[φ]


1

4TΔφn +1 − φn − 1

2 Δ fn + hn( ) − 12 Δ fn +1 + hn +1( )( )

2

n=1

M

∑ ⎡

⎣ ⎢

⎤

⎦ ⎥


Use Grassman variables

Stratonovich, again

€

P[φ] =J[φ]


1

4TΔφn +1 − φn − 1

2 Δ fn + hn( ) − 12 Δ fn +1 + hn +1( )( )

2

n=1

M

∑ ⎡

⎣ ⎢

⎤

⎦ ⎥



€

ψ,ψ :

ψψ = −ψ ψ , ψ 2 =ψ 2 = 0,

ψ1ψ 2 = −ψ 2ψ1, etc.

Stratonovich, again

€

P[φ] =J[φ]


1

4TΔφn +1 − φn − 1

2 Δ fn + hn( ) − 12 Δ fn +1 + hn +1( )( )

2

n=1

M

∑ ⎡

⎣ ⎢

⎤

⎦ ⎥



€

ψ,ψ :

ψψ = −ψ ψ , ψ 2 =ψ 2 = 0,

ψ1ψ 2 = −ψ 2ψ1, etc.

Taylor series expansionsterminate:

Stratonovich, again

€

P[φ] =J[φ]


1

4TΔφn +1 − φn − 1

2 Δ fn + hn( ) − 12 Δ fn +1 + hn +1( )( )

2

n=1

M

∑ ⎡

⎣ ⎢

⎤

⎦ ⎥



€

ψ,ψ :

ψψ = −ψ ψ , ψ 2 =ψ 2 = 0,

ψ1ψ 2 = −ψ 2ψ1, etc.

€

exp(aψ ψ ) =1+ aψ ψ


Stratonovich, again

€

P[φ] =J[φ]


1

4TΔφn +1 − φn − 1

2 Δ fn + hn( ) − 12 Δ fn +1 + hn +1( )( )

2

n=1

M

∑ ⎡

⎣ ⎢

⎤

⎦ ⎥



€

ψ,ψ :

ψψ = −ψ ψ , ψ 2 =ψ 2 = 0,

ψ1ψ 2 = −ψ 2ψ1, etc.

“integrals”:

€



Stratonovich, again

€

P[φ] =J[φ]


1

4TΔφn +1 − φn − 1

2 Δ fn + hn( ) − 12 Δ fn +1 + hn +1( )( )

2

n=1

M

∑ ⎡

⎣ ⎢

⎤

⎦ ⎥



€

ψ,ψ :

ψψ = −ψ ψ , ψ 2 =ψ 2 = 0,

ψ1ψ 2 = −ψ 2ψ1, etc.

€

dψdψ

1

ψ

ψ

ψ ψ

⎛

⎝

⎜ ⎜ ⎜ ⎜

⎞

⎠

⎟ ⎟ ⎟ ⎟

∫ =

0

0

0

1

⎛

⎝

⎜ ⎜ ⎜ ⎜

⎞

⎠

⎟ ⎟ ⎟ ⎟

“integrals”:

€



determinants

€

dψdψ ∫ exp ψ iij

∑ Aijψ j

⎛

⎝ ⎜ ⎜

⎞

⎠ ⎟ ⎟= detA

determinants

€


∑ Aijψ j

⎛

⎝ ⎜ ⎜

⎞

⎠ ⎟ ⎟= detA

dx

2π∫ exp − 1

2 x i

ij

∑ Aij x j

⎛

⎝ ⎜ ⎜

⎞

⎠ ⎟ ⎟= detA( )

−1/ 2cf for real x

determinants

€


∑ Aijψ j

⎛

⎝ ⎜ ⎜

⎞

⎠ ⎟ ⎟= detA

dx

2π∫ exp − 1

2 x i

ij

∑ Aij x j

⎛

⎝ ⎜ ⎜

⎞

⎠ ⎟ ⎟= detA( )

−1/ 2

dzdz*

2π∫ exp − 1

2 zi*

ij

∑ Aijz j

⎛

⎝ ⎜ ⎜

⎞

⎠ ⎟ ⎟= detA( )

−1

cf for real x

and for complex z

determinants

€


∑ Aijψ j

⎛

⎝ ⎜ ⎜

⎞

⎠ ⎟ ⎟= detA

dx

2π∫ exp − 1

2 x i

ij

∑ Aij x j

⎛

⎝ ⎜ ⎜

⎞

⎠ ⎟ ⎟= detA( )

−1/ 2

dzdz*

2π∫ exp − 1

2 zi*

ij

∑ Aijz j

⎛

⎝ ⎜ ⎜

⎞

⎠ ⎟ ⎟= detA( )

−1

cf for real x

and for complex z

€

J[φ] = det∂η

∂φ

⎛

⎝ ⎜

⎞

⎠ ⎟= det

∂

∂t− ′ f (φ)

⎛

⎝ ⎜

⎞

⎠ ⎟

= DψDψ exp dt dd xψ (t)∂

∂t− ′ f (φ)

⎛

⎝ ⎜

⎞

⎠ ⎟ψ (t)∫

⎡

⎣ ⎢

⎤

⎦ ⎥∫

so representJ as

determinants

€


∑ Aijψ j

⎛

⎝ ⎜ ⎜

⎞

⎠ ⎟ ⎟= detA

dx

2π∫ exp − 1

2 x i

ij

∑ Aij x j

⎛

⎝ ⎜ ⎜

⎞

⎠ ⎟ ⎟= detA( )

−1/ 2

dzdz*

2π∫ exp − 1

2 zi*

ij

∑ Aijz j

⎛

⎝ ⎜ ⎜

⎞

⎠ ⎟ ⎟= detA( )

−1

cf for real x

and for complex z

€

J[φ] = det∂η

∂φ

⎛

⎝ ⎜

⎞

⎠ ⎟= det

∂

∂t− ′ f (φ)

⎛

⎝ ⎜

⎞

⎠ ⎟

= DψDψ exp dt dd xψ (t)∂

∂t− ′ f (φ)

⎛

⎝ ⎜

⎞

⎠ ⎟ψ (t)∫

⎡

⎣ ⎢

⎤

⎦ ⎥∫

so representJ as

“ghost” variables

Stratonovich generating functional

€

Z[h,θ] = DφD ˆ φ DψDψ exp −S[φ, ˆ φ ,ψ ,ψ ,h,θ]( )∫(one variable)


€

Z[h,θ] = DφD ˆ φ DψDψ exp −S[φ, ˆ φ ,ψ ,ψ ,h,θ]( )∫

€

−S[φ, ˆ φ ,ψ ,ψ ,h,θ] = d∫ t −T ˆ φ 2(t) + i ˆ φ (t) − ˙ φ (t) + f (φ) + h(t)( ) + iθ(t)φ(t)[ ]

+ dt ψ (t) ∂t − ′ f (φ)( )ψ (t)∫

(one variable)


€


€


+ dt ψ (t) ∂t − ′ f (φ)( )ψ (t)∫

(one variable)

(field)

€

−S[φ, ˆ φ ,ψ ,ψ ,h,θ] =

d∫ t dd x −T ˆ φ 2 + i ˆ φ − ˙ φ − r0φ − u0φ3 +∇ 2φ + h( ) +ψ ∂t + r0 + 3u0φ

2 −∇ 2( )ψ + iθφ[ ]


€


€


+ dt ψ (t) ∂t − ′ f (φ)( )ψ (t)∫

(one variable)

(field)

€

−S[φ, ˆ φ ,ψ ,ψ ,h,θ] =


2 −∇ 2( )ψ + iθφ[ ]

free action:

€

−S0 = d∫ t dd x −T ˆ φ 2 − i ˆ φ ∂t + r0 −∇ 2( )φ +ψ ∂t + r0 −∇ 2

( )ψ[ ]


€


€


+ dt ψ (t) ∂t − ′ f (φ)( )ψ (t)∫

(one variable)

(field)

€

−S[φ, ˆ φ ,ψ ,ψ ,h,θ] =


2 −∇ 2( )ψ + iθφ[ ]

free action:

€

−S0 = d∫ t dd x −T ˆ φ 2 − i ˆ φ ∂t + r0 −∇ 2( )φ +ψ ∂t + r0 −∇ 2

( )ψ[ ]

interactions:

€

−S1 = u0 d∫ t dd x −i ˆ φ φ3 + 3ψ φ2ψ( )

ghost correlations:

€

ψ −p,−ωψ pω 0=

−1

−iω + r0 + p2

ghost correlations:

€


−1

−iω + r0 + p2

Now when we expand

€


= DφD ˆ φ ∫ φ(1) ˆ φ (2)exp −S1( )exp −S0( ) = φ(1) ˆ φ (2)exp −S1( )0

ghost correlations:

€


−1

−iω + r0 + p2

Now when we expand

€



we get

€

φ(1) ˆ φ (2) 1− S1 +L( )0

= C0(1,2)

+iu0 d∫ 3 φ(1) ˆ φ (2) ˆ φ (3)φ3(3)0

− 3u0 d3∫ φ(1) ˆ φ (2)ψ (3)φ2(3)ψ (3)0

ghost correlations:

€


−1

−iω + r0 + p2

Now when we expand

€



we get

€

φ(1) ˆ φ (2) 1− S1 +L( )0

= C0(1,2)

+iu0 d∫ 3 φ(1) ˆ φ (2) ˆ φ (3)φ3(3)0

− 3u0 d3∫ φ(1) ˆ φ (2)ψ (3)φ2(3)ψ (3)0

€

3 φ(1) ˆ φ (2)0

d3∫ ψ (3)ψ (3)0

φ2(3)0

+6 d3 φ(1)φ(3)0

ψ (3)ψ (3)∫0

φ(3) ˆ φ (2)0

new terms

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

cancellation of closed loops

€

3 φ(1) ˆ φ (2)0

d3∫ ˆ φ (3)φ(3)0

φ2(3)0

+6 d3 φ(1)φ(3)0

φ(3) ˆ φ (3)∫0

φ(3) ˆ φ (2)0

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


€

3 φ(1) ˆ φ (2)0

d3∫ ˆ φ (3)φ(3)0

φ2(3)0

+6 d3 φ(1)φ(3)0

φ(3) ˆ φ (3)∫0

φ(3) ˆ φ (2)0


that were zero with Ito convention but not Stratonovich


€

3 φ(1) ˆ φ (2)0

d3∫ ˆ φ (3)φ(3)0

φ2(3)0

+6 d3 φ(1)φ(3)0

φ(3) ˆ φ (3)∫0

φ(3) ˆ φ (2)0


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


€

Φ=φ+ξ ψ +ψ ξ −iξ ξ ˆ φ

€

ξ,ξ Grassman numbers

the superfield

€

f (φ) = −∂V (φ)

∂φ


Then if

€


€


the superfield

€

f (φ) = −∂V (φ)

∂φ


Then if

€

Z = DΦexp −S Φ[ ]( )∫

S = dt dξ dξ ∫ ∂Φ

∂ξT

∂Φ

∂ξ −ξ

∂Φ

∂t

⎛

⎝ ⎜

⎞

⎠ ⎟+ V Φ(t,ξ ,ξ )( )

⎡

⎣ ⎢

⎤

⎦ ⎥

≡ dτ∫ ∂Φ

∂ξT

∂Φ

∂ξ −ξ

∂Φ

∂t

⎛

⎝ ⎜

⎞

⎠ ⎟+ V Φ(τ )( )

⎡

⎣ ⎢

⎤

⎦ ⎥

€


€


the generating functional can be written

How does this happen?

Expand the potential term:



€

Φ=φ+ξ ψ +ψ ξ + iξ ξ ˆ φ ⇒

V (Φ) = V (φ) + (ξ ψ +ψ ξ + iξ ξ ˆ φ ) ′ V (φ) + 12 (ξ ψ +ψ ξ + iξ ξ ˆ φ )2 ′ ′ V (φ)



€



Integrate over the “Grassman time”



€



€

dξ dξ ∫ V (Φ) = i ˆ φ ′ V (φ) +ψψ ′ ′ V (φ) = −i ˆ φ f (φ) +ψ ′ f (φ)ψ




€



€

dξ dξ ∫ V (Φ) = i ˆ φ ′ V (φ) +ψψ ′ ′ V (φ) = −i ˆ φ f (φ) +ψ ′ f (φ)ψ


which are the terms in the action involving f.

Lecture 13: Field-theoretic formulation of Langevin models Outline: Functional (path) integral...

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Transcript of Lecture 13: Field-theoretic formulation of Langevin models Outline: Functional (path) integral...