Pensieve header: Nilpotent Gaussian integration.

Initialization

CCF[ℰ_] := ExpandDenominator@ExpandNumerator@Together[ℰ];
CCF[ℰ_] := Factor[ℰ];
CF[ω_.ℰ_𝔼]:=CF[ω]CF/@ℰ;
CF[ℰ_List]:=CF/@ℰ;
CF[sd_SeriesData] := MapAt[CF,sd,3];
CF[ℰ_] := Module[{vs=Cases[ℰ,(Subscript[(x|p), __]),∞]⋃{x,p,ϵ},ps,c},
Total[CoefficientRules[Expand[ℰ], vs] /. (ps_->c_) :> CCF[c](Times@@(vs^ps))] ];

Integration

Using Picard Iteration!

𝔼/:𝔼[A_]𝔼[B_]:=𝔼[A+B]

$π=Identity;

What must a Feynman Ring F have? (Over some set of labels S with elements x,y,...)
* A vector space over ℚ.
* Has a symmetric linear Z|->(∂_(x,y))Z and a symmetric bilinear ((Subscript[Z, 1]),(Subscript[Z, 2]))|->((∂_x)(Subscript[Z, 1]))((∂_y)(Subscript[Z, 2])) that satisfy the axioms of (roughly) a connected circuit algebra.
* Has (Subscript[q, x,y]):F->ℚ in some sense dual to some (Subscript[θ, x,y])∈F.
* Has (Subscript[Ev, vs->0]):F->F.
Further axioms must be worked out.
Goals.
* Define ∫.
* Prove a Fubini theorem.
* Prove a theorem about the injectivity of the Laplace transform.

Unprotect[Integrate];
Integrate::sing="How dare you ask me to integrate a singular Gaussian!";
∫ω_.𝔼[L_]\[DifferentialD](vs_List):=Module[{n,L0,Q,Δ,G,Z,e,λ,a,b},
n=Length@vs; L0=L/.ϵ->0;
Q=Table[(-(∂_(vs[[a]],vs[[b]]))L0)/.Thread[vs->0]/.(Subscript[(p|x), __])->0,{a,n},{b,n}];
If[(Δ=CF@Det[Q])==0,Message[Integrate::sing]; Return[]];
Z=CF@$π[L+vs.Q.vs/2]; G=Inverse[Q]/2;
While[
e=CF@$π[((∂_λ)Z)-Sum[G[[a,b]](((∂_(vs[[a]],vs[[b]]))Z)+((∂_(vs[[a]]))Z)((∂_(vs[[b]]))Z)),
{a,n},{b,n}]];
0=!=e,Z-=(∫_0^λ)e \[DifferentialD]λ
];
PowerExpand@Factor[ω(((Δ(((2π))^n)))^(-1/2))]𝔼[CF[Z/.λ->1/.Thread[vs->0]]]
];
Protect[Integrate];

∫𝔼[I λ (Subscript[x, 1]^2)/2]\[DifferentialD]{(Subscript[x, 1])}

∫𝔼[-I λ (Subscript[x, 1]^2)/2]\[DifferentialD]{(Subscript[x, 1])}

∫𝔼[(I/2){(Subscript[x, 1]),(Subscript[x, 2])}.(({{a, b}, {b, c}})).{(Subscript[x, 1]),(Subscript[x, 2])}]\[DifferentialD]{(Subscript[x, 1]),(Subscript[x, 2])}

∫𝔼[-λ (Subscript[x, 1]^2)/2]\[DifferentialD]{(Subscript[x, 1])}

∫𝔼[-(Subscript[x, 1]^2)/2+ξ (Subscript[x, 1])]\[DifferentialD]{(Subscript[x, 1])}

∫𝔼[-(1/2){(Subscript[x, 1]),(Subscript[x, 2])}.(({{a, b}, {b, c}})).{(Subscript[x, 1]),(Subscript[x, 2])}+{(Subscript[ξ, 1]),(Subscript[ξ, 2])}.{(Subscript[x, 1]),(Subscript[x, 2])}]\[DifferentialD]{(Subscript[x, 1]),(Subscript[x, 2])}

I1=∫𝔼[-(1/2){(Subscript[x, 1]),(Subscript[x, 2])}.(({{a, b}, {b, c}})).{(Subscript[x, 1]),(Subscript[x, 2])}+{(Subscript[ξ, 1]),(Subscript[ξ, 2])}.{(Subscript[x, 1]),(Subscript[x, 2])}]\[DifferentialD]{(Subscript[x, 1])}

∫I1 \[DifferentialD]{(Subscript[x, 2])}

∫𝔼[ξ x+η y+z(x-y)+(x^2)]\[DifferentialD]{x,z}

Integration of ϵ-Series

Block[{$π=Normal[#+((O[ϵ])^7)]&},
∫𝔼[-(x^2)/2+ϵ (x^3)/6]\[DifferentialD]{x}]

Block[{$π=Normal[#+((O[ϵ])^7)]&},
∫𝔼[-(ϕ^2)/2+ϵ (ϕ^4)/24]\[DifferentialD]{ϕ}]

Block[{$π=Normal[#+((O[ϵ])^5)]&},
∫𝔼[p x+ϵ (p^2)x]\[DifferentialD]{p,x}]

Block[{$π=Total@Select[MonomialList[#,{ϵ, x,p}],
mon|->And[
Exponent[mon,ϵ]<=2,
Exponent[mon,x]==Exponent[mon,p]
]
]&},
∫𝔼[p x+a (x^2)p+ϵ b (x^3)(p^3)]\[DifferentialD]{p,x}]

Block[{$π=Total@Select[MonomialList[#,{ϵ,x,p}],
mon|->And[
Exponent[mon,ϵ]<4,
Exponent[mon,x]-Exponent[mon,p]<=3
]
]&},
∫𝔼[p x+a (x^2)p+ϵ b (p^2)x]\[DifferentialD]{p,x}]

Block[{$π=Total@Select[MonomialList[#,{ϵ,x,p}],
mon|->And[
Exponent[mon,ϵ]<4,
Exponent[mon,x]-Exponent[mon,p]<=3-Exponent[mon,ϵ]
]
]&},
∫𝔼[p x+a (x^2)p+ϵ b (p^2)x]\[DifferentialD]{p,x}]

MatrixForm@Table[
∫𝔼[(Subscript[x, 1])(Subscript[p, 2])+(Subscript[x, 2])(Subscript[p, 3])+(Subscript[x, 3])(Subscript[p, 1])+(Subscript[ξ, i])(Subscript[x, i])+(Subscript[π, j])(Subscript[p, j])]\[DifferentialD]{(Subscript[x, 1]),(Subscript[x, 2]),(Subscript[x, 3]),(Subscript[p, 1]),(Subscript[p, 2]),(Subscript[p, 3])},
{i,3},{j,3}]