Pensieve header: Formal Gaussian integration over an arbitrary “Feynman Ring”; last to use Picard iteration.

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.

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[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))] ];

The Basic Feynman Ring

𝒮={x,(Subscript[x, _]),y,z};
(Subscript[q, x_,y_])[f_]:=((∂_(x,y))f)/.Thread[𝒮->0];
(Subscript[θ, x_,y_]):=x y;
f_≡0:=f===0;
(Subscript[Ev, vs_List->0])[f_]:=CF[f/.Thread[vs->0]]

The ϵSeries Feynman Ring

𝒮={x,y,z,ϕ,(Subscript[x, _]),(Subscript[p, _]),(Subscript[(UnderBar[x]), _]),(Subscript[(UnderBar[p]), _])};(Subscript[q, x_,y_])[ser_ϵSeries]:=((∂_(x,y))ser[[1]])/.Thread[𝒮->0];
(Subscript[θ, x_,y_]):=x y;
ϵSeries/:D[ser_ϵSeries,vs___]:=D[#,vs]&/@ser;
ϵSeries/:s1_ϵSeries+s2_ϵSeries:=ϵSeries@@Table[s1[[kk]]+s2[[kk]],{kk,Min[Length@s1,Length@s2]}];
ϵSeries/:t_+ser_ϵSeries:=MapAt[(#+t)&,ser,1];
ϵSeries/:s1_ϵSeries*s2_ϵSeries:=ϵSeries@@Table[Sum[s1[[ii+1]]s2[[kk-ii+1]],{ii,0,kk}],{kk,0,Min[Length@s1,Length@s2]-1}];
ϵSeries/:c_*ser_ϵSeries:=(c#)&/@ser;
ser_ϵSeries ≡ 0 := And@@((#===0)&/@ser);
ϵSeries/:Integrate[ser_ϵSeries,pars__]:=ϵSeries@@(Integrate[#,pars]&/@ser);
ϵSeries/:(Subscript[Ev, vs_List->0])[ser_ϵSeries]:=ser/.Thread[vs->0];
CF[ser_ϵSeries]:=CF/@ser;

Integration

Using Picard Iteration!

𝔼/:𝔼[A_]𝔼[B_]:=𝔼[A+B]

𝔼[sd_SeriesData]/;(List@@sd)[[{1,2,4,6}]]==={ϵ,0,0,1}:=𝔼[ϵSeries@@PadRight[sd[[3]],sd[[5]],0]]

Unprotect[Integrate];
Integrate::sing="How dare you ask me to integrate a singular Gaussian!";
∫ω_.𝔼[L_]\[DifferentialD](vs_List):=Module[{n,Q,Δ,G,Z,e,λ,a,b},
n=Length@vs;
Q=Table[(Subscript[q, vs[[a]],vs[[b]]])[L],{a,n},{b,n}];
If[(Δ=CF@Det[-Q])==0,Message[Integrate::sing]; Return[]];
Z=CF[L-Sum[Q[[a,b]](Subscript[θ, vs[[a]],vs[[b]]]),{a,n},{b,n}]/2];
G=Inverse[Q]/2;
While[e=D[Z,λ];Do[e+=G[[a,b]](D[Z,vs[[a]],vs[[b]]]+D[Z,vs[[a]]]D[Z,vs[[b]]]),{a,n},{b,n}];
e=CF[e];
!(e≡0),Z=CF[Z-(∫_0^λ)e \[DifferentialD]λ]
];
PowerExpand@Factor[ω(((Δ(((2π))^n)))^(-1/2))]𝔼[CF[(Subscript[Ev, vs->0])[Z]/.λ->1]]
];
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 ϵSeries

∫𝔼[-(x^2)/2+ϵ (x^3)/6+((O[ϵ])^7)]\[DifferentialD]{x}

∫𝔼[-(ϕ^2)/2+ϵ (ϕ^4)/24+((O[ϵ])^7)]\[DifferentialD]{ϕ}

∫𝔼[p x+ϵ (p^2)x+((O[ϵ])^5)]\[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}]