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
€
Z[β,h] = Dφexp∫ −βE[φ]( )
generating functionals
In the equilibrium case, the partition function
€
Z[β,h] = Dφexp∫ −βE[φ]( )
€
= Dφexp − 12 dd x∫ r0φ
2(x) + 12 u0φ
4 (x) + (∇φ(x))2 − 2h(x)φ(x)[ ]{ }∫
generating functionals
In the equilibrium case, the partition function
€
Z[β,h] = Dφexp∫ −βE[φ]( )
€
= Dφexp − 12 dd x∫ r0φ
2(x) + 12 u0φ
4 (x) + (∇φ(x))2 − 2h(x)φ(x)[ ]{ }∫is a generating functional:
generating functionals
In the equilibrium case, the partition function
€
Z[β,h] = Dφexp∫ −βE[φ]( )
€
= Dφexp − 12 dd x∫ r0φ
2(x) + 12 u0φ
4 (x) + (∇φ(x))2 − 2h(x)φ(x)[ ]{ }∫is a generating functional:
€
∂ logZ
∂β= E
generating functionals
In the equilibrium case, the partition function
€
Z[β,h] = Dφexp∫ −βE[φ]( )
€
= Dφexp − 12 dd x∫ r0φ
2(x) + 12 u0φ
4 (x) + (∇φ(x))2 − 2h(x)φ(x)[ ]{ }∫is a generating functional:
€
∂ logZ
∂β= E
€
∂2 logZ
∂β 2= E 2 − E
2
generating functionals
In the equilibrium case, the partition function
€
Z[β,h] = Dφexp∫ −βE[φ]( )
€
= Dφexp − 12 dd x∫ r0φ
2(x) + 12 u0φ
4 (x) + (∇φ(x))2 − 2h(x)φ(x)[ ]{ }∫is a generating functional:
€
δ logZ
δh(x)= φ(x)
€
∂ logZ
∂β= E
€
∂2 logZ
∂β 2= E 2 − E
2
generating functionals
In the equilibrium case, the partition function
€
Z[β,h] = Dφexp∫ −βE[φ]( )
€
= Dφexp − 12 dd x∫ r0φ
2(x) + 12 u0φ
4 (x) + (∇φ(x))2 − 2h(x)φ(x)[ ]{ }∫is a generating functional:
€
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
generating functionals
In the equilibrium case, the partition function
€
Z[β,h] = Dφexp∫ −βE[φ]( )
€
= Dφexp − 12 dd x∫ r0φ
2(x) + 12 u0φ
4 (x) + (∇φ(x))2 − 2h(x)φ(x)[ ]{ }∫is a generating functional:
€
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)
Dynamics (single variable)
Start from the equation of motion
€
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,
Dynamics (single variable)
Start from the equation of motion
€
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
∫
Dynamics (single variable)
Start from the equation of motion
€
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
€
φn +1 − φn = Δ f (φn ) + hn( ) + η nIto:
Stratonovich:
equations of motion
€
φn +1 − φn = Δ f (φn ) + hn( ) + η nIto:
€
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 = Δ f (φn ) + hn( ) + η nIto:
€
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 = Δ f (φn ) + hn( ) + η nIto:
€
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):
€
P[φ] =J[φ]
(4πTΔ)M / 2exp −
1
4TΔφn +1 − φn − f (φn ) + hn[ ]Δ( )
2
n=1
M
∑ ⎡
⎣ ⎢
⎤
⎦ ⎥
equations of motion
€
φn +1 − φn = Δ f (φn ) + hn( ) + η nIto:
€
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):
€
P[φ] =J[φ]
(4πTΔ)M / 2exp −
1
4TΔφn +1 − φn − f (φn ) + hn[ ]Δ( )
2
n=1
M
∑ ⎡
⎣ ⎢
⎤
⎦ ⎥ (Ito)
equations of motion
€
φn +1 − φn = Δ f (φn ) + hn( ) + η nIto:
€
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):
€
P[φ] =J[φ]
(4πTΔ)M / 2exp −
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 = Δ f (φn ) + hn( ) + η nIto:
€
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):
€
P[φ] =J[φ]
(4πTΔ)M / 2exp −
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:
______diagonal
equations of motion
€
φn +1 − φn = Δ f (φn ) + hn( ) + η nIto:
€
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):
€
P[φ] =J[φ]
(4πTΔ)M / 2exp −
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:
______diagonal
______________1 belowdiagonal
equations of motion
€
φn +1 − φn = Δ f (φn ) + hn( ) + η nIto:
€
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):
€
P[φ] =J[φ]
(4πTΔ)M / 2exp −
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
______________1 belowdiagonal
equations of motion
€
φn +1 − φn = Δ f (φn ) + hn( ) + η nIto:
€
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):
€
P[φ] =J[φ]
(4πTΔ)M / 2exp −
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
______________1 belowdiagonal
(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]
(4πTΔ)M / 2exp −
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]
(4πTΔ)M / 2exp −
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]
(4πTΔ)M / 2exp −
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]
(4πTΔ)M / 2exp −
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]
(4πTΔ)M / 2exp −
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]
(4πTΔ)M / 2exp −
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]
(4πTΔ)M / 2exp −
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
(4πTΔ)M / 2exp −
1
4TΔφn +1 − φn − f (φn ) + hn[ ]Δ( )
2
n=1
M
∑ ⎡
⎣ ⎢
⎤
⎦ ⎥back to Ito:
Martin-Siggia-Rose
€
P[φ | h] =1
(4πTΔ)M / 2exp −
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
(4πTΔ)M / 2exp −
1
4TΔφn +1 − φn − f (φn ) + hn[ ]Δ( )
2
n=1
M
∑ ⎡
⎣ ⎢
⎤
⎦ ⎥back to Ito:
€
exp − 14 a2
( ) =dy
2πexp −y 2 + iya( )
−∞
∞
∫ ⇒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
(4πTΔ)M / 2exp −
1
4TΔφn +1 − φn − f (φn ) + hn[ ]Δ( )
2
n=1
M
∑ ⎡
⎣ ⎢
⎤
⎦ ⎥back to Ito:
€
exp − 14 a2
( ) =dy
2πexp −y 2 + iya( )
−∞
∞
∫ ⇒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
∑ ⎡
⎣ ⎢
⎤
⎦ ⎥
generating function (multivariate characteristic function)
Martin-Siggia-Rose
€
P[φ | h] =1
(4πTΔ)M / 2exp −
1
4TΔφn +1 − φn − f (φn ) + hn[ ]Δ( )
2
n=1
M
∑ ⎡
⎣ ⎢
⎤
⎦ ⎥back to Ito:
€
exp − 14 a2
( ) =dy
2πexp −y 2 + iya( )
−∞
∞
∫ ⇒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
∑ ⎡
⎣ ⎢
⎤
⎦ ⎥
generating function (multivariate characteristic function)
€
Z[h,θ] = dφn
n=1
M
∏∫ e iθ nφn
n=1
M
∏ ⎛
⎝ ⎜
⎞
⎠ ⎟P[φ]
Martin-Siggia-Rose
€
P[φ | h] =1
(4πTΔ)M / 2exp −
1
4TΔφn +1 − φn − f (φn ) + hn[ ]Δ( )
2
n=1
M
∑ ⎡
⎣ ⎢
⎤
⎦ ⎥back to Ito:
€
exp − 14 a2
( ) =dy
2πexp −y 2 + iya( )
−∞
∞
∫ ⇒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
∑ ⎡
⎣ ⎢
⎤
⎦ ⎥
generating function (multivariate characteristic function)
€
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:
€
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”
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
€
f (x, t) = − r0 −∇ 2( )φ(x, t) − u0φ
3(x, t)
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)
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:
€
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
_____inter-actionterm L1
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
_____inter-actionterm L1
________“source” terms
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
_____inter-actionterm L1
________“source” terms
note:
€
Z[h,0] = DφD ˆ φ exp −S[φ, ˆ φ ,h,θ]( )∫ =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
_____inter-actionterm L1
________“source” terms
note:
€
Z[h,0] = DφD ˆ φ exp −S[φ, ˆ φ ,h,θ]( )∫ =1 (normalization of P(ϕ|h))
correlation functions
correlation functions
magnetization
correlation functions
€
M(x, t) = φ(x, t) = −i limθ →0
δZ
δθ(x, t)magnetization
correlation functions
€
M(x, t) = φ(x, t) = −i limθ →0
δZ
δθ(x, t)magnetization
correlation functions:
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
δθ(x, t)magnetization
correlation functions:
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
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
correlation functions:
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
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
correlation functions:
€
=δM(x, t)
δh( ′ x , ′ t )
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
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
correlation functions:
€
=δM(x, t)
δh( ′ x , ′ t )= susceptibility / response function
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
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
correlation functions:
€
=δM(x, t)
δh( ′ x , ′ t )= susceptibility / response function
€
ˆ φ (x, t) = −limθ →0
δZ
δh(x, t)= 0
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
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
correlation functions:
€
=δM(x, t)
δh( ′ x , ′ t )= susceptibility / response function
€
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
⎛
⎝
⎜ ⎜ ⎜ ⎜
⎞
⎠
⎟ ⎟ ⎟ ⎟
(invert the matrix in the exponent in S0)
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
⎛
⎝
⎜ ⎜ ⎜ ⎜
⎞
⎠
⎟ ⎟ ⎟ ⎟
(invert the matrix in the exponent in S0)
back to time domain:
€
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
⎛
⎝
⎜ ⎜ ⎜ ⎜
⎞
⎠
⎟ ⎟ ⎟ ⎟
(invert the matrix in the exponent in S0)
back to time domain:
€
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)( )
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)( )
€
=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
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.
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
average of product of 6 Gaussian variables: Use Wick’s theoremsum of products of all pairwise averages)
€
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
average of product of 6 Gaussian variables: Use Wick’s theoremsum of products of all pairwise averages)
€
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)
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
average of product of 6 Gaussian variables: Use Wick’s theoremsum of products of all pairwise averages)
€
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)
but most of these vanish:
€
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
average of product of 6 Gaussian variables: Use Wick’s theoremsum of products of all pairwise averages)
€
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)
but most of these vanish:
€
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)
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
Can generate a diagrammatic expansion like that in Lect 7
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
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)
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
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)
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
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)
C(3,3)The surviving term:
…
€
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]
(4πTΔ)M / 2exp −
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]
(4πTΔ)M / 2exp −
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]
(4πTΔ)M / 2exp −
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.
Use Grassman variables
Stratonovich, again
€
P[φ] =J[φ]
(4πTΔ)M / 2exp −
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.
Use Grassman variables
€
ψ,ψ :
ψψ = −ψ ψ , ψ 2 =ψ 2 = 0,
ψ1ψ 2 = −ψ 2ψ1, etc.
Stratonovich, again
€
P[φ] =J[φ]
(4πTΔ)M / 2exp −
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.
Use Grassman variables
€
ψ,ψ :
ψψ = −ψ ψ , ψ 2 =ψ 2 = 0,
ψ1ψ 2 = −ψ 2ψ1, etc.
Taylor series expansionsterminate:
Stratonovich, again
€
P[φ] =J[φ]
(4πTΔ)M / 2exp −
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.
Use Grassman variables
€
ψ,ψ :
ψψ = −ψ ψ , ψ 2 =ψ 2 = 0,
ψ1ψ 2 = −ψ 2ψ1, etc.
€
exp(aψ ψ ) =1+ aψ ψ
Taylor series expansionsterminate:
Stratonovich, again
€
P[φ] =J[φ]
(4πTΔ)M / 2exp −
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.
Use Grassman variables
€
ψ,ψ :
ψψ = −ψ ψ , ψ 2 =ψ 2 = 0,
ψ1ψ 2 = −ψ 2ψ1, etc.
“integrals”:
€
exp(aψ ψ ) =1+ aψ ψ
Taylor series expansionsterminate:
Stratonovich, again
€
P[φ] =J[φ]
(4πTΔ)M / 2exp −
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.
Use Grassman variables
€
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”:
€
exp(aψ ψ ) =1+ aψ ψ
Taylor series expansionsterminate:
determinants
€
dψdψ ∫ exp ψ iij
∑ Aijψ j
⎛
⎝ ⎜ ⎜
⎞
⎠ ⎟ ⎟= detA
determinants
€
dψdψ ∫ exp ψ iij
∑ Aijψ j
⎛
⎝ ⎜ ⎜
⎞
⎠ ⎟ ⎟= detA
dx
2π∫ exp − 1
2 x i
ij
∑ Aij x j
⎛
⎝ ⎜ ⎜
⎞
⎠ ⎟ ⎟= detA( )
−1/ 2cf for real x
determinants
€
dψdψ ∫ exp ψ iij
∑ 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
€
dψdψ ∫ exp ψ iij
∑ 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
€
dψdψ ∫ exp ψ iij
∑ 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)
Stratonovich generating functional
€
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)
Stratonovich generating functional
€
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)
(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[ ]
Stratonovich generating functional
€
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)
(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[ ]
free action:
€
−S0 = d∫ t dd x −T ˆ φ 2 − i ˆ φ ∂t + r0 −∇ 2( )φ +ψ ∂t + r0 −∇ 2
( )ψ[ ]
Stratonovich generating functional
€
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)
(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[ ]
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:
€
ψ −p,−ωψ pω 0=
−1
−iω + r0 + p2
Now when we expand
€
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
ghost correlations:
€
ψ −p,−ωψ pω 0=
−1
−iω + r0 + p2
Now when we expand
€
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
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:
€
ψ −p,−ωψ pω 0=
−1
−iω + r0 + p2
Now when we expand
€
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
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
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
that were zero with Ito convention but not Stratonovich
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
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
€
Φ=φ+ξ ψ +ψ ξ −iξ ξ ˆ φ
€
ξ,ξ Grassman numbers
the superfield
€
f (φ) = −∂V (φ)
∂φ
Define a combination of the real and Grassman fields
Then if
€
Φ=φ+ξ ψ +ψ ξ −iξ ξ ˆ φ
€
ξ,ξ Grassman numbers
the superfield
€
f (φ) = −∂V (φ)
∂φ
Define a combination of the real and Grassman fields
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 )( )
⎡
⎣ ⎢
⎤
⎦ ⎥
€
Φ=φ+ξ ψ +ψ ξ −iξ ξ ˆ φ
€
ξ,ξ Grassman numbers
the generating functional can be written
How does this happen?
Expand the potential term:
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)
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”
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)
€
dξ dξ ∫ V (Φ) = i ˆ φ ′ V (φ) +ψψ ′ ′ V (φ) = −i ˆ φ f (φ) +ψ ′ f (φ)ψ
Integrate over the “Grassman time”
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)
€
dξ dξ ∫ V (Φ) = i ˆ φ ′ V (φ) +ψψ ′ ′ V (φ) = −i ˆ φ f (φ) +ψ ′ f (φ)ψ
Integrate over the “Grassman time”
which are the terms in the action involving f.
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