Yang-Mills Measure and Axial Gauge Fixing on R4 Adrian P. C. Lim 7 Email: [email protected] 1 0 2 Abstract n a Letgbeasemi-simpleLiealgebra. Forag-valued1-formA,considertheYang-Millsaction J 0 S (A)= |dA+A∧A|2 YM 1 ZR4 using the standard metric on T∗R4. Using axial gauge fixing, we want to make sense of the ] R following path integral, P . Tr Texp A e−21SYM(A) DA, th ZA∈AR4,g/G (cid:20)ZC (cid:21) a wherebyDAissomeLebesguetypeofmeasureonthespaceofg-valued1-forms,modulogauge m transformations AR4,g/G. Here T is the time ordering operator. [ We will construct an Abstract Wiener space for which we can define the Yang-Mills path 2 integral rigorously, both in the Abelian and non-Abelian cases. Subsequently, we will then v derive the Wilson Area Law formula, in both the Abelian and non-Abelian cases from these 9 definitions. OneofthemostimportantapplicationsoftheAreaLawformulawillbetoexplain 2 why thepotential measured between a quarkand antiquark is a linear potential. 5 1 MSC 2010: 81T13, 81T08 0 Keywords: Yang-Mills measure, axial gauge fixing, area law, quark confinement,vortex . 1 0 7 1 Preliminaries 1 : v i Consider a 4-manifold M and a principal bundle P over M, with structure group G. We assume X that G is compact and semi-simple. Without loss of generality we will assume that G is a Lie ar subgroupof U(N¯), N¯ N and P M is a trivial bundle. We will identify the Lie algebra g of G with a Lie subalgebra∈of the Lie a→lgebra u(N¯) of U(N¯) throughout this article. Suppose we write Tr≡TrMat(N¯,C). Then we can define a positive, non-degenerate bilinear form by hA,Bi=−TrMat(N¯,C)[AB] (1.1) for A,B g. Let Eα N be an orthonormal basis in g, which will be fixed throughout this ∈ { }α=1 article. The vector space of all smooth g-valued 1-forms on the manifold M will be denoted by A . M,g DenotethegroupofallsmoothG-valuedmappingsonM byG,calledthe gaugegroup. Thegauge group induces a gauge transformation on A , A G A given by M,g M,g M,g × → A Ω:=AΩ =Ω 1dΩ+Ω 1AΩ − − · 1 for A A , Ω G. The orbit of an element A A under this operation will be denoted by M,g M,g ∈ ∈ ∈ [A] and the set of all orbits by A/G. Let Λq(T M) be the q-th exterior power of the cotangent bundle over M. Fix a Riemannian ∗ metric g on M and this in turn defines an inner product , on Λq(T M), for which we can q ∗ h· ·i defineavolumeformdω onM. ThisallowsustodefineaHodgestaroperator actingonk-forms, ∗ :Λk(T M) Λ4 k(T M) such that for u,v Λk(T M), we have ∗ − ∗ ∗ ∗ → ∈ u v = u,v dω. (1.2) q ∧∗ h i An inner product on the set of smooth sections Γ(Λk(T M)) is then defined as ∗ u,v = u,v dω. (1.3) q h i h i ZM Given u E Λq(T M) g, we write ∗ ⊗ ∈ ⊗ u E 2 = Tr[E E] u u= Tr[E E] u,u dω. q | ⊗ | − · ∧∗ − · h i Hence for A A , the Yang-Mills action is given by M,g ∈ S (A)= dA+A A2. (1.4) YM | ∧ | ZM Note that this action is invariant under gauge transformations. Let C be a simple closed curve in the manifold M. The holonomy operator of A, computed along the curve C, is given by Texp A , (cid:20)ZC (cid:21) whereby T is the time ordering operator. See Definition 6.6 for the definition of T. It is of interest to make sense of the following path integral, 1 Tr Texp A e−12SYM(A) DA, (1.5) Z ZA∈AM,g/G (cid:20)ZC (cid:21) whereby DA is some Lebesgue type of measure on the space of g-valued 1-forms, modulo gauge transformations and Z = e−12SYM(A) DA. ZA∈AM,g/G Notation 1.1 Let Λp(Rn) be the p-th exterior power of the vector space Rn, n = 3,4. We also denote the smooth sections of a bundle P by Γ(P). In this article, when n = 3, p = 1; and when n=4, p=2. Fromnowon,weonlyconsiderM =R4 andtaketheprincipalbundleP overR4 tobethetrivial bundle. OnR4,fixthecoordinateaxisandchooseglobalcoordinates x0,x1,x2,x3 andlet e 3 be the standard orthonormal basis in R4. We will also choose the st{andard Riema}nnian m{etrai}cao=n0 R4. Now let T R4 R4 denote the trivial cotangent bundle over R4, i.e. T R4 = R4 Λ1(R4) and Λ1(R3) den∗ote→the subspace in Λ1(R4) spanned by dx1,dx2,dx3 . There∗is an∼obvio×us inner product defined on Λ1(R3), i.e. dxi,dxj = 0 if i = {j, 1 otherwise}, which it inherits from the standard metric on T R4. h i 6 ∗ 2 Using axial gauge fixing, every A AR4,g/G can be gauge transformed into a g-valued 1-form, ∈ of the form A= 3 a dxj Eα, subject to the conditions α j=1 j,α⊗ ⊗ Pa (P0,x1,0,0)=0, a (0,x1,x2,0)=0, a (0,x1,x2,x3)=0. 1,α 2,α 3,α Hence it suffices to consider the Yang-Mills integral in Expression (1.5) to be over the space of g-valued 1-forms of the form A= 3 a dxj Eα, whereby a :R4 R is smooth. α j=1 j,α⊗ ⊗ j,α → Its curvature is then given by P P dA+A A= a dxi dxj Eα+ a a dxi dxj [Eα,Eβ] i;j,α i,α j,β ∧ ⊗ ∧ ⊗ ⊗ ∧ ⊗ α 1 i<j 3 α,β 1 i<j 3 X ≤X≤ X ≤X≤ 3 + a dx0 dxj Eα, 0:j,α ⊗ ∧ ⊗ α j=1 XX for a :=( 1)ij[∂ a ∂ a ], a :=∂ a . i;j,α i j,α j i,α 0:j,α 0 i,α − − Notation 1.2 Note that Λ2(T∗R4) ∼= R4×Λ2(R4). Using the global coordinates {x0,x1,x2,x3}, we fix an orthonormal basis dx1 dx2,dx3 dx1,dx1 dx2,dx0 dx1,dx0 dx2,dx0 dx3 in Λ2(R4). Using the standar{d me∧tric on R4,∧the corres∧ponding vo∧lume form∧is given by∧dω =} dx0 dx1 dx2 dx3. ∧ ∧ ∧ From the Hodge star operator and the above volume form, we can define an inner product on the set of smooth sections Γ(Λ2(T R4)) as in Equation (1.3). ∗ For i,j,k =1,2,3, we will define ǫ =( 1)σ(ijk) if all are distinct, σ(ijk) is a permutation ijk | | − in S , and σ(ijk) is the number of transpositions; 0 otherwise. 3 | | Definition 1.3 Define cαβ = Tr Eγ[Eα,Eβ] . γ − (cid:2) (cid:3) Then, dA+A A ∧ = a dxi dxj + a a cαβ dxi dxj i;j,γ ⊗ ∧ i,α j,β γ ⊗ ∧ γ (cid:20)1 i<j 3 α<β1 i<j 3 X ≤X≤ X ≤X≤ 3 + a dx0 dxj Eγ. 0:j,γ ⊗ ∧ ⊗ j=1 (cid:21) X Thus, dA+A A2 = a2 + a a a a cαβcαˆβˆ ZR4| ∧ | Xi<jZR4(cid:20)Xα i;j,α Xγ α<Xβ,αˆ<βˆ i,α j,β i,αˆ j,βˆ γ γ +2 a a a cαβ dω+ a2 dω. (1.6) i;j,γ i,α j,β γ 0:j,α α<β,γ (cid:21) j ZR4 α X X X Itisconventionalwisdomtointerpretexp 1 dω a2 +a2 DAasaGaus- −2 i<j R4 α i;j,α 0:j,α sian measure. h P R P (cid:0) (cid:1)i 3 1.1 Abelian Case Consider first a 2-dimensional Euclidean space R2, with Abelian group G = U(1), and u(1) = ∼ R √ 1. Let S be a surface with boundary C =∂S. Write A= 2 A dxi Γ(Λ1(T R2)), A⊗:R2− R. Using Stokes’ Theorem, i=1 i⊗ ∈ ∗ i → P 2 2 A dxi = A dxi = dA= dA 1 = dA,1 . i i S S ZC i=1 ⊗ Z∂S i=1 ZS ZR2 · h i X X Here, , is the L2 inner product on the space of Lebesgue integrable functions on R2. Thus, the h· ·i Yang-Mills path integral becomes 1 e dA2/2DA e√−1hdA,1Sie−12|dA|2DA. A −| | ZA Now, make a heuristic chanRge of variables, A dA, hence we have 7→ 1 detd 1 e dA2/2D[dA] detd−1 e√−1hdA,1Sie−21|dA|2D[dA] − A −| | ZA 1 R = e dA2/2D[dA] e√−1hdA,1Sie−21|dA|2D[dA]. A −| | ZA This is a Gaussianintegralof the formgRivenin Lemma A.1, hence we candefine the path integral as exp 1 2/2 =exp[ S /2], whereby S is the area of the surface S. S −| | −| | | | Now(cid:2), we mov(cid:3)e up to R4, still using G = U(1). The above argument still applies, except that now 1 has norm0. This will yield 1 for any surface S, which means we have to redefine our path S integral to obtain non-trivial results. Let χ be the evaluation map, i.e. given a (smooth) function f : R4 R, we will write x → f(x)= f,χ . Tothe physicists,χ is justtheDiracDelta function. Furthermore,givena2-form x x writtenhin theiform F f dxi dxj, with f being smooth functions on R4; when we ≡ 0 i<j 3 ij ∧ ij write F,χ dxa dxb , we≤me≤an h x⊗ ∧ iP F,χ dxa dxb = f dxi dxj,χ dxa dxb x ij x ⊗ ∧ ∧ ⊗ ∧ * + 0 i<j 3 (cid:10) (cid:11) ≤X≤ = f dxa dxb,χ dxa dxb =f (x) R. (1.7) ab x ab ∧ ⊗ ∧ ∈ (cid:10) (cid:11) Choose a surface S such that ∂S = C. Suppose σ : [0,1]2 R4 is any parametrization for → S. Using axial gauge fixing, we only consider 1-forms of the form A = 3 A dxi. Write i=1 i ⊗ A :=( 1)ij[∂ A ∂ A ], A =∂ A . We also write A =A . By Stokes’ Theorem, i;j i j j i 0:j j 0 0;j 0:j − − P 3 3 A dxi = A dxi = dA i i ⊗ ⊗ ZC i=1 Z∂S i=1 ZS X X 3 = dsdt [A (σ)Jσ ](s,t)+ [A (σ)Jσ ](s,t)  i;j | ij| 0:j | 0j|  Z[0,1]2 1 i<j 3 j=1 ≤X≤ X   = dsdt dA, χ Jσ (s,t) dxi dxj := dA,ν˜ , (1.8) Z[0,1]2 * 0 i<j 3 σ(s,t)| ij| ⊗ ∧ + h Si ≤X≤ 4 where ν˜ = dsdt χ Jσ (s,t) dxi dxj, S σ(s,t)| ij| ⊗ ∧ Z[0,1]20 i<j 3 ≤X≤ is a linear functional on the space of smooth sections of 2-forms in Λ2(T R4). And Jσ is defined ∗ ij in Definition A.3. Here, we wish to point out that the inner product in question is given by Equation (1.7) and therefore, dA, χ Jσ (s,t) dxi dxj = dA, χ dxi dxj Jσ (s,t) (1.9) * σ(s,t)| ij| ⊗ ∧ + * σ(s,t)⊗ ∧ +| ij| 0 i<j 3 0 i<j 3 ≤X≤ ≤X≤ = A ,χ Jσ (s,t) i;j σ(s,t) | ij| 0 i<j 3 ≤X≤ (cid:10) (cid:11) = A (σ(s,t)) Jσ (s,t). i;j | ij| 0 i<j 3 ≤X≤ Thus, we want to make sense of 1 e√−1hdA,ν˜Sie−21|dA|2DA Z ZA forsomenormalizationconstant. We haveto doachangeofvariables,i.e. A dA. Thenwehave 7→ 1 e√−1hdA,ν˜Sie−12|dA|2det(d−1)D[dA], Z ZA which is some undefined constant det(d 1). After dividing away this constant, we are left with − 1 Z¯ e√−1hdA,ν˜Sie−21|dA|2D[dA], ZA which we can define as a Gaussian integral, for some normalization constant Z¯. Now, apply a similar argument as in the R2 case, the Yang-Mills path integral will yield exp ν˜ 2/2 . S −| | There are a few problems with this argument. Firstofall, χx is not anhone(cid:2)stfunction(cid:3), in fact it is a generalizedfunction. Secondly,when we constructaGaussianmeasureonΓ(Λ2(T R4)), we ∗ need to enlarge the space and consider generalized sections in Λ2(T R4). As such, the term ν 2 ∗ S | | will face serious problems as it is not clear how to define a product of generalized functions. Because of the above problems, if we wish to carry out the above heuristic argument, we cannot define the path integral over real-valued 2-forms. We will instead construct an Abstract Wienerspace,consistingofΛ1(R3)-valuedholomorphicfunctionsinC4. Thepathintegralgivenin Equation(1.5)depends onthe curveC M R4,ormorecorrectlyachoiceofasurfaceS whose boundary is ∂S = C. Hence, we need t⊂o emb≡ed the surface S R4 inside C4, by embedding R4 inside C4 in the standard way, so that we can define the path in⊂tegral using the Abstract Wiener space setting. Such an approach was used in [Lim11] and [Lim12] to construct the Chern-Simons Path integral and obtain link invariants in R3 respectively. This work is motivated by trying to apply the Abstract Wiener Space construction used in [Lim11]. However, there are significant differences in the Chern-Simons Path integral and the Yang-Mills Pathintegral. Hence, we need to modify the construction used in [Lim11] accordingly, so that a rigorous definition can be given to the Yang-Mills Path integral. See Definition 3.2. 5 Thenon-AbeliancasewillbedealtwithinSection4. SeeDefinition7.6forarigorousdefinition ofthe Yang-MillsPathintegralfor the non-Abeliancase. Fromboth the Abelian andnon-Abelian cases, we will then derive the Wilson Area Law Formulas from these definitions. The results are given by Theorems 3.3 and 8.3. Finally, Section 9 will focus on applications of the Area Law formula. We wish to make the following final remark. Dimension 3 is key for the Chern-Simons Path integral formalism to work effectively. In the Yang-Mills Path integral formalism, dimension 4 is thekey. Sowheredidweusethefactthatthedimensionis4? ItisactuallyusedinAppendix A.2. Notation 1.4 Suppose we have two Hilbert spaces, H and H . We consider the tensor product 1 2 H H . The inner product on the tensor product H H is given by 1 2 1 2 ⊗ ⊗ u u ,v v = u ,v u ,v . h 1⊗ 2 1⊗ 2iH1⊗H2 h 1 1iH1h 2 2iH2 This definition of theinner product on thetensorproduct ofHilbert spaces willbeassumedthrough- out this article. See also subsection 2.4. Finally, we always use , to denote an inner product. h· ·i 2 Construction of Wiener Measure Throughout the rest of this article, we adopt the following notation. Notation 2.1 For x R4, we let φ (x) = κ4e κ2x2/2/(2π)2, which is a Gaussian function κ − | | ∈ with variance 1/κ2. Define a function ψz ≡ ψ(z), where ψ(z) = √12πe−P4i=1|zi|2/2 and z ≡ (z ,z ,z ,z ) C4. 0 1 2 3 ∈ Remark 2.2 Throughout this article, this Gaussian function will play a key role in our calcula- tions. As mentioned earlier on, the Dirac Delta function is a generalized function, but we can approximate it using a Gaussian function φ . The larger the value of κ, the better is this approxi- κ mation. The variance 1/κ2, denotes how well we can resolve a point in R4. In our construction of the Yang-Mills path integral, we will make use of this Gaussian function in this construction, thus all our Yang-Mills path integral define later, will depend on the parameter κ. 2.1 Schwartz Space Notation 2.3 We let p denote the n-tuple (m ,m ,...,m ), m ,...m 0 are integers with r 1 2 n 1 n zP1mnj1=z12mm2·j··=znmrn.. LAentdPrwdeenwortietethper!se:t=ofma1ll!msu2c!h··n·-mtunp!.les,Fio.re.z = (z1,z2,.≥..zn) ∈ Cn, zpr := n P = (m ,m ,...,m ) m =r . r  1 2 n j   (cid:12) Xj=1  (cid:12) (cid:12) Let P= ∞r=0Pr.   ThereSis an ordering which we will adopt in the rest of the article. We will write pr pr¯, if in ≤ the order of priority, r r¯, followed by m m¯ , m m¯ ,...,m m¯ . 1 1 2 2 n n ≤ ≤ ≤ ≤ 6 ConsidertheSchwartzspaceS (R4),withtheGaussianfunctionφ ,√φ (x)=κ2e κ2x2/4/(2π). κ κ κ − | | Any f S (R4) can be written in the form κ ∈ f(x)=p(x) φ (x), κ wherebypisapolynomialin(x0,x1,x2,x3). Theinpnerproduct , definedontheSchwartzspace h· ·i is given by f,g = f g dλ, (2.1) h i ZR4 · dλ is Lebesgue measure on R4. Let S (R4) be the smallest Hilbert space containing S (R4), using κ κ this inner product. The Hermite polynomials h forman orthogonalsetin L2(R,dµ) with the Gaussianmea- i i 0 sure dµ(x0) e−x20/2dx0/√2π{. L}et≥ ≡ H (x):=h (x0)h (x1)h (x2)h (x3), p =(i,j,k,l) P , pr i j k l r ∈ r be a product of Hermite polynomials and Hκ :=H (κ). pr pr · We have the normalizedHermite polynomials H /√p ! with respect to the Gaussianmeasure pr r e−P3i=0|xi|2/2dλ/(2π)2. Then ∞ H (κx0,κx1,κx2,κx3) φ / p !:p P pr κ r r ∈ r r[=0n p p o is an orthonormalbasis for S (R4). κ Notation 2.4 We will write 3 S (R4) Λ1(R3)= f dxa : f S (R4) κ a a κ ⊗ ( ⊗ ∈ ) a=1 X and S (R4) Λ2(R4)= f dxa dxb : f S (R4) . κ ⊗  ab⊗ ∧ ab ∈ κ  0≤Xa<b≤3  Here, S (R4) Λ1(R3)is thespaceofsmooth1-forms over R4 andthis spaceconsistsofΛ1(R3)- κ valued functions⊗in R4, integrable with respect to Lebesgue measure. In a similar fashion, S (R4) Λ2(R4) is the space of smooth 2-forms over R4 and this space κ consists of Λ2(R4)-valued functio⊗ns in R4, integrable with respect to Lebesgue measure. Note that these Schwartz spaces are dependent on κ. Recall we use the standard metric on TR4 and the volume form on R4 is given by dω = dx0 dx1 dx2 dx3. Using the Hodge star operator and the above volume form, we will define an i∧nner p∧roduct∧on S (R4) Λ2(R4) by κ ⊗ f dxa dxb, fˆ dxa dxb = f ,fˆ . ab ab ab ab * ⊗ ∧ ⊗ ∧ + 0≤Xa<b≤3 0≤Xa<b≤3 0≤Xa<b≤3D E 7 Definition 2.5 Define a bilinear form on S (R4) Λ1(R3), by df,dg , f,g S (R4) Λ1(R3). κ κ Note that df,dg S (R4) Λ2(R4), and we comput⊗e the bilinear fhorm ofidf and∈dg using⊗the inner κ product in S (R4∈) Λ2(R⊗4). κ ⊗ Proposition 2.6 The bilinear form d,d on S (R4) Λ1(R3) is an inner product. κ h · ·i ⊗ Proof. The only thing we have to show is that dω,dω = 0 imply ω = 0. Now, recall using h i axialgauge fixing, ω = 3 ω dxi. Therefore,0= dω 2 ∂ ω 2 and thus ∂ ω 2 =0 for each i=1 i⊗ | | ≥| 0 | | 0 i| i = 1,2,3. This means that ω is independent of x . But since ω is L2 integrable over x , hence i 0 i 0 P ω 0 for each i. Therefore ω 0. i ≡ ≡ Complete S (R4) Λ1(R3) into a Hilbert space, denoted by S (R4) Λ1(R3). In future, we κ κ ⊗ ⊗ will denote the inner product on this Hilbert space by d,d . h · ·i 2.2 Segal Bargmann Transform In the construction of the Abstract Wiener space, we need to enlarge the space S (R4) Λ1(R3), κ ⊗ by choosing a suitable measurable norm. But by doing so, we will end up with distribution valued forms. We do not want to consider this,asthiswillleadtodifficultiesindefiningν˜ giveninEquation(1.8). Toavoidthisproblem,we S willusetheSegalBargmannTransformandmaptheSchwartzspacetothespaceofΛ1(R3)-valued holomorphic functions, over in C4. Thus, path integralis not defined onthe space of Λ1(R3)-valued Schwartzfunctions in R4, but rather, over the space of Λ1(R3)-valued holomorphic functions in C4. Let z =(z ,z ,z ,z ) C4, each z C. Consider the real vector space spanned by zn : z 0 1 2 3 i C , integrable with re∈spect to the ∈Gaussian measure, equipped with a sesquilinear{comple∈x }∞n=0 inner product, given by 1 zr,zr′ = zr zr′e−|z2| dx dp, z =x+√ 1p. (2.2) h i π C · − Z Note that z means complex conjugate. Denote this (real) inner product space by H2(C), which consists of polynomials in z. An orthonormal basis is given by zn : n 0 . √n! ≥ (cid:26) (cid:27) Notation 2.7 Let H2(C4) H2(C) 4, which consists of polynomials in z =(z ,z ,z ,z ). Write ⊗ 0 1 2 3 ≡ 3 H2(C4) Λ1(R3)= f dxa : f H2(C4) . a a ⊗ ( ⊗ ∈ ) a=1 X Notethatthis is aspaceof complex-valued holomorphic sections inthetrivial bundle Λ1(R3) C4, → integrable using a Gaussian measure. Let H2(C4) be the smallest Hilbert space containing H2(C4). Also denote H2(C4) Λ2(R4)= f dxa dxb : f H2(C4) . ⊗  ab⊗ ∧ ab ∈  0≤Xa<b≤3    8 Notethat this is aspace of complex valued holomorphic sections in the trivial bundle Λ2(R4) C4, → integrable using a Gaussian measure. We will further define an inner product on H2(C4) Λ2(R4) by ⊗ f dxa dxb, fˆ dxa dxb = f ,fˆ . ab ab ab ab * ⊗ ∧ ⊗ ∧ + 0≤Xa<b≤3 0≤Xa<b≤3 0≤Xa<b≤3D E Here, f and fˆ are in H2(C4). ab ab We will continue to use , to denote the inner product on H2(C4) Λ2(R4) and write dλ = 4 h· ·i ⊗ π14e−P3i=0|zi|2 3i=0dxidpi. Note that Q f ,fˆ = f fˆ dλ . ab ab ab ab 4 D E ZC4 Recall the inner product we are using in S (R4) Λ1(R3) is d,d . We are now going to κ construct an isometry between S (R4) Λ1(R3) and H⊗2(C4) Λ1(hR3·),·uising d,d . κ ⊗ ⊗ h · ·i The Hermite polynomials satisfy the following property h (x)=xh (x) h (x). n+1 n − ′n Therefore, d x h (x)e x2/4 = xh (x) h (x) h (x) e x2/4. n − n n+1 n − dx − − 2 (cid:16) (cid:17) h i But we also have h =nh , which means that xh (x)=h (x)+nh (x). Thus, ′n n 1 n n+1 n 1 − − d 1 n h (x)e x2/4 = h (x)+ h (x) h (x) e x2/4 n − n+1 n 1 n+1 − dx 2 2 − − (cid:16) (cid:17) (cid:20) (cid:21) n 1 = h (x) h (x) e x2/4. n 1 n+1 − 2 − − 2 (cid:20) (cid:21) Definition 2.8 Define an operator d by a n 1 dazan = 2zan−1− 2zan+1, (2.3) for a=0,1,2,3. Definition 2.9 For any f H2(C4), define a ∈ 3 3 d f dxa := d f dx0 dxa+ ( 1)ij[d f d f ]dxi dxj. a 0 0 i j j i ⊗ ∧ − − ∧ a=1 a=1 1 i<j 3 X X ≤X≤ Furthermore, define a bilinear form , in H2(C4) Λ1(R3) by d,κ h· ·i ⊗ f dxa, g dxb =κ2 d f dxa,d g dxb . (2.4) a b a b * ⊗ ⊗ + * ⊗ ⊗ + Xa Xb d,κ Xa Xb 9 Remark 2.10 Just as we have the differential operator d acting on 1-forms in S (R4) Λ1(R3), κ the analogous operator will be d acting on Λ1(R3)-valued holomorphic functions, whi⊗ch are in H2(C4) Λ1(R3). ⊗ The SegalBargmanntransformΨ actually mapsHermite polynomialsin R4 to complexholo- κ morphic functions in C4, equipped with the Gaussian measure as defined in Equation (2.2). Definition 2.11 (Segal Bargmann Transform) Refer to Notation 2.3. Using the Segal Bargmann Transform Ψ :S (R4) H2(C4), we map κ κ → zpr zi0zi1zi2zi3 Ψ :H (κx0,κx1,κx2,κx3) φ / p ! 0 1 2 3 , κ pr κ r 7−→ √pr! ≡ √i0!i1!i2!i3! p p if p =(i ,i ,i ,i ). Clearly, it is an isometry. r 0 2 2 3 We can extend this isometry to tensor and direct products and by abuse of notation, use the same symbol. That is, Ψ :S (R4) Λ2(R4) H2(C4) Λ2(R4) by κ κ ⊗ → ⊗ Ψ f dxa dxb = Ψ [f ] dxa dxb. κ a,b κ a,b  ⊗ ∧  ⊗ ∧ 0 a<b 3 0 a<b 3 ≤X≤ ≤X≤   By abuse of notation, we use the same symbol to define an isometry that sends this Schwartz space S (R4) Λ1(R3)into H2(C4) Λ1(R3), overinC4. Note that H2(C4) Λ1(R3) is complex- κ valued holomo⊗rphic sections of the t⊗rivial bundle Λ1(R3) C4, integrable us⊗ing a Gaussian mea- sure. Alternatively, one can view this space as Λ1(R3)-val→ued holomorphic functions in C4, which are integrable with respect to the Gaussian measure. By definition of d , we note that a Ψ (df)=κd(Ψ f), f S (R4) Λ1(R3). κ κ κ ∈ ⊗ Thus we have the following result. Proposition 2.12 We have , is an inner product. Thus, welet denote thenorm using d,κ d,κ this inner product and (H2(Ch4·)·i Λ1(R3), , ) denote this inner pr|o·|duct space. d,κ ⊗ h· ·i Proof. It follows from Ψ ω,Ψ ω = dω,dω . κ κ d,κ h i h i Henceforth,complete(H2(C4) Λ1(R3), , )intoaHilbertspace,denotedbyH2(C4) Λ1(R3). d,κ ⊗ h· ·i ⊗ Note that the Segal Bargmann Transform Ψ extends into an isometry between S (R4) Λ1(R3) κ κ and H2(C4) Λ1(R3). ⊗ ⊗ 2.3 Abstract Wiener space Let (H, , ) be a (real) infinite dimensional Hilbert space. h· ·i Definition 2.13 10