## 1 Introduction

Although canonical quantisation provides the basic formalism of quantum field theory, the corresponding Schrödinger Representation, in which the field operators are diagonal, has not received commensurate attention. This is partly due to the popularity of the functional integral which displays space-time symmetries manifestly, and partly because the existence of wave functionals was only shown by Symanzik as late as 1981, . Nonetheless there has been growing interest in the subject as a result of the search for new tools in field theory, and also because the Schrödinger Representation is implicit in much recent work on field theories defined on space-times with boundaries, see for example .

The vacuum wave-functional (VWF), , may be constructed as a functional integral over the Euclidean space-time which has the quantisation surface as boundary. is then a functional of the values the field takes on the boundary. As we will see, these boundary values act as source terms in the functional integral. Symanzik showed that, at least in perturbation theory, this functional has a finite limit as the cut-off is removed, subject to the inclusion of the usual counter-terms together with additional ones localised to which result in the boundary values of the field undergoing an additional field renormalisation, . These boundary counter-terms are absent in fewer than three dimensions, as they are for Yang-Mills theory in four dimensions due to gauge invariance. He also proved the existence of the Schrödinger equation for in four dimensions.

In  it was shown that the vacuum functional of the scaled Yang-Mills field extends to an analytic function of in the complex -plane with the negative real axis removed. This also applies to scalar field theory. This allows the vacuum functional to be reconstructed for arbitrary in terms of the scaled field for large using Cauchy’s theorem. The scaled field is slowly varying for large , and for such a field we would expect to be able to expand in powers of derivatives divided by the lightest glueball mass (in our appendix C we give an argument to justify the possibility of performing this local expansion). Thus can be obtained for arbitrary from a knowledge of this derivative expansion. The existence of this expansion was originally considered by Greensite  who found the leading terms from a Monte-Carlo simulation of lattice gauge theory. We emphasize that this procedure is valid for arbitrary , it does not amount to restricting attention to slowly varying field configurations, it is simply a way of parametrising the function space of sources in the functional integral in terms of a derivative expansion.

satisfies the Schrödinger equation, but this takes a special form when using the derivative expansion for due to the employment of Cauchy’s theorem. This was described in  where a non-perturbative approximation scheme was outlined. The semiclassical expansion of this equation was shown to agree with the direct semi-classical evaluation of via Feynman diagrams in , and extended to Yang-Mills theory in  where the beta-function was correctly reproduced from the derivative expansion. This is not as obvious as it might appear because a naive insertion of a local expansion into the usual Schrödinger equation will not converge for momenta greater than the mass of the lightest particle and so will not lead to the correct behavior as the ultra-violet cut-off in that equation is removed. What was missing from this work was a method for constructing vacuum expectation values (VEVs) directly from the derivative expansion, which is the subject of this paper. We will show that when these are written as functional integrals over the boundary values of fields they are analytic in an ultra-violet momentum cut-off in the plane cut as above. Again, Cauchy’s theorem may be used to compute VEVs for large cut-off from a knowledge of the corresponding functional integral for small cut-off, which in turn can be obtained from the derivative expansion, or some other systematic approach. Notice that if we try to compute the VEV in the most obvious way, by expanding the logarithm of the vacuum functional in a local expansion and doing the usual perturbative approach then we would get a sum of contractions which would, in general, lead to ultra-violet divergences of all orders. These divergences cannot be absorbed into renormalisation of the wave-functional to obtain finite VEVs because the wave-functional is already finite according to Symanzik’s work. (The only possibility for such renormalisation is if the inner product involves a non-trivial weight functional with coefficients that can be chosen to cancel divergences. The form of these weights is determined by the hermiticity of the Hamiltonian operator, and is very restricted. In 1+1 dimensional scalar field theory, and Yang-Mills theory such weights are absent.) The origin of these ultra-violet divergences is that we would be attempting to compute the integral for field configurations beyond the convergence radius of the derivative expansion, and this is inconsistent. In this paper we propose a method for computing VEVs in which the same expansion is employed but with a cut-off that lies inside the convergence radius of the series. Typically this means that the cut-off is smaller than the mass of the lightest particle. It therefore does not appear to be an ultraviolet cut-off. However we will be able to send the cut-off to infinity in this expansion because, as we will show, that VEVs are analytic in the cut-off when we continue to complex values. Thus we can use Cauchy’s theorem to relate the large cut-off behaviour which we need to compute, to the small cut-off behaviour which we can calculate using the local expansion of the funcctional integral.

We concentrate throughout on the toy model of -theory in 1+1 dimensions as this is particularly straightforward given the absence of boundary counter-terms resulting in there being no wave-function renormalization. The absence of boundary counter-terms is shared by Yang-Mills theory which also has a correspondingly simple Schrödinger equation. Super-renormalizable theories are, in any case, of interest in their own right by virtue of their connection with integrable theories and with String Theory. We will only discuss the VEV of operators which will be diagonal in field configuration space (we do not expect that our conclusions will change if we consider more general operators, , as we can see in  where the analyticity of is shown). Finally, with an analytical continuation it is often difficult to estimate truncation errors, but we will see that they can be controlled.

Several authors [1, 7, 8, 9, 10, 11] have devised perturbative and non-perturbative aproaches to compute the vacuum wave-functional. In section 2 we describe its representation as a functional integral. In the next section we give a general discussion of the construction of VEVs in the Schrödinger Representation in terms of Feynman diagrams. We display the mechanism whereby the Feynman diagram expansion of , which makes use of a propagator on the space-time with Dirichlet boundary conditions, leads to the usual Feynman diagrams for VEVs on the full space-time with the standard propagator. We end the section by giving an operator approach which uses the results of appendix A. In section 4 we translate the calculations of the previous section into the language of first quantisation in which the vacuum functional can be expressed in terms of random paths that are reflected at the quantisation surface. In section 5 we describe the analyticity of VWF and VEVs and describe the resummation of the series in the cut-off. Section 6 ilustrates our method in the simpler context of non-relativistic quantum mechanics and we have left to the appendix B mathematical details of our method of analytic continuation. In the section 7 we will discuss the computation of the equal-time two point function through diagrams in a dimensionally reduced effective theory. The last section is devoted to our conclusions.

## 2 Representations for the vacuum wave functional

The VWF is the inner product of the vacuum and an eigenbra of the field operator restricted to the quantization surface (which we take to be ) belonging to the eigenvalue :

 ⟨φ|^ϕ(x,0)=φ(x)⟨φ| (1)

The -dependence of the eigenbra may be made explicit by writing

 ⟨φ|=⟨D|exp(i∫dxφ(x)^π(x,0)) (2)

where is annihilated by , i.e. it is the state , D stands for Dirichlet, and is canonically conjugate to . The canonical commutation relations then yield (2) together with

 δδφ(x)⟨φ|=i⟨φ|^π(x) (3)

if we apply the Euclidean time evolution operator to any state, , not orthogonal to the vacuum, then for large times

 exp(−^HT)|υ⟩∼|0⟩e−E0T⟨0|υ⟩(T→∞) (4)

where is the energy of the vacuum. Thus

 Ψ[φ]=limT→∞N⟨D|ei∫dxφ(x)^π(x,0)e−^HT|υ⟩ (5)

Where is a normalization constant depending on . Using this, as we will explicitly show later, may be written as the functional integral

 ∫Dϕe−SE[ϕ]−∫dx(φ(x)˙ϕ(x,0)+Λφ2(x)) (6)

where is the Euclidean action for the space . is a regularization of that arises from the differentiation of the time ordered product that is represented by the functional integral, i.e.

 T(^π(x,t)^π(x′,t′))=∂2∂t∂t′T(^φ(x,t)^φ(x′,t′))−iδ(x−x′)δ(t−t′) (7)

On the boundary the integration variable should vanish, reflecting the fact that .

Alternatively, we can obtain this path integral representation for the VWF, by beginning with

 Ψ0[ϕ]=Nlimτ→∞eτE0∫Dφ(x,t)e−SE(φ,˙φ)∣∣φ(x,0)=ϕ(x) (8)

where can be anything, and performing a functional change of variables in the path integral in such a way that we do not have the -field dependence in the integration limit. Our change of variables is formally

 ~φ(x,t)=φ(x,t)−2θ(t)ϕ(x)D~φ=DφSE[φ,˙φ]=∫0−Tdt∫dxL(~φ+2θ(t)ϕ,˙~φ+2δ(t)ϕ) (9)

where is the step function and we take . Naively the terms do not contribute to the potential. Therefore our path integral on removing the tilde can be written as

 ∫Dφ(x,t)e−SE(φ,˙φ+2δ(t)ϕ)∣∣φ(x,0)=0 (10)

In the scalar case we will have

 SE=∫0−∞dt∫dx{12(˙φ)2+V(φ)}+∫dx˙φ(x,0)ϕ+∫dxδ(0)ϕ2 (11)

Therefore, the logarithm of the VWF () is given by the sum of connected diagrams constructed from the new action (11) and with a boundary at where the field vanishes. The new action contains a source term on the boundary and a term (to be regulated) coupled to the the boundary fields. This argument has been rather too formal. To be more careful we should smooth the functions in (9), taking them to be non-constant in a region of size , and this will regulate . So we replace by . With a cutoff this function will be given by

 θΛ(t)=i2π∫Λ−Λdω1ω+iϵe−iωt, (12)

and we have

We can also see how to get (6) from (5) together with the -terms within the canonical operator formalism. We will use the following identity

 eAeB=Te∫0−1dteBt(B+A)e−Bt (13)

where and are arbitrary matrices (or operators) and is the time-ordering operator. As usual, the -ordering implies that the first term in the exponential is to be thought of as and then, at the end set to a constant. This relation can be derived by considering the operator , then we calculate and then we integrate it back to get the integral equation

 U(t)=1+∫t0dt′U(t′)(B+A(−t′))

with . Once we solve the integral equation in terms of a -ordered exponential, we set and we get (13).

Now we will consider

 ⟨D|ei∫dxφ(x)^π(x,0)e−T^H|υ⟩=⟨D|ei∫dxφ(x)^π(x,0)e−ϵ^He−(T−ϵ)^H|υ⟩ (14)

and we will use our relation (13) to combine the term into a single exponential, but before that we will follow some intermediate steps. Firstly we take (and to shorten the notation )

 e−ϵ^Heiϕ^π=Te−ϵ∫0−1dt(−iϵϕ^π+eiϕ^πt^He−iϕ^πt)=Te−∫0−ϵdt(−iϵϕ^π+^H(^π,^φ+ϕtϵ)) (15)

which (after hermitian conjugate, , and later ) will become

 eiϕ^πe−ϵ^H=Te−∫0−ϵdt(−iϵϕ^π+^H(^π,^φ+(tϵ+1)ϕ)) (16)

Therefore

 ⟨ϕ|e−T^H|χ⟩=⟨D|eiϕ^πe−T^H|χ⟩=⟨D|Te−∫0−Tdt′^H′(t′)|χ⟩ (17)

with , where the is defined by

 θϵ(t)=⎧⎪ ⎪⎨⎪ ⎪⎩0if t≤−ϵt2ϵ+12if −ϵ

and the is obtained by taking its derivative. Notice that we needed to put a factor in front because

 ⟨q|e−ϵ(12^p2+V(^q))|q1⟩ = ⟨q=0|e−ϵ(12^p2−iϵq^p+V(^q+2θϵ(t)q))|q1⟩= (18) = Ne−12(−q1ϵ+qϵ)2−V(q1+2θϵ(t)q)=Ne−S(q,˙q)

where the (in the last equality) is interpreted as and as . As we see we get the same action as before. Finally we can write the VWF as

 Ψ[φ] = limT→∞N⟨D|Te−∫0−Tdt′^H(t′)|ϕ⟩= (19) = limT→∞N∫0,t=0ϕ,t=−TDχ(x,t)e−SE(χ(x,t)+2θϵ(t)φ(x),˙χ(x,t)+2δϵ(t)φ(x))

Notice that we could have got the following relation

 eiϕ^πe−T^H=Te−∫0−Tdt(−iTϕ^π+^H(^π,^φ+(tT+1)ϕ)) (20)

which could be derived with (9) using . Lastly, we may also construct a relation

 eiϕ^πe−T^H=Te−∫0−Tdt(^H−iTϕ^π(t)) (21)

with but this time we cannot use (9) to obtain the corresponding path integral version. We have seen from a range of methods that the (time-independent) VWF will be given by

 Ψ[ϕ(x)]=N∫Dφ(x,t)e−SE−∫dx˙φ(x,0)ϕ(x)−∫dxδϵ(0)ϕ2∣∣φ(x,0)=0 (22)

We can also obtain a path integral representation for the VWF (with -independent boundary conditions) without any delta function by using a different shift in the field in (9). Consider shifting by a solution to the free field equations denoted by , where will be defined by

 ΩΛ(t)=sinh(t√−∂2+m2)sinh(Λ√−∂2+m2). (23)

Thus which yields

 =∫Dφe−S0(φ)−∫V(φ+ΩΛ(t)ϕ)−S0(ΩΛ(t)ϕ)∣∣φ(x,0)=φ(x,Λ)=0. (24)

This provides an alternative derivation of the starting point used in  to set up a non-perturbative algorithm to compute the VWF by resumming the perturbative expansion of (24). Another possibility is to shift the field by a solution to the Euler-Lagrange equations of the full action with boundary conditions and (where ). In the background field technique a (the expectation value is taken with the boundary condition ) is taken (which at first order coincides with the one which minimises the action) and then the gives the exponential of the effective action evaluated at the configuration . This is the method used in  where the gauge theory case is analyzed and plays the role of their induced background field, although they do not expand in the background field, and in  where they study general scalar theories with curved boundaries using dimensional regularization for the space-time dimensions. In this way they give a method to use dimensional (space-time) regularization with the Schrödinger representation where the time is given by the coordinate perpendicular to the boundary of the -dimensional space-time. Note that usually  the space-like dimensions are regulated differently (for example with dimensional regularization) from the time (for example by time-splitting).

We end this section by deriving the important path integral representation introduced by Symanzik , where the Dirichlet boundary conditions are only introduced through a boundary term. Again we can derive this representation in an easy way, following steps similar to those of eq. (9). Consider (as before, we understand the delta functions to be regularized)

 ∫Dφ(x,t)e−0∫−∞dt∫dx{12(˙φ)2+V(φ)}∫DC(x)e∫dxδ(0)(ϕ−C)2∫DC(x)e∫dxδ(0)(ϕ−C)2 (25)

where and . Now absorb the denominator into a proportionallity constant and commute the and integrals. Shifting the variables gives

 N∫DC(x)Dφ(x,t)e−0∫−∞dt∫dx{12˙φ2+V(φ)+2˙φδ(t)(ϕ−C)+2(δ(t))2(ϕ−C)2}e∫dxδ(0)(ϕ−C)2 (26)

where now and . After the cancellation of the exponential our VWF will be given by

 Ψ0[ϕ]=N′∫DC(x)Dφ(x,t)e−0∫−∞dt∫dx{12˙φ2+V(φ)}−∫dx˙φ(x,0)(ϕ(x)−φ(x,0)), (27)

where the integral has to be understood as . We can now interpret with the conditions and as an ordinary with and free boundary at . This formula agrees with Symanzik’s one (eq. (2.10) of ) but for an arbitrary potential. We expect that quantum corrections will renormalize it.

As Symanzik has discussed, placing source terms on the boundary leads to divergences, . These appear in perturbation theory because the field is placed at the same point as the image charges that enforce the boundary conditions on propagators. In order to regulate these divergences we should split in time the fields (thus the fields in (28) will be defined at different ordered times, but with their ). These divergences appear as the coefficient of local operators forming a boundary operator expansion [9, 12] (analogous to the usual operator product expansion). These coefficients will scale with some non-trivial dimension (in perturbation theory they are calculated as a series in the coupling), which (order by order in the coupling) will lead to logarithms of . In a perturbatively renormalizable theory (where the anomalous dimensions cannot make relevant an irrelevant operator) we need to only consider the operators of dimension smaller or equal than the product of operators at the boundary because . Therefore, in order to use the Schrödinger representation (where is implied), we should subtract the previous divergences (Wilson-boundary coefficients) in the original lagrangian. Extension of the validity of the boundary operator expansion to the non-perturbative domain suggests that we may have to consider additional relevant fields, we assume that this is not the case. Symanzik  showed that in theory in dimensions only two counterterms where needed ( and ) and conjectured that in a general perturbatively renormalizable field theory in four dimensions we only need operators of dimension less or equal than three. Because in lower dimensions we find fewer UV infinities, we will limit our discussion to scalar theory in dimensions, where there is no wave-function renormalization and (28) is valid.

## 3 Feynman diagram expansion of VEVs

The purpose of this section is to describe how the Feynman diagram expansion of the VWF, , generates the usual diagrams of equal time Green’s functions via the following relation

 ⟨ϕ(x1,0)⋅⋅⋅ϕ(xn,0)⟩=∫Dϕ(x)ϕ(x1)⋅⋅⋅ϕ(xn)|Ψ0[ϕ]|2 (28)

When we use representation (8) for in (28) we formally obtain the usual path integral for the time ordered product of field operators, as we should. If instead we first perturbatively compute the vacuum functionals using (22) then we get some unusual diagrams that generate an effective action which will be used to compute (28) with new propagators and (non-local) vertices. It is of interest to see how these combine to produce the usual result for the VEV. This will also allow us to use ordinary Feynman diagrams to compute (28) (which will be used in the section 7 to compute an equal-time propagator).

Given the comments at the end of the last section we take the Euclidean action in 1+1 dimensions as

 SE=∫dxdt(12(˙ϕ2+ϕ′2+M2(Λ)ϕ2)+g4!ϕ4) (29)

In perturbation theory the only divergent diagrams with external legs are tadpoles which can be removed by normal ordering. This enables the dependence of on the cut-off to be calculated. We will regulate with a cut-off on the spatial component of the momentum (in dimensions this is almost sufficent, as we will see), so that

 M2(Λ)=M2+ℏδM2−gℏ4∫p2<Λdp02πdp1p02+p2+M2==M2+ℏδM2−gℏ4∫p2<Λdp2π1√p2+M2==M2+ℏM2c (30)

The Feynman diagram expansion of the VWF can now be constructed so that its logarithm, , is a sum of connected diagrams in which is the source for restricted to the boundary . The major difference from usual Feynman diagrams encountered in free space is that the propagator satisfies Dirichlet boundary conditions, which means that it should vanish when either end lies on the boundary. Such a propagator is given by the method of images as

 G(x,t,y,t′)=G0(x,t,y,t′)−GI(x,t,y,t′) (31)

where is the free-space propagator and

 GI(x,t,y,t′)=G0(x,−t,y,t′)=G0(x,t,y,−t′)

The tree-level diagrams that contribute up to the term in are given in figure (1). The heavy line denotes the boundary, , and the dots denote the differentiation with respect to that results from being coupled to . When the propagator ends on the boundary this differentiation leads to and the image propagator contributing equally:

 ∂∂tG(x,t,y,t′)|t=0=2∂∂tG0(x,t,y,t′)|t=0 (32)

The term in (6) cancels a divergence in the first diagram of figure (1) since this is

 ∫dxdyφ(x)φ(y)∂2∂t∂t′2G0(x,t,y,t′)|t=t′=0==∫dp2π2~φ(p)~φ(−p)∫dp0p20p20+p2+m2 (33)

and the term leads to a subtraction so that the integral is replaced by

 ∫dp0(p20p20+p2+m2−1)=−π√p2+m2 (34)

All the diagrams that occur in figure (1) involve integrals over the time-like components of Euclidean momenta as this integration forces the source terms to be on the boundary, but this is the only divergent integration since the coupling, , has dimensions of . Another way to deal with this divergence is take the two times and to be distinct, say and . Then the last term of (7) is so with this prescription and the integral in (34) is replaced by

 ∫dp0p20p20+p2+m2eip0ϵ (35)

The exponent allows the contour to be closed in the lower half-plane giving (34), as before. The integrals can be done in a straightforward way by contour integration (with semi-circle in the upper plane) using

 δ(∑ip0i)=−1πIm1∑ip0i+iϵ (36)

Now, we expand as

 W[φ]=1ℏ∞∑n=1∫dp1⋯dp2n~φ(p1)⋯~φ(p2n)Γ2n(p1,⋯,p2n)δ(p1+⋯+p2n) (37)

and the tree-level contributions to the kernels, , are given by

 Γ′2(p,−p)=−14π√p2+M2=−ω(p)4πΓ′4(p1,⋯,p4)=−g(2π)34!(ω(p1)+⋯+ω(p4)) (38)

and, for , the recursion relation

 Γ′2r(p1,⋯,p2r)=4π∑2ri=1ω(pi)r−1∑n=2n(r+1−n)×S{Γ′2n(k,p1,⋯,p2n−1)Γ′2(r+1−n)(q,p2n,⋯,p2r)} (39)

where symmetrises the momenta, , and . Figure 2: One-loop diagrams for Ψ0[ϕ], up to fourth order in ϕ.

The one-loop diagrams up to fourth order in are shown in figure (2), where the cross denotes the mass counter-term given by the 1-loop term in (30), which we chose to be (so ). These yield the contribution

 Γℏ2(p,−p)=ℏg32π2psinh−1(pM)−ℏδM28πω(p) (40)

and

 Γℏ4(p1,⋯,p4)=−g2ℏ(2π)34!(ω(p1)+⋯+ω(p4))S{∫∞0dq2ω(q)+∑4i=1ω(pi)⋅(−1ω(q)(ω(q)+ω(p1))3(ω(q)+ω(p1)+ω(p2)+ω(q+p1+p2))⋅1(ω(q)+ω(p3)+ω(p4)+ω(−q+p3+p4)))+12ω(p1)∑4i=1ω(pi)} (41)

We will now study how the perturbative calculation of yields the Feynman diagram expansion of equal time Green’s functions when substituted in (28). Keeping only in the exponent and expanding the other contributions to , which we call , yields the Fourier transform of the equal time Green’s functions as

 ∫dx1⋯dxn⟨0|^φ(x1,0)⋯^φ(xn,0)|0⟩e−i∑ni=1pixi=∫D~φe2∫dp~φ(p)Γ2(p,−p)~φ(p)∑(2~W[φ])nn!~φ(p1)⋯~φ(pn) (42)

So we have to contract with the in the diagrams contributing to using the inverse of which is the Fourier transform of the equal time propagator in free-space (a cut-off in spatial momenta will be implied, but not for the time-like momenta which will be integrated out)

 ∫dp2πeipx12√p2+M2=∫dp02πdp2π1p20+p2+M2eipx=G0(x,0,0,0) (43) Figure 3: Diagrams contributing to the equal time two-point function to one-loop. Figure 4: Usual Feynman diagram calculation of the two-point function. The double line is the free space propagator.

If we denote this by a line above the boundary then the diagrams contributing to the equal time two-point function are, to one-loop order, those shown in figure (3), which is to be compared with the usual Feynman diagram calculation of the VEV of two fields in terms of the free space propagator which we denote by a double line in figure (4). We can understand the equivalence of these two sets of diagrams, and those of other VEVs, by studying the gluing together of two free-space propagators in dimensions on a -dimensional plane. Consider

 ∫dyG0(x1,t1,y,t)∂∂tG(y,t,x2,t2)=∫dpdp0dqdq0dy(2π)2(D+1)iq0ei[p(x1−y)+p0(t1−t)+q(y−x2)+q0(t−t2)](p20+p2+M2)(q20+q2+M2)=∫dpdp0dq0dy(2π)(D+2)iq0ei[p(x−y)+p0(t1−t)+q0(t−t2)](p20+ω2(p))(q20+ω2(p)) (44)

The contour may be closed in the upper or lower half-plane, depending on the sign of , , giving

 12ϵ(t−t2)∫dpdp0(2π)D+1ei[p0(t1−t)+ω(p)|t−t2|+p(x−y)]p20+ω2(p) (45)

and a similar treatment of the integration leads to

 12ϵ(t−t2)∫dp(2π)Dei[ω(p)(|t1−t|+|t−t2|)+p(x−y)]2ω(p) (46)

Given that

 G(x,t1,y,t2)=∫dp(2π)D12ω(p)ei[ω(p)|t1−t2|+p(x−y)], (47)

it follows that

 2∫dyG0(x1,t1,y,t)∂∂tG(y,t,x2,t2)=⎧⎪ ⎪ ⎪ ⎪⎨⎪ ⎪ ⎪ ⎪⎩G0(x1,t1,x2,t2)t1>t,t>t2−G0(x1,t1,x2,t2)t1t2−GI(x1,t1,x2,t2)t1>t,t

where is the “image propagator” equal to the free space propagator for the points and the reflection of in the plane at time . In short, if the two points are on opposite sides of the plane at time , the two propagators are “glued” to form the usual propagator, up to a sign, if they are on the same side the gluing produces the image propagator. In the next section we will interpret this relation in terms of the geometry of random paths. It should not be confused with the self-reproducing property of heat-kernels, but plays a nonetheless fundamental role in field theory. For example, applying it twice leads to

 ∫dx2dx3G0(x1,t1,x2,t2)(−∂2∂t2∂t34G0(x2,t2,x3,t3))G0(x3,t3,x4,t4)==G0(x1,t1,x4,t4)fort1>t2>t3>t4 (49) Figure 5: The propagator at equal times is the inverse of 2Γ2.

Taking all the to zero gives a relation which may be expressed graphically as in figure (5), which shows that the propagator at equal times is the inverse of , with the time-splitting regularization we discussed earlier.
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