1. 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

2. generating functionals
In the equilibrium case, the partition function

3. generating functionals
In the equilibrium case, the partition function

4. generating functionals
In the equilibrium case, the partition function

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

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

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

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

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

10. 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

11. Dynamics (single variable)
Start from the equation of motion

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

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

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

15. equations of motion
Ito:

16. equations of motion
Ito:

17. equations of motion
Ito:
Stratonovich:

18. equations of motion
Ito:
Stratonovich:

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

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

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

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

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

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

25. 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

26. 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)

27. Stratonovich

28. Stratonovich

29. Stratonovich
Jacobian:

30. Stratonovich
Jacobian:

31. Stratonovich
Jacobian:

32. Stratonovich
Jacobian:

33. Stratonovich
Jacobian:

34. Stratonovich
Jacobian:
(back to this later)

35. Martin-Siggia-Rose
back to Ito:

36. Martin-Siggia-Rose
back to Ito:

37. Martin-Siggia-Rose
back to Ito:
use

38. Martin-Siggia-Rose
back to Ito:
use

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

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

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

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

43. a field theory:
Δ -> 0:
“action”

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

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

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

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

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

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

50. 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))

51. correlation functions

52. correlation functions
magnetization

53. correlation functions
magnetization

54. correlation functions
magnetization
correlation functions:

55. correlation functions
magnetization
correlation functions:

56. correlation functions
magnetization
correlation functions:

57. correlation functions
magnetization
correlation functions:

58. correlation functions
magnetization
correlation functions:
= susceptibility / response function

59. correlation functions
magnetization
correlation functions:
= susceptibility / response function

60. correlation functions
magnetization
correlation functions:
= susceptibility / response function

61. free action

62. free action
free action:

63. free action
free action:

64. free action
free action:
generating functional:

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

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

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

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

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

70. 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

71. perturbation theory

72. perturbation theory
want to evaluate quantities like

73. perturbation theory
want to evaluate quantities like
expand

74. perturbation theory
want to evaluate quantities like
expand

75. 1st order:

76. 1st order:

77. 1st order:
average of product of 6 Gaussian variables: Use Wick’s theorem
sum of products of all pairwise averages)

78. 1st order:
average of product of 6 Gaussian variables: Use Wick’s theorem
sum of products of all pairwise averages)
(graphs on
blackboard)

79. 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:

80. 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:

81. 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)

82. Feynman graphs
The surviving term:

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

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

85. 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

86. 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

87. 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)

88. 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.

89. Stratonovich, again

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

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

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

93. 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:

94. 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:

95. 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”:

96. 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”:

97. determinants

98. determinants cf
for real x

99. determinants cf
for real x
and
for complex z

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

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

102. Stratonovich generating functional
(one variable)

103. Stratonovich generating functional
(one variable)

104. Stratonovich generating functional
(one variable)
(field)

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

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

107. ghost correlations:

108. ghost correlations:
Now when we expand

109. ghost correlations:
Now when we expand
we get

110. ghost correlations:
Now when we expand
we get
new terms

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

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

113. 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

114. 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

115. the superfield
Define a combination of the real and Grassman fields

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

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

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

119. How does this happen?
Expand the potential term:

120. How does this happen?
Expand the potential term:

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

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

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