Noncommutative geometry and path integrals

Mikhail Kapranov

Department of Mathematics, Yale University, 10 Hillhouse Avenue, New Haven, CT, 06520 USA

mikhail.kapranov@yale.edu

To Yuri Ivanovich Manin on his 70th birthday

Introduction

(0.1) A monomial in noncommutative variables X and Y, say, X^{i}Y^{j}X^{k}Y^{l}..., can be visualized as a lattice path in the plane, starting from 0, going i steps in the horizontal direction, j steps in the vertical one, then again k steps in the horizontal one, and so on. Usual commutative monomials are often visualized as lattice points, for example x^{a}y^{b} corresponds to the point (a, b). To lift such a monomial to the noncommutative domain, is therefore the same as to choose a “history” for (a, b), i.e., a lattice path originating at 0 and ending at (a, b).

This correspondence between paths and noncommutative monomials can be extended to more general piecewise smooth paths, if we deal with exponential functions instead. Let us represent our commutative variables as x = e^{z}, y = e^{w}, then a monomial will be replaced by the exponential e^{az+bw} and we are free to take a and b to be any real numbers. To lift this exponential to the noncommutative domain, i.e. to a series in Z, W where X = e^{Z}, Y = e^{W}, one needs to choose a path γ in ℝ^{2} joining 0 with (a, b). One can easily see this by approximating γ by lattice paths with step 1/M, M → ∞, and working with monomials in X^{1/M} = e^{Z/M} and Y^{1/M} = e^{W/M}. Denote this exponential series E_{γ}(Z, W).

This suggests the possibility of a “noncommutative Fourier transform” (NCFT) identifying appopriate spaces of functions of noncommuting variables (say, of matrices of intederminate size) with spaces of ordinary functions or measures on the space of paths. For example, to a measure μ on the space Π of paths (or some completion of it) we want to associate the function F(μ) of Z, W given by

(0.1.1)  F(μ)(Z, W) = ∫_{γ∈Π} E_{γ}(Z, W)dμ(γ),

The basic phenomenon here seems to be that the two types of functional spaces (noncommutative functions of n variables vs. ordinary commutative functions but on the space of paths in ℝ^{n}), have, on some fundamental level, the same size.

The goal of this paper and the ones to follow [K1-2] is to investigate this idea from several points of view.

(0.2) The concept of NCFT seems to implicitly underlie the very foundations of quantum mechanics such as the equivalence of the Lagrangian and Hamiltonian approaches to the theory. Indeed, the Lagrangian point of view deals with path integrals while the Hamiltonian one works with noncommuting operators. Further, it is very close to the concept of the “Wilson loop” functional (trace of the holonomy) in Yang-Mills theory [Po]. Note that the exponential E_{γ}, being itself the holonomy of a certain formal connection, is invariant under reparametrization of the path. Quantities invariant under reparametrization are particularly important in string theory, and the reparametrization invariance of the Wilson loop led to conjectural relations between strings and N → ∞ limit of Yang-Mills theory [Po].

As the integral transform F should, intuitively, act between spaces of the same size, it does not lead to any loss of information and can therefore be viewed as “path integration without integration”. The actual integration occurs when we restrict the function F(μ) to the commutative locus, i.e., make Z and W commute. Alternatively, instead of allowing Z, W to be arbitrary matrices, we take them to be scalars. Then all paths having the same endpoint will contribute to make up a single Fourier mode of the commutativized function. We arrive at the following conclusion: the natural homomorphism R → R_{ab} of a noncommutative ring to its maximal commutative quotient is the algebraic analog of path integration.

(0.3) The idea that the space of paths is related to the free group and to its various versions has been clearly enunciated by K.-T. Chen [C1] in the 1950’s and can be traced throughout almost all of his work [C0]. Apparently, a lot more can be said about this classical subject. Thus, the universal connection with values in the free Lie algebra (known to Chen and appearing in (2.1) below) leads to beautiful non-holonomic geometry on the free nilpotent Lie groups G_{n,d}, which is still far from being fully understood, see [G].

Well known examples of measures on path spaces are provided by probability theory and we spend some time in §4 below to formulate various results from probabilistic literature in terms of NCFT. Most importantly, the Fourier transform of the Wiener measure on paths in ℝ^{n} is the noncommutative Gaussian series exp (− ∑_{i=1}^{n} Z_{j}^{2}) where Z_{i} are considered as noncommuting variables. We should mention here the recent book by Baudoin [Ba] who considered the idea of associating a noncommutative series to a stochastic process. It is clearly the same type of construction as our NCFT except in the framework of probability theory: parametrized paths, positive measures etc.

(0.4) I would like to thank R. Beals, E. Getzler, H. Koch, Y.I. Manin and M. A. Olshanetsky for useful discussions. I am also grateful to the referee for several remarks that helped improve the exposition. This paper was written during my stay at the Max-Planck-Institute für Mathematik in Bonn and I am grateful to the Institute for support and excellent working conditions. This work was also partially supported by an NSF grant.

1 Noncommutative monomials and lattice paths.

(1.1) Noncommutative polynomials and the free semigroup. Consider n noncommuting (free) variables X_{1}, ..., X_{n} and form the algebra of noncommutative polynomials in these variables. This algebra will be denoted by ℂ⟨X_{1}, ..., X_{n}⟩. It is the same as the tensor algebra

T (V) = ⨁_{d=1}^{∞} V^{⊗d}, V = ℂ^{n} = ⨁_{i=1}^{n} ℂ · X_{i}.

A noncommutative monomial in X = (X_{1}, ..., X_{n}) is, as described in the Introduction, the same as a monotone lattice path in ℝ^{n} starting at 0. We denote by F_{n}^{+} the set of all such paths and write X^{γ} for the monomial corresponding to a path γ. The set F_{n}^{+} is a semigroup with the following operation. If γ, γ′ are two monotone paths as above starting at 0, then γ ∘ γ′ is obtained by translating γ so that its beginning meets the end of γ′ and then forming the composite path. It is clear that F_{n}^{+} is the free semigroup on n generators. Thus a typical noncommutative polynomial is written as

(1.1.1)  f(X_{1}, ...X_{n}) = f(X) = ∑_{γ∈F_{n}^{+}} a_{γ}X^{γ}.

Along with ℂ⟨X_{1}, ..., X_{n}⟩ we will consider the algebra ℂ[x_{1}, ..., x_{n}] of usual (commutative) polynomials in the variables x_{1}, ..., x_{n}. A typical such polynomial will be written as

(1.1.2)  g(x_{1}, ..., x_{n}) = g(x) = ∑_{α∈Z_{+}^{n}} b_{α}x^{α}, x^{α} = x_{1}^{α_{1}}...x_{n}^{α_{n}}.

The two algebras are related by the commutativization homomorphism

(1.1.3)  c : ℂ⟨X_{1}, ..., X_{n}⟩ → ℂ[x_{1}, ..., x_{n}],

which takes X_{i} to x_{i}. For a path γ ∈ Γ_{n} let e(γ) ∈ ℤ_{+}^{n} denote the end point of γ. Then we have

(1.1.4)  c(X^{γ}) = x^{e(γ)}.

This means that at the level of coefficients the commutativization homomorphism is given by the summation over paths with given endpoints: if g(x) = c(f(X)), then

(1.1.5)  b_{α} = ∑_{e(γ)=α} a_{γ}.

(1.2) Noncommutative power series. Let I ⊂ ℂ⟨X_{1}, ..., X_{n}⟩ be the span of monomials of degree ≥ 1. Then clearly I is a 2-sided ideal and I^{d} is the span of monomials of degree ≥ d. We define the algebra ℂ⟨⟨X_{1}, ..., X_{n}⟩⟩ as the completion of ℂ⟨X_{1}, ..., X_{n}⟩ in the I-adic topology. Explicitly, elements of ℂ⟨⟨X_{1}, ..., X_{n}⟩⟩ can be seen as infinite formal linear combinations of noncommutative monomials, i.e., expressions of the form ∑_{γ∈F_{N}^{+}} a_{γ}X^{γ}. For example,

(1.2.1)  e^{X_{1}} · e^{X_{2}} = ∑_{i,j=0}^{∞} \frac{X_{1}^{i}X_{2}^{j}}{i!j!} , \frac{1}{1 − (X_{1} + X_{2})} = ∑_{γ∈F_{2}^{+}} X^{γ}

are noncommutative power series. We will be also interested in convergence of noncommutative series. A series f(X) = ∑_{γ∈F_{n}^{+}} a_{γ}X^{γ} will be called entire, if

(1.2.2)  lim_{γ→∞} R^{l(γ)}|a_{γ}| = 0, ∀R > 0.

Here l(γ) is the length of the path γ and the limit is taken over the countable set F_{n}^{+} (so no ordering of this set is needed). We denote by ℂ⟨⟨X_{1}, ..., X_{n}⟩⟩^{ent} the set of entire series. It is clear that this set is a subring.

(1.2.3) Proposition. The condition (1.2.2) is equivalent to the property that for any N and for any square matrices X_{1}^{0}, ..., X_{n}^{0} of size N the series of matrices ∑ a_{γ}(X^{0})^{γ} obtained by specializing X_{i} → X_{i}^{0}, converges absolutely.

(1.3) Noncommutative Laurent polynomials. By a noncommutative Laurent monomial in X_{1}, ..., X_{n} we will mean a monomial in positive and negative powers of the X_{i} such as, e.g., X_{1}X_{2}X_{1}^{−1}X_{2}^{5}. In other words, this is an element of F_{n}, the free noncommutative group on the generators X_{i}. A noncommutative Laurent polynomial is then a finite formal linear combination of such monomials i.e., an element of the group algebra of F_{n}. We will denote this algebra by

(1.3.1)  ℂ⟨X_{1}^{±1}, ..., X_{n}^{±1}⟩ = ℂ[F_{n}].

As before, a noncommutative Laurent monomial corresponds to a lattice path in ℝ^{n} beginning at 0 but not necessarily monotone. These paths are defined up to cancellation of pieces consisting of a sub-path and the same sub-path run in the opposite direction immediately afterwards.

We retain the notation X^{γ} for the monomial corresponding to a path γ. We also write (−γ) for the path inverse to γ, so X^{−γ} = (X^{γ})^{−1}.

(1.4) Noncommutative Fourier transform: discrete case. The usual (commutative) Fourier transform relates the spaces of functions on a locally compact abelian group G and its Pontryagin dual Ĝ. The “discrete” case G = ℤ^{n}, Ĝ = (S^{1})^{n} corresponds to the theory of Fourier series.

In the algebraic formulation, the discrete Fourier transform identifies the space of finitely supported functions

(1.4.1)  b : ℤ^{n} → ℂ, α ↦ b_{α}, | Supp(b)| < ∞,

with the space ℂ[x_{1}^{±1}, ..., x_{n}^{±1}] of Laurent polynomials. It is given by the well known formulas

(1.4.2)  (b_{α}) ↦ f, f(x) = ∑_{α∈ℤ^{n}} b_{α}x^{α},

(1.4.3)  f ↦ (b_{α}), b_{α} = ∫_{|x_{1}|=...=|x_{n}|=1} f(x)x^{−α}d^{∗}x_{1}...d^{∗}x_{n},

where d^{∗}x is the Haar measure on S^{1} with volume 1. Our goal in this section is to give a generalization of these formulas for noncommutative Laurent polynomials.

Instead of (1.4.1) we consider the space of finitely supported functions

(1.4.4)  a : F_{n} → ℂ, γ ↦ a_{γ}, |Supp(a)| < ∞.

The discrete noncommutative Fourier transform is the identification of this space with ℂ⟨X_{1}^{±1}, ..., X_{n}^{±1}⟩ via

(1.4.5)  (a_{γ}) ↦ f, f(X) = ∑_{γ∈F_{n}} a_{γ}X^{γ}.

This identification ceases to look like a tautology if we regard a noncommutative Laurent polynomial as a function f which to any n invertible elements X_{1}^{0}, ..., X_{n}^{0} of any associative algebra A associates an element f(X_{1}^{0}, ..., X_{n}^{0}) ∈ A. We want then to recover the coefficients a_{γ} in terms of the values of f on various elements of various A. Most importantly, we will consider A = Mat_{N}(ℂ), the algebra of matrices of size N and let N be arbitrary. To get a generalization of (1.4.3) we replace the unit circle |x| = 1 by the group of unitary matrices U(N) ⊂ Mat_{N}(ℂ). Let d^{∗}X be the Haar measure on U(N) of volume 1.

The following result is a consequence of the so-called “asymptotic freedom theorem for unitary matrices” due to Voiculescu [V], see also [HP] for a more elementary exposition.

(1.4.6) Theorem. If f(X) = ∑_{γ∈F_{n}} a_{γ}X^{γ} is a noncommutative Laurent polynomial, then we have

a_{γ} = lim_{N→∞} \frac{1}{N} tr ∫_{X_{1},...,X_{n}∈U(N)} f(X_{1}, ..., X_{n}) X^{−γ} d^{∗}X_{1}...d^{∗}X_{n}.

As for the commutative case, the theorem is equivalent to the following orthogonality relation. It is this relation that is usually called the “asymptotic freedom” in the literature.

(1.4.7) Reformulation. Let γ ∈ F_{n} be a nontrivial lattice path. Then

lim_{N→∞} \frac{1}{N} tr ∫_{X_{1},...,X_{n}∈U(N)} X^{γ} d^{∗}X_{1}...d^{∗}X_{n} = 0.

Note that for γ = 0 the integral is equal to 1 for any N.

Passing to the N → ∞ limit is unavoidable here since for any given N there exist nonzero noncommutative polynomials which vanish identically on Mat_{N}(ℂ). An example is provided by the famous Amitsur-Levitsky polynomial

f(X_{1}, ..., X_{2N}) = ∑_{σ∈S_{2N}} sgn(σ)X_{σ(1)} · ... · X_{σ(2N)}.

2 Noncommutative exponential functions.

(2.1) The universal connection and noncommutative exponentials. Let us introduce the “logarithmic” variables Z_{1}, ..., Z_{n}, so that we have the embedding

(2.1.1)  ℂ⟨X_{1}, ..., X_{n}⟩ ⊂ ℂ⟨⟨Z_{1}, ..., Z_{n}⟩⟩, X_{i} ↦ e^{Z_{i}}.

The algebra ℂ⟨⟨Z_{1}, ..., Z_{n}⟩⟩ is a projective limit of finite-dimensional algebras, namely

(2.1.2)  ℂ⟨⟨Z_{1}, ..., Z_{n}⟩⟩ = lim_{⟵}_{d} ℂ⟨Z_{1}, ..., Z_{n}⟩/I^{d},

where the ideal I is as in (1.2).

Consider the space ℝ^{n} with coordinates y_{1}, ..., y_{n}. On this space we have the following 1-form with values in ℂ⟨⟨Z_{1}, ..., Z_{n}⟩⟩:

(2.1.3)  Ω = ∑_{i} Z_{i} · dy_{i} ∈ Ω^{1}(ℝ^{n}) ⊗ ℂ⟨⟨Z_{1}, ..., Z_{n}⟩⟩.

We consider the form as a connection on ℝ^{n}. One can see it as the universal translation invariant connection on ℝ^{n}, an algebraic version of the connection of Kobayashi on the path space, see [Si], §3.

Let γ be any piecewise smooth path in ℝ^{n}. We define the noncommutative exponential function corresponding to γ to be the holonomy of the above connection along γ:

(2.1.4)  E_{γ}(Z) = E_{γ}(Z_{1}, ..., Z_{n}) = P exp ∫_{γ} Ω ∈ ℂ⟨⟨Z_{1}, ..., Z_{n}⟩⟩.

The holonomy can be understood by passing to finite-dimensional quotients as in (2.1.2) and solving an ordinary differential equation with values in each such quotient.

It is clear that E_{γ}(Z) becomes unchanged under parallel translations of γ, since the form Ω is translation invariant. So in the following we will always assume that γ begins at 0.

Further, E_{γ}(Z) is invariant under reparametrizations of γ: this is a general property of the holonomy of any connection. So let us give the following definition.

(2.1.5) Definition. Let M be a C^{∞}-manifold. An (oriented) unparametrized path in M is an equivalence class of pairs (I, γ : I → M) where I is a smooth manifold with boundary diffeomorhic to [0, 1] and γ is a piecewise smooth map I → M. Two such pairs (I, γ) and (I′, γ′) are equivalent if there is an orientation preserving piecewise smooth homeomorphism ϕ : I → I′ such that γ = γ′ ∘ ϕ.

We will denote an unparametrized path simply by γ.

(2.1.6) Example. Let γ be a straight segment in ℝ^{2} joining (0,0) and (1,1). Let also δ be the path consisting of the horizontal segment [(0, 0), (0, 1)] and the vertical segment [(0, 1), (1, 1)]. Let σ be the path consisting of the vertical segment [(0, 0), (1, 0)] and the horizontal segment [(1, 0), (1.1)]. Then

E_{γ}(Z_{1}, Z_{2}) = e^{Z_{1}+Z_{2}}, E_{δ}(Z_{1}, Z_{2}) = e^{Z_{1}}e^{Z_{2}}, E_{σ}(Z) = e^{Z_{2}}e^{Z_{1}}.

More generally, if γ is a lattice path corresponding to the integer lattice ℤ^{n}, then E_{γ}(Z) = X^{γ} is the noncommutative monomial in X_{i} = e^{Z_{i}} associated to γ as in §1.

Let γ, γ′ be two unparametrized paths in ℝ^{n} starting at 0. Their product γ ∘ γ′ is the path obtained by translating γ so that its beginning meets the end of γ′ and then forming the composite path. The set of γ’s with this operation forms a semigroup. For a path γ we denote by γ^{−1} the path obtained by translating γ so that its end meets 0 and then taking it with the opposite orientation. Finally, we denote by Π_{n} the set of paths as above modulo cancellations, i.e., forgetting sub-paths of a given path consisting of a segment and then immediately of the same segment run in the opposite direction. Clearly the set Π_{n} forms a group which we will call the group of paths in ℝ^{n}.

The standard properties of the holonomy of connections imply the following:

(2.1.7) Proposition. (a) We have

E_{γ∘γ′}(Z) = E_{γ}(Z) · E_{γ′}(Z), E_{γ^{−1}}(Z) = E_{γ}(Z)^{−1}

(equalities in ℂ⟨⟨Z_{1}, ..., Z_{n}⟩⟩).

(b) The series E_{γ}(Z) is entire, i.e., it converges for any given N by N matrices Z_{1}^{0}, ..., Z_{n}^{0}.

(c) If Z_{1}^{0}, ..., Z_{n}^{0} are Hermitian, then E_{γ}(iZ_{1}^{0}, ..., iZ_{n}^{0}) is unitary.

The property (a) implies that E_{γ}(Z) depends only on the image of γ in the group Π_{n}. Further, let us consider the commutativisation homomorphism

(2.1.8)  c : ℂ⟨⟨Z_{1}, ..., Z_{n}⟩⟩ → ℂ[[z_{1}, ..., z_{n}]].

The following is also obvious.

(2.1.9) Proposition. If a = (a_{1}, ..., a_{n}) is the endpoint of γ, then

c(E_{γ}(Z)) = e^{(a,z)}

is the usual exponential function.

Thus there are as many ways to lift e^{(a,z)} into the noncommutative domain as there are paths in ℝ^{n} joining 0 and a.

(2.2) Idea of a noncommutative Fourier transform. The above observations suggest that there should be a version of Fourier transform which would identify an appropriate space of measures on Π_{n} with an appropriate space of functions of n noncommutative variables Z_{1}, ..., Z_{n}, via the formula

(2.2.1)  μ ↦ f(Z_{1}, ..., Z_{n}) = ∫_{γ∈Π_{n}} E_{γ}(iZ_{1}, ..., iZ_{n})Dμ(γ).

The integral in (2.2.1) is thus a path integral. The concept of a “function of noncommuting variables” is of course open to interpretation. Several such interpretations are currently being considered in Noncommutative Geometry.

In the present paper we adopt a loose point of view that a function of n noncommutative variables is an element of an algebra R equipped with a homomorphism ℂ⟨Z_{1}, ..., Z_{n}⟩ → R. We will assume that this homomorphism realizes R as some kind of completion, or localization (or both) of ℂ⟨Z_{1}, ..., Z_{n}⟩. In other words, that R does not have “superfluous” elements, independent of the images of the Z_{i}. See [Ta] for an early attempt to define noncommutative functions in the analytic context.

(2.2.2) Examples. We can take R = ℂ⟨⟨Z_{1}, ..., Z_{n}⟩⟩^{ent}, the algebra of entire power series. Alternatively we can take R to be the skew field of “noncommutative rational functions” in Z_{1}, ..., Z_{n} constructed by P. Cohn [Coh]. Thus expressions such as

exp(Z_{1}^{2} + Z_{2}^{2}), (Z_{1}^{2} + Z_{2}^{2})^{−1} , (Z_{1}Z_{2} − Z_{2}Z_{1})^{−1} + Z_{3}^{−2}Z_{1}

are considered as noncommutative functions.

It will be important for us to be able to view a “function” f(Z_{1}, ..., Z_{n}) as above as an actual function defined on appropriate subsets of n-tuples of N by N matrices for each N and taking values in matrices of the same size.

Similarly, the group Π_{n} can also possibly be replaced by various related objects (completions). In this paper we will consider several approaches such as completion by a pro-algebraic group or completion by continuous paths.

Alternatively, functions on Π_{n} should correspond to “noncommutative measures”, or distributions on the space of noncommutative functions. Examples of such “measures” are being studied in Free Probability Theory [HP]

[NS] [VDN]. See §6 below.

Note that we have a surjective homomorphism of groups

(2.2.3)  e : Π_{n} → ℝ^{n}, γ ↦ e(γ).

Here e(γ) is the endpoint of γ. One important property of the NCFT is the following principle which is just a consequence of Proposition 2.1.9: under the Fourier transform the integration over paths with given beginning and end, i.e., the pushdown of measures on Π_{n} to measures on ℝ^{n}, corresponds to a simple algebraic operation: the commutativization homomorphism

(2.2.4)  c : R → R/([R, R]),

where R is a noncommutative algebra and the RHS is the maximal commutative quotient of R.

(2.3) Relation to Chen’s iterated integrals. Let us recall the main points of Chen’s theory. Let M be a smooth manifold, γ an unparametrized path and ω a smooth 1-form on M.

Along with the ”definite integral” ∫_{γ} ω, we can consider the ”indefinite integral” which is a function “on γ”, or, more precisely, on the abstract interval I such that γ is a map I → M. For any t ∈ I we have the sub-path γ_{≤t} going from the beginning of I until t and we have the function

∫_{(γ)} ω : I → ℂ, t ↦ ∫_{γ_{≤t}} ω.

If now ω_{1} and ω_{2} are two smooth 1-forms on M, we can form a new 1-form on γ by multiplying (the restriction of) ω_{2} and the function ∫_{(γ)} ω_{1}. Then this form can be integrated along γ. The result is called the iterated integral

∫\overrightarrow{γ} ω_{1} · ω_{2} = ∫_{γ}(ω_{2} · ∫_{(γ)} ω_{1}).

Note that if we think of γ as a map γ : I → M, then the iterated integral is equal to

∫_{t_{1}≤t_{2}∈I} γ^{∗}(ω_{1})(t_{1})γ^{∗}(ω_{2})(t_{2}).

Note that integration over all t_{1}, t_{2} ∈ I would give the product (∫_{γ} ω_{1})·(∫_{γ} ω_{2}).

Similarly, one defines the d-fold iterated integral of d smooth 1-forms ω_{1}, ..., ω_{d} on M by induction:

∫\overrightarrow{γ} ω_{1} · ... · ω_{d} = ∫_{γ}(ω_{d} · ∫\overrightarrow{(γ)} ω_{1} · ... · ω_{d−1}),

where the (d − 1)-fold indefinite iterated integral is defined as the function on I of the form

t → ∫\overrightarrow{γ_{≤t}} ω_{1} · ... · ω_{d−1}.

As before the iterated integral is equal to the integral over the d-simplex:

∫\overrightarrow{γ} ω_{1} · ... · ω_{d} = ∫_{t_{1}≤...≤t_{d}∈I} γ^{∗}ω_{1}(t_{1})...γ^{∗}ω_{d}(t_{d}).

The concept of iterated integrals extends in an obvious way to 1-forms with values in any associative (pro-)finite-dimensional ℂ-algebra R. The well known Picard series for the holonomy of a connection consists exactly of such iterated integrals. We state this as follows.

(2.3.1) Proposition. Let R be any (pro-)finite-dimensional associative ℂalgebra, and A be a smooth 1-form on M with values in R considered as a connection form. Then the parallel transport along an unparametrized path γ has the form

P exp ∫_{γ} A = ∑_{d=0}^{∞} ∫\overrightarrow{γ} A · ... · A.

Here the term corresponding to d = 0 is set to be equal to 1.

Let us specialize this to M = ℝ^{n}, R = ℂ⟨⟨Z_{1}, ..., Z_{n}⟩⟩ and Ω = ∑ Z_{i}dy_{i}. We obtain:

(2.3.2) Corollary. The coefficient of the series E_{γ}(Z_{1}, ..., Z_{n}) at any noncommutative monomial Z_{i_{1}}...Z_{i_{d}} is equal to the iterated integral

∫\overrightarrow{γ} dy_{i_{1}} · ... · dy_{i_{d}}.

Thus E_{γ} is the generating function for all the iterated integrals involving constant 1-forms on ℝ^{n}.

(2.3.3) Example. By the above

E_{γ}(Z) = 1 + ∑ a_{i}Z_{i} + ∑ b_{ij}Z_{i}Z_{j} + ...

where a_{i} = ∫_{γ} dy_{i} is the ith coordinate of the endpoint of γ and

b_{ij} = ∫_{γ}(dy_{i} · ∫_{(γ)} dy_{j}) = ∫_{γ} y_{j}dy_{i}.

Suppose that γ is closed, so a_{i} = 0. Then b_{ii} = 0 and for i ≠ j we have that b_{ij} is the oriented area encirlced by γ after the projection to the (i, j)-plane.

The following was proved by Chen [C2].

(2.3.4) Theorem. The homomorphism Π_{n} → ℂ⟨⟨Z_{1}, ..., Z_{n}⟩⟩^{∗} sending γ to E_{γ} is injective. In other words, if a path γ has all iterated integrals as above equal to 0, then γ is (equivalent modulo cancellations to) a constant path (situated at 0).

(2.4) Group-like and primitive elements. Let F L(Z_{1}, ..., Z_{n}) be the free Lie algebra generated by Z_{1}, ..., Z_{n}. It is characterized by the obvious universal property, see [R] for background. This property implies that we have a Lie algebra homomorphism

(2.4.1)  h : F L(Z_{1}, ..., Z_{n}) → ℂ⟨Z_{1}, ..., Z_{n}⟩,

and this homomorphism identifies ℂ⟨Z_{1}, ..., Z_{n}⟩ with the universal enveloping algebra of F L(Z_{1}, ..., Z_{n}). Further, let us consider the Hopf algebra structure on ℂ⟨Z_{1}, ..., Z_{n}⟩ given on the generators by

(2.4.2)  Δ(Z_{i}) = Z_{i} ⊗ 1 + 1 ⊗ Z_{i}.

The following result, originally due to K. Friedrichs, is a particular case of a general property of enveloping algebras.

(2.4.3) Theorem. The image of h consists precisely of all primitive elements, i.e., of elements f such that Δ(f) = f ⊗ 1 + 1 ⊗ f.

We will also use the term Lie elements for primitive elements of ℂ⟨Z_{1}, ..., Z_{n}⟩.

Further, consider the noncommutative power series algebra ℂ⟨⟨Z_{1}, ..., Z_{n}⟩⟩. It is naturally a topological Hopf algebra with respect to the comultiplication given by (2.4.2) on generators and extended by additivity, multiplicativity and continuity.

The free Lie algebra is graded:

(2.4.4)  F L(Z_{1}, ..., Z_{n}) = ⨁_{d≥1} F L(Z_{1}, ..., Z_{n})_{d},

where F L(Z_{1}, ..., Z_{n})_{d} is the span of Lie monomials containing exactly d letters. We denote by

(2.4.4)  𝔤_{n} = ∏_{d≥1} F L(Z_{1}, ..., Z_{n})_{d}

its completion, i.e., the set of formal Lie series. This is a complete topological Lie algebra. We clearly have an embedding of 𝔤_{n} into ℂ⟨⟨Z_{1}, ..., Z_{n}⟩⟩ induced by the embedding of the graded components as above. Further, degree-by-degree considerations and Theorem 2.4.3 imply the following:

(2.4.5) Corollary. A noncommutative power series f ∈ ℂ⟨⟨Z_{1}, ..., Z_{n}⟩⟩ lies in 𝔤_{n} if and only if it is primitive, i.e., Δ(f) = f ⊗ 1 + 1 ⊗ f with respect to the topological Hopf algebra structure defined above.

Along with primitive (or Lie) series in Z_{1}, ..., Z_{n} we will consider group-like elements of ℂ⟨⟨Z_{1}, ..., Z_{n}⟩⟩, i.e., series Φ satisfying

(2.4.6)  Δ(Φ) = Φ ⊗ Φ.

The completed tensor product ℂ⟨⟨Z_{1}, ..., Z_{n}⟩⟩ˆ⊗ℂ⟨⟨Z_{1}, ..., Z_{n}⟩⟩ consists of series in 2n variables Z′_{i} = Z_{i} ⊗ 1 and Z′′_{i} = 1 ⊗ Z_{i} which satisfy [Z′_{i}, Z′′_{j}] = 0 and no other relations. Thus a series Φ(Z_{1}, ..., Z_{n}) is group-like if it satisfies the exponential property: (2.4.7) F (Z′_{1}+Z′′_{1}, ..., Z′_{n}+Z′′_{n}) = F (Z′_{1}, ..., Z′_{n})·F (Z′′_{1}, ..., Z′′_{n}), provided [Z′_{i}, Z′′_{j}] = 0, ∀i, j.

We denote by G_{n} the set of primitive elements in ℂ⟨⟨Z_{1}, ..., Z_{n}⟩⟩. Elementary properties of cocommutative Hopf algebras and elementary convergence arguments in the adic topology imply the following:

(2.4.8) Proposition. (a) G_{n} is a group with respect to the multiplication.

(b) The exponential series defines a bijection

exp : 𝔤_{n} → G_{n},

with the inverse given by the logarithmic series.

(c) The image of any series Φ ∈ G_{n} under the commutativization homomorphism (2.1.8) is a formal series of the form e^{(a,z)} for some a ∈ ℂ^{n}.

(d) If Φ ∈ G_{n}, then

Φ(−Z_{1}, ..., −Z_{n}) = Φ(Z_{1}, ..., Z_{n})^{−1}

(equality of power series).

(2.4.9) Example. The above proposition implies that the series

log(e^{Z_{1}} · e^{Z_{2}}) ∈ ℂ⟨⟨Z_{1}, Z_{2}⟩⟩

is in fact a Lie series. It is known as the Campbell-Hausdorff series and its initial part has the form

log(e^{Z_{1}} · e^{Z_{2}}) = Z_{1} + Z_{2} + \frac{1}{2}[Z_{1}, Z_{2}] + ...

Let G_{n}(ℝ) ⊂ G_{n} be the set of group-like series with real coefficients. Further, the Lie algebra F L(Z_{1}, ..., Z_{n}) is in fact defined over rational numbers. In particular, it makes sense to speak about its real part. By taking the completion as above, we define the real part of the completed free algebra 𝔤_{n}(ℝ). It is clear that the exponential series establishes a bijection between 𝔤_{n}(ℝ) and G_{n}(ℝ).

The following fact was also pointed out by Chen [C2].

(2.4.9) Theorem. If γ ∈ Π_{n} is a path in ℝ^{n} as above, then E_{γ}(Z) is grouplike. Moreover, it lies in the real part G_{n}(ℝ).

Note that a typical element Φ = Φ(Z_{1}, ..., Z_{n}) ∈ G_{n} is a priori just a formal power series and does not have to converge for any given matrix values of the Z_{i} (unless they are all 0). At the same time, series of the form Φ = E_{γ}, γ ∈ Π_{n}, converge for all values of the Z_{i}. This leads to the proposal, formulated by Chen [C3] to view series from G_{n} with good covergence properties as corresponding to “generalized paths”, i.e., paths perhaps more general than piecewise C^{∞} ones. The theory of stochastic integrals, see below, provides a step in a similar direction.

(2.5) Finite-dimensional approximations to G_{n} and 𝔤_{n}. Let us recall a version of the Malcev theory for nilpotent Lie algebras. Let k be a field of characteristic 0. A Lie algebra 𝔤 over k is called nilpotent of degree d if all d-fold iterated commutators in 𝔤 vanish. Let U(𝔤) be the universal enveloping algebra of 𝔤. It is a Hopf algebra with the comultiplication given by Δ(x) = x ⊗ 1 + 1 ⊗ x for x ∈ 𝔤. The subspace I in U(𝔤) generated by all nontrivial Lie monomials in elements of 𝔤, is an ideal, with U(𝔤)/I = k.

(2.5.1) Lemma. If 𝔤 is nilpotent of some degree, then ⋂ I^{n} = 0.

Thus the I-adic completion

(2.5.2)  Û(𝔤) = lim_{⟵} U(𝔤)/I^{n}

is a complete topological algebra containing U(𝔤). As before, the standard Hopf algebra structure on U(𝔤) gives rise to a topological Hopf algebra structure on Û(𝔤). We then have the following fact.

(2.5.3) Theorem. (a) 𝔤 is the set of primitive elements of Û(𝔤)

(b) The set G of group-like elements in Û(𝔤) is the nilpotent group associated, via the Malcev theory, to the Lie algebra 𝔤.

(c) If k = ℝ or ℂ, then G is the simply connected real or complex Lie group with Lie algebra 𝔤.

(d) The exponential map establishes a bijection between 𝔤 and G.

Let now k = ℂ and

(2.5.4)  𝔤_{n,d} = F L(X_{1}, ..., X_{n})/F L(X_{1}, ..., X_{n})_{≥d+1}.

This is a finite-dimensional Lie algebra known as the free nilpotent Lie algebra of degree d generated by n elements. It satisfies the obvious universal property. Then

𝔤_{n} = lim_{⟵}_{n} 𝔤_{n,d}.

So 𝔤_{n} is the free pro-nilpotent Lie algebra on n generators.

Let R_{n,d} be the quotient of R_{n} = ℂ⟨⟨Z_{1}, ..., Z_{n}⟩⟩ by the closed ideal generated by all the (d + 1)-fold commutators of the Z_{i}. For example, R_{n,1} = ℂ[[Z_{1}, ..., Z_{n}]] is the usual (commutative) power series algebra.

The topological Hopf algebra structure on R_{n} descends to R_{n,d}, and we easily see the following:

(2.5.5) Proposition. R_{n,d} is isomorphic to Û(𝔤_{n,d}) as a topological Hopf algebra.

We denote by G_{n,d} ⊂ R_{n,d}^{∗} the group of group-like elements of R_{n,d}. Then the above facts imply:

(2.5.6) Theorem. (a) G_{n,d} is the simply connected complex Lie group with Lie algebra 𝔤_{n,d}.

(b) G_{n} is the projective limit of G_{n,d}.

Thus G_{n,d} is the “free unipotent complex algebraic group of degree d with n generators” while G_{n} is the free prounipotent group with n generators.

As above, taking k = ℝ, we get the real parts G_{n,d}(ℝ) and 𝔤_{n,d}(ℝ). The homomorphism E : Π_{n} → G_{n}(ℝ) gives rise, for any d ≥ 1 to the homomorphism

(2.5.7)  ϵ_{n,d} : Π_{n} → G_{n,d}(ℝ)

whose target is a finite-dimensional Lie group.

(2.5.8) Proposition. For any d ≥ 1 the homomorphism ϵ_{n,d} is surjective.

In other words, the group G_{n} can be seen as a (pro-)algebraic completion of the path group Π_{n}. Proof: Let Π_{n}^{rect} ⊂ Π_{n} be the subgroup of rectangular paths, i.e., paths consisting of segments each going in the direction of some particular coordinate. As a group, Π_{n}^{rect} is the free product of n copies of ℝ. Let Z_{i,d} ∈ 𝔤_{n,d} be the image of Z_{i}. Then the image of Π_{n}^{rect} in G_{n,d}(ℝ) is the subgroup generated by the 1-parameter subgroups exp(t · Z_{i,d}), t ∈ ℝ, i = 1, ..., n. As the Z_{i,d} generate 𝔤_{n,d} as a Lie algebra, the corresponding 1-parameter subgroups generate G_{n,d}(ℝ) as a group. Therefore ϵ_{n,d}(Π_{n}^{rect}) = G_{n,d}(ℝ).

(2.6) Complex exponentials. Consider the complexification ℂ^{n} of the space ℝ^{n} from (2.1). The form Ω from (2.1.3) is then a holomorphic form on ℂ^{n} with values in ℂ⟨⟨Z_{1}, ..., Z_{n}⟩⟩. In particular, we have the noncommutative exponential function

E_{γ}(Z) ∈ G_{n} ⊂ ℂ⟨⟨Z_{1}, ..., Z_{n}⟩⟩

for any unparametrized path γ in ℂ^{n} starting at 0. Because Ω is holomorphic, E_{γ}(Z) is, in addition to invariance under cancellations, also invariant under deformations of sub-paths of γ inside holomorphic curves. Let Π_{n}^{ℂ} be the quotient of Π_{2n}, the group of paths in ℂ^{n} = ℝ^{2n} by the equivalence relation generated by such deformations. Obviously, Π_{n}^{ℂ} is a group, and the correspondence γ ↦ E_{γ} gives rise to a homomorphism

(2.6.1)  E : Π_{n}^{ℂ} → G_{n}.

Unlike the real case, it seems to be unknown whether (2.6.1) is injective. As before, we see that the composite homomorphism

(2.6.2)  ϵ_{n,d}^{ℂ} : Π_{n}^{ℂ} → G_{n,d}

is surjective.

(2.6.3) Example. Let C be a complex analytic curve, c_{0} ∈ C be a point, and ϕ : C → ℂ^{n} a holomorphic map such that ϕ(c_{0}) = 0. Denote by p : C̃ → C the universal covering of C corresponding to the base point c_{0}. In other words, C̃ is ths space of pairs (c, γ), where c ∈ C and γ is a homotopy class of paths joinig c_{0} and c. Then, by the above, ϕ induces a map ϕ̃ : C̃ → Π_{n}^{ℂ}. The composition

ϖ_{d} = ϵ_{n,d}^{ℂ} ∘ ϕ̃: C̃ → G_{n,d}

can be called the period map of degree d. The restriction of ϖ_{d} to p^{−1}(c_{0}) = π_{1}(C, c_{0}) is a homomorphism

m_{d} : π_{1}(C, c_{0}) → G_{n,d}

called the monodromy homomorphism of degree d. We get then the “Albanese map”

α_{d} : C → G_{n,d}/Im(m_{d}).

The particular case when C is the maximal Abelian covering of a smooth projective curve of genus n, and ϕ is the Abel-Jacobi map, corresponds to the setting of Parshin [Pa]. Iterated integrals of modular forms were studied by Manin [Ma].

In the subsequent paper [K1] we will use complex noncommutative exponentials to construct invariants of degenerations of families of curves in an algebraic variety.

3 Generalities on NCFT.

(3.0) Formal FT on nilpotent groups. Let us start with the general situation of (2.5) with k = ℝ. Thus 𝔤 is a finite dimensional nilpotent real Lie algebra and G is the corresponding simply connected Lie group. Then G is realized inside Û(𝔤) as the set of group-like elements. In general, we can think of elements of Û(𝔤) as some kind of formal series (infinite formal linear combinations of elements of a Poincare-Birkhoff-Witt basis of U(𝔤)).

To keep the notation straight, we denote by E_{g} ∈ Û(𝔤) the element corresponding to g ∈ G.

(3.0.1) Example. Let G = ℝ^{n} with coordinates y_{1}, ..., y_{n}, then Û(𝔤) is the ring ℂ[[z_{1}, ..., z_{n}]] of formal Taylor series. If g = (y_{1}, ..., y_{n}) ∈ G, then E_{g} = E_{g}(z) = exp (∑_{i} y_{i}z_{i}) is the exponential series with the vector of exponents (y_{1}, ..., y_{n}).

The above example motivates the following definition. Let μ be a measure on G, or, more generally, a distribution (understood as a generalized measure, i.e., as a functional on the space of C^{∞}-functions). Its formal Fourier transform is the element (formal series) given by

(3.0.2)  F̂(μ) = ∫_{g∈G} E_{g}dμ ∈ Û(𝔤),

whenever the integral is defined.

Recall that for two distributions μ, ν on a Lie group G their convolution is defined by

(3.0.4)  μ ∗ ν = m_{∗}(μ ⊠ ν),

where m : G × G → G is the multiplication, and μ ⊠ ν is the Cartesian product of μ and ν. Here we assume that the pushdown under m is defined. The following is then straightforward.

(3.0.5) Proposition. For two (generalized) measures μ, ν on G we have

F̂(μ ∗ ν) = F̂(μ) · F̂(ν),

(product in Û(𝔤)).

(3.1) Pro-measures and formal NCFT. We now specialize the above to the case when G = G_{n,d}(ℝ). In other words, we consider the projective system of Lie groups

(3.1.1)  ⋯ → G_{n,3}(ℝ) → G_{n,2}(ℝ) → G_{n,1}(ℝ) = ℝ^{n}

with projective limit G_{n}(ℝ). For d ≥ d′ let

(3.1.2)  p_{dd′} : G_{n,d}(ℝ) → G_{n,d′}(ℝ)

be the projection. By a pro-measure on G_{n}(ℝ) we will mean a compatible system of measures on the G_{n,d}(ℝ). In other words, a pro-measure is a system μ_{•} = (μ_{d}) such that each μ_{d} is a measure on G_{n,d}(ℝ) such that for any d ≥ d′ the pushdown (p_{dd′})_{∗}(μ_{d}) is defined as a measure on G_{n,d′}(ℝ) and is equal to μ_{d′}. Equivalently, this means that for any continuous function f on G_{n,d′}(ℝ) we have

(3.1.3)  ∫_{G_{n,d′} (ℝ)} f · dμ_{d′} = ∫_{G_{n,d}(ℝ)}(f ∘ p_{dd′}) · dμ_{d},

whenever the LHS is defined.

More generally, by a pro-distribution we mean a system of distributions on the G_{n,d}(ℝ) (understood as generalized measures, i.e., as functionals on C^{∞}-functions) compatible in the similar sense, i.e., satisfying (3.1.3) for C^{∞}- functions f.

For Φ = Φ(Z_{1}, ..., Z_{n}) ∈ G_{n} we denote by Φ_{i_{1},...,i_{p}} the coefficient of Φ at Z_{i_{1}} ⋯ Z_{i_{p}}. It is clear that Φ_{i_{1},...,i_{p}} depends only on the image of Φ in G_{n,p}, so it makes sense to speak about Ψ_{i_{1},...,i_{p}} for Ψ ∈ G_{n,d}, d ≥ p.

Let μ_{•} be a pro-distribution on G_{n}(ℝ). Its formal Fourier transform is the formal series F̂(μ_{•}) ∈ ℂ⟨⟨Z_{1}, ..., Z_{n}⟩⟩ defined as follows:

(3.1.4)  F̂(μ_{•}) = ∑_{p=0}^{∞} ∑_{i_{1},...,i_{p}}(∫_{Ψ∈G_{n,d}(ℝ)} Ψ_{i_{1}...i_{p}} · dμ_{d})Z_{i_{1}} ⋯ Z_{i_{p}}.

Here for each p the number d is any integer greater or equal to p, and we assume that all the integrals converge.

The convolution operation extends, in an obvious way, to pro-distributions on G_{m}(ℝ) and we get:

(3.1.6) Proposition 1. If μ_{•}, ν_{•} are two pro-distributions, then

F̂(μ_{•} ∗ ν_{•}) = F̂(μ_{•}) · F̂(ν_{•})

(product in ℂ⟨⟨Z_{1}, ..., Z_{n}⟩⟩).

(3.2) Delta-functions. In classical analysis the Fourier transform of δ^{(m)}, the mth derivative of the delta function, is the monomial z^{m}. We now give a noncommutative analog of this fact.

First of all, let δ_{d} be the delta function on G_{n,d}(ℝ) supported at 1. Then δ_{•} = (δ_{d}) is a pro-distribution, and

(3.2.1)  F̂(δ_{•}) = 1 ∈ ℂ⟨⟨Z_{1}, ..., Z_{n}⟩⟩.

Next, first derivatives of the delta function at a point on a C^{∞} manifold correspond to elements of the complexified tangent space to the manifold at this point. This, if ξ ∈ F L(Z_{1}, ..., Z_{n}), and ξ_{d} is the image of ξ in 𝔤_{n,d} = T_{1}G_{n,d}(ℝ) ⊗ ℂ, then we have the distribution ∂_{ξ_{d}}(δ_{d}) on G_{n,d}(ℝ), and these distributions form a pro-distribution ∂_{ξ}(δ_{•}).

Further, for any Lie group G with Lie algebra 𝔤 the iterated derivatives of the delta function at 1 correspond to elements of U(𝔤 ⊗ ℂ), the universal enveloping algebra. Thus for any ψ ∈ U(𝔤_{n,d}) we have a punctual distribution D_{ψ}(δ_{d}) on G_{n,d}(ℝ).

Let now f ∈ ℂ⟨Z_{1}, ..., Z_{n}⟩ be a noncommutative polynomial. Recall that ℂ⟨Z_{1}, ..., Z_{n}⟩ is the enveloping algebra of F L(Z_{1}, ..., Z_{d}). Thus for any d we have the image of f in U(𝔤_{n,d}), which we denote by f_{d}. As before, the distributions D_{f_{d}}(δ_{d}) form a pro-distribution which we denote D_{f}(δ_{•}).

(3.2.2) Theorem. We have F̂(D_{f}(δ_{•})) = f. In other words, F̂ takes iterated derivatives of the delta function into (noncommutative) polynomials.

Let L_{f,d} be the left invariant differential operator on G_{n,d}(ℝ) corresponding to f_{d} ∈ U(𝔤_{n,d})). Similarly, let R_{f,d} be the right invariant differential operator corresponding to f_{d}. Recall that distributions (volume forms) form a right module over the ring of differential operators. In other words, if P is a differential operator acting on functions by ϕ ↦ P ϕ, then we write the action of the adjoint operator on volume forms by ω ↦ ωP. Thus, if μ_{•} = (μ_{d}) is a pro-distribution, and f ∈ ℂ⟨Z_{1}, ..., Z_{n}⟩, then we have pro-distributions μ_{•}L_{f} = (μ_{d}L_{f,d}) and μ_{•}R_{f} = (μ_{d}R_{f,d}). Since applying R_{f,d} or L_{f,d} to a distribution is the same as the right or left convolution with D_{f_{d}}(δ_{d}), Proposition 3.1.6 implies the following.

(3.2.3) Proposition. If ϕ ∈ ℂ⟨⟨Z_{1}, ..., Z_{n}⟩⟩ is the Fourier transform of μ_{•}, then for any f ∈ ℂ⟨Z_{1}, ..., Z_{n}⟩ the product f · ϕ is the Fourier transform of μ_{•}L_{f}, and ϕ · f is the Fourier transform of μ_{•}R_{f}.

(3.3) Measures and convergent NCFT. Let p_{d} : G_{n}(ℝ) → G_{n,d}(ℝ) be the projection. By a cylindric open set in G_{n}(ℝ) we mean a set of the form p_{d}^{−1}(U), where d ≥ 1 and U ⊂ G_{n,d}(ℝ) is an open set. These sets form thus a basis of the projective limit topology on G_{n}(ℝ). We denote by 𝔖 the σ-algebra of sets in G_{n}(ℝ) generated by cylindric open sets. Its elements will be simply called Borel subsets in G_{n}(ℝ).

(3.3.1) Example. Let

G_{n}(ℝ)^{ent} = G_{n}(ℝ) ∩ ℂ⟨⟨Z_{1}, ..., Z_{n}⟩⟩^{ent}

be the subgroup formed by entire series, see (1.2.2). Since for Φ ∈ G_{n}(ℝ) each given coefficient of f depends on the image of Φ in some G_{n,d}(ℝ), the condition (1.2.2) implies that G_{n}(ℝ)^{ent} is a Borel subset. Note further that for Φ ∈ G_{n}(ℝ)^{ent} and any Hermitian matrices Z_{1}^{0}, ..., Z_{n}^{0} (of any size N) the matrix Φ(iZ_{1}^{0}, ..., iZ_{n}^{0}) is unitary. This follows from the reality of the coefficients in Φ and from Proposition 2.4.8(d).

By a measure on G_{n}(ℝ) we mean a complex valued, countably additive measure on the σ-algebra 𝔖. If μ is such a measure, we define its Fourier transform to be the function of indeterminate Hermitian N by N matrices Z_{1}, ..., Z_{n} (with indeterminate N) given by

(3.3.2)  F(μ)(Z_{1}, ..., Z_{n}) = ∫_{Φ∈G_{n}(ℝ)^{ent}} Φ(iZ_{1}, ..., iZ_{n})dμ(Φ).

As usual, by a probability measure on G_{n}(ℝ) we mean a real, nonnegativevalued measure on 𝔖 of total volume 1.

Given a pro-measure μ_{•} = (μ_{d}) on G_{n}(ℝ), the correspondence

(3.3.3)  p_{d}^{−1}(U) ↦ μ_{d}(U), U ∈ G_{n,d}(ℝ),

defines a finite-additive function on cylindric open sets in G_{n}(ℝ). The following fact is a version of the basic theorem of Kolmogoroff ( [SW], Thm. 1.1.10) that a stochastic process is uniquely determined by its finite-dimensional distributions.

(3.3.4) Theorem. If μ_{•} is a probability pro-measure (i.e., each μ_{d} is a probability measure), then the correspondence (3.3.3) extends to a unique probability measure μ = lim_{⟵} μ_{d} on G_{n}(ℝ), so that μ_{d} = p_{d∗}(μ).

Thus, probability measures and probability pro-measures are in bijection.

Proof: The original theorem of Kolmogoroff is about probability measures on an infinite product of measure spaces. Now, the projective limit G_{n}(ℝ) is a closed subset in the infinite product ∏_{d} G_{n,d}(ℝ). We can then apply Kolmogoroff’s theorem to this product and get a probability measure supported on this subset.

4 Noncommutative Gaussian and the Wiener measure.

(4.1) Informal overview. By the noncommutative Gaussian we mean the following noncommutative power series

(4.1.1)  Ξ(Z) = exp(−\frac{1}{2} ∑_{i=1}^{n} Z_{i}^{2}) ∈ ℂ⟨⟨Z_{1}, ..., Z_{n}⟩⟩^{ent}.

As the series is entire, we will denote by the same symbol Ξ(Z_{1}, ..., Z_{n}) its value on any given square matrices Z_{1}, ..., Z_{n}. In the classical (commutative) analysis the Fourier transform of a Gaussian is another Gaussian. In this section we present a noncommutative extension of this fact. Informally the answer can be formulated as follows.

(4.1.2) Informal theorem. “The” measure on the space of paths whose Fourier transform gives Ξ(Z) is the Wiener measure.

We write “the” in quotes because so far there is no uniqueness result for NCFT, so (4.1.2) can be read in one direction: that the NCFT of the Wiener measure is Ξ(Z). Still, there are two more issues one has to address in order to make (4.1.2) into a theorem. First, the Wiener measure (see below for a summary) is defined on the space of parametrized paths, while NCFT is defined for measures on the space of unparametrized paths. This can be addressed by considering the pushdown of the Wiener measure (i.e., by performing the integration over the space of parametrized paths).

Second, and more importantly, the Wiener measure is defined on the space of continuous paths, and piecewise smooth paths form a subset of measure 0. On the other hand, the series E_{γ}(Z) is a solution of a differential equation involving the time derivatives of γ and so is a priori not defined if γ is just a continuous path. This difficulty is resolved by using the theory of stochastic integrals and stochastic differential equations which indeed provides a way of associating E_{γ}(Z) to all continuous γ except those forming a set of Wiener measure 0.

Once these two modifications are implemented, (4.1.2) becomes an instance of the familiar principle in the theory of stochastic differential equations: that the direct image of the Wiener measure under the map given by the solution of a stochastic differential equation, is the heat measure for the corresponding (hypo)elliptic operator, see [Bel] [Ok] [Bi1].

(4.2) The hypo-Laplacians and their heat kernels. Let Z_{i,d} be the image of Z_{i} in 𝔤_{n,d}, and L_{i,d} be the left invariant vector field on G_{n,d}(ℝ) corresponding to Z_{i,d}. We consider L_{i,d} as a first order differential operator on functions. The dth hypo-Laplacian is the operator

(4.2.1)  Δ_{d} = ∑_{i=1}^{n} L_{i,d}^{2}

in functions on G_{n,d}(ℝ). For d ≥ d′ the operators Δ_{d} and Δ_{d′} are compatible:

(4.2.2)  Δ_{d}(p_{dd′}^{∗}f) = p_{dd′}^{∗}(Δ_{d′}f), ∀f ∈ C^{2}(G_{d′}(ℝ)),

where the projection p_{dd′} is as in (3.1.2). This follows because a similar compatibility holds for each L_{i,d} and L_{i,d′}.

For d > 1 the number of summands in (4.2.1) is less than the dimension of G_{n,d}(ℝ), so Δ_{d} is not elliptic. However, Δ_{d} is hypoelliptic [Ho], i.e., every distribution solution of Δ_{d}u = 0 is real analytic. This follows from Theorem 1.1 of Hörmander [Ho], since the Z_{i,d} generate 𝔤_{n,d} as a Lie algebra. Further, it is obvious that Δ_{d} is positive:

(4.2.3)  (Δ_{d}u, u) ≥ 0, u ∈ C_{0}^{∞}(G_{n,d}(ℝ)).

General properties of positive hypoelliptic operators [Ho] imply that the heat operator exp(−tΔ_{d}), t > 0, is given by a positive C^{∞} kernel. Because this operator is left invariant, we get part (a) of the following theorem:

(4.2.4) Theorem. (a) The operator exp(Δ_{d}/2) is given by convolution with a uniquely defined probability measure θ_{d} on G_{n,d}(ℝ). This measure is infinitely differentiable with respect to the Haar measure.

(b) For d ≥ d′ the measures θ_{d} and θ_{d′} are compatible: (p_{dd′})_{∗}(θ_{d}) = θ_{d′}.

Part (b) above follows from (4.2.2).

Thus we obtain a probability pro-measure θ_{•} = (θ_{d}) on G_{n}(ℝ) and hence a probability measure θ = lim_{⟵} θ_{d}.

(4.2.5) Examples. (a) the group G_{n,1} is identified with the space ℝ^{n} from (2.1) with coordinates y_{1}, ..., y_{n}, and Z_{i,1} = ∂/∂y_{i}. Therefore Δ_{1} is the standard Laplacian on ℝ^{n}, and

θ_{1} = \frac{dy_{1}...dy_{n}}{(2π)^{n/2}} exp(−\frac{1}{2} ∑_{i=1}^{n} y_{i}^{2})

is the usual Gaussian measure on ℝ^{n}. Each θ_{d}, d > 1 is thus a lift of this measure to G_{n,d}.

(b) For d = 2 an explicit formula for θ_{2} was obtained by Gaveau in [G]. Here we consider the case n = 2 where the formula was also obtained by Hulanicki [Hu]. In this case 𝔤_{2,2} is the Heisenberg Lie algebra with basis consisting of Z_{1,2}, Z_{2,2} and the central element h = [Z_{1,2}, Z_{2,2}]. Denoting y_{1}, y_{2}, v the corresponding exponential coordinates on G_{2,2}, we have

θ_{2} = \frac{dy_{1}dy_{2}dv}{(2π)^{2}} ∫_{τ=−∞}^{∞} \frac{2τ}{sinh(2τ)} · exp(iτv − (y_{1}^{2} + y_{2}^{2})\frac{2τ}{tanh(2τ)})dτ.

In fact, all known formulas in the literature (see [BGG] for a survey) involve integration over auxuliary parameters.

(4.2.6) Theorem. The formal Fourier transform of the pro-measure θ_{•} is equal to the noncommutative Gaussian Ξ(Z).

Proof: This follows from the fact that the delta-pro-distribution D_{Z_{i}}(δ_{•}) corresponding to the generator Z_{i} ∈ 𝔤_{n} is taken by F into the monomial Z_{i}. For each d the corresponding distribution takes a function f on G_{n,d}(ℝ) into the value of L_{i}(f) at the unit element of G_{n,d}(ℝ). Further, convolution of such pro-distributions corresponds to composition of left invariant differential operators in the spaces of functions of the G_{n,d}(ℝ). So the system of the heat kernel operators on the G_{n,d}(ℝ), d ≥ 1, given by exp (−\frac{1}{2} ∑ L_{i}^{2}) has, as a pro-distribution, the Fourier transform equal to exp (−\frac{1}{2} ∑ Z_{i}^{2}). ⊓ ⊔

(4.3) The Wiener measure. Let P_{n} be the space of continuous parametrized paths γ : [0, 1] → ℝ^{n} such that γ(0) = 0. The Wiener measure w on P_{n} is first defined on cylindrical open sets C(t_{1}, ..., t_{m}, U_{1}, ..., U_{m}), where 0 < t_{1} < ..., < t_{m} < 1 and U_{i} ⊂ ℝ^{n} is open. By definition,

C(t_{1}, ..., t_{m}, U_{1}, ..., U_{m}) = {γ : γ(t_{i}) ∈ U_{i}, i = 1, ..., m},

and

(4.3.1)  w(C(t_{1}, ..., t_{m}, U_{1}, ..., U_{m})) =

= ∫_{(y^{(1)},...,y^{(m)})∈U_{1}×...×U_{m}} ∏_{i=0}^{m} \frac{exp(−∥y^{(i+1)} − y^{(i)}∥^{2}/2(t_{i+1} − t_{i}))}{(2π(t_{i+1} − t_{i}))^{1/2}}dy^{(1)}...dy^{(m)}.

Here we put t_{0} = 0, t_{m+1} = 1 and y^{(0)} = 0. Further, it is proved that w extends to a probability measure on the σ-algebra generated by the above subsets.

The Brownian motion is the family of ℝ^{n}-valued functions (random variables) on P_{n} parametrized by t ∈ [0, 1]:

(4.3.2)  b(t) = (b_{1}(t), ..., b_{n}(t)), b(t) : P_{n} → ℝ^{n}, b(t)(γ) = γ(t).

let P_{n}^{sm} ⊂ P_{n} be the subset of piecewuse smooth paths. Then it is well known that w(P_{n}^{sm}) = 0.

As well known, the Wiener measure has the following intuitive interpretation

(4.3.3)  dw(γ) = exp (− ∫_{0}^{1} ∥γ′(t)∥^{2}dt) Dγ, Dγ = ∏_{t=0}^{1} dγ(t).

In other words, Dγ is the (nonexistent) Lebesgue measure on the infinitedimensional vector space of all paths, while the integral in the exponential is the action of a free particle.

(4.5) Reminder on stochastic integrals. Let ω = ∑_{i=1}^{n} ϕ_{i}(y)dy_{i} be a 1form on ℝ^{n} with (complex valued) C^{∞} coefficients. If γ : [0, 1] → ℝ^{n} is a piecewise smooth path, then we can integrate ω along γ, getting a number

(4.5.1)  ∫_{γ} ω = ∫_{0}^{1} γ^{∗}(ω) = ∫_{0}^{1} ∑_{i} ϕ_{i}(γ(t))γ′_{i}(t)dt.

This gives a map (function)

(4.5.2)  ∫ (ω) : P_{n}^{sm} ⟶ ℂ.

If γ(t) is just a continuous path without any differentiability assumptions, then (4.5.1) is not defined, so there is no immediate extension of the map (4.5.2) to the space P_{n}. The theory of stochastic integrals provides several (a priori different) ways to construct such an extension. The two best known approaches are the Ito and Stratonovich integrals over the Brownian motion, see [SW] [KW]. They are functions

(4.5.3)  ∫ ^{Ito}(ω), ∫ ^{Str}(ω) : P_{n} → ℂ,

defined everywhere outside some subset of Wiener measure 0, and measurable with respect to this measure.

To construct them, see [Ok], p. 14-16, one has to consider Riemann sum approximations to the integral but restrict to Riemann sums of some particular type. For a piecewise smooth path γ the integral is the limit of sums

(4.5.4)  ∑_{i=1}^{n} ∑_{ν=1}^{m} ϕ_{i}(γ(ξ_{ν}))((γ_{i}(t_{ν}) − γ_{i}(t_{ν−1})),

where 0 = t_{0} < t_{1} < ... < t_{m} = 1 is a decomposition of [0, 1] into intervals, and ξ_{ν} ∈ [t_{ν−1}, t_{ν}] are some chosen points. In the smooth case the limit exists provided max(t_{ν} −t_{ν−1}) goes to 0 (in particular, the choice of ξ_{ν} is inessential).

Now, to obtain ∫^{Ito}(ω), one chooses the class of Riemann sums with

(4.5.5)  t_{ν} = ν/m, ξ_{ν} = t_{ν−1}, m = 2^{q}, q → ∞.

In other words, for each q the above sum defines a function S_{q}^{Ito}(ω) : P_{n} → ℂ, and

(4.5.6)  ∫ ^{Ito}(ω) = lim_{q→∞} S_{q}^{Ito}(ω).

To obtain ∫^{Str}(ω), one chooses the class of Riemann sums with

(4.5.7)  t_{ν} = ν/m, ξ_{ν} = (t_{ν−1} + t_{ν})/2, m = 2^{q}, q → ∞.

Each such sum gives a function S_{q}^{Str}(ω) : P_{n} → ℂ, and

(4.5.8)  ∫ ^{Str}(ω) = lim_{q→∞} S_{q}^{Str}(ω).

It is known that ∫^{Str}(ω) is invariant under smooth reparametrizations of the path considered as transformations acting on P_{n} as well as satisfies a transparent change of variables formula.

The more common notation for the stochastic integrals (considered as random variables on P_{n}) is:

(4.5.9)  ∫ ^{Ito}(ω) = ∫_{0}^{1} ω(b(t))db(t)), ∫ ^{Str}(ω) = ∫_{0}^{1} ω(b(t)) ∘ db(t)),

where b(t) is the Brownian motion (4.3.2). Thus db(t) and ∘db(t) stand for the two ways (due to Ito and Stratonovich) of regularizing the (a priori divergent) differential of the Brownian path b(t). See [Ok] for the relation between the two regularization schemes. By restricting to the truncated path [0, s], s ≤ 1, one defines the stochastic integrals ∫_{0}^{s} in each of the above setting.

(4.6) Stochastic holonomy. Let G be a Lie group which we suppose to be embedded as a closed subgroup of GL_{N}(ℂ) for some N and let 𝔤 ⊂ Mat_{N}(ℂ) be the Lie algebra of G. Let A = ∑ A_{i}(y)dy_{i} be a smooth 𝔤-valued 1-form on ℝ^{n}, which we consider as a connection in the trivial G-bundle over ℝ^{n}. If γ : [0, 1] → ℝ^{n} is a piecewise smooth path, then we have the holonomy of A along γ:

(4.6.1)  Hol_{γ}(A) = P exp ∫_{γ} A ∈ G.

It is the value at t = 1 of the solution U(t) ∈ GL_{N}(ℂ) of the differential equation

(4.6.2)  \frac{dU}{dt} = U(t)(∑_{i} A_{i}(γ(t)) · γ′_{i}(t)), U(0) = 1.

The holonomy defines thus a map

(4.6.3)  Hol(A) : P_{n}^{sm} → G.

As before, (4.6.2) and thus Hol_{γ}(A) have no immediate sense without some differentiability assumptions on A.

The theory of stochastic differential equations [Ok] [KW] resolves this difficulty by replacing the above differential equation by an integral equation and understanding the integral in a regularized sense as in (4.5). Thus, one defines the Ito and Stratonovich stochastic holonomies which are measurable maps

(4.6.3)  Hol^{Ito}(A), Hol^{Str}(A) : P_{n} → G,

defined outside a subset of Wiener measure 0. For example, Hol^{Str}(A) is defined as the value at t = 1 of the G-valued stochastic process U(t) satisfying the Stratonovich integral equation

(4.6.4)  U(t) = 1 + ∫_{0}^{t} U(s)(∑_{i} A_{i}(b(s)) ∘ db_{i}(s)).

We will be particularly interested in the case when A_{i} are constant, i.e., our connection is translation invariant. In this case B(t) = ∑ A_{i}b_{i}(t) is a (possibly degenerate) Brownian motion on 𝔤 and U(t) is the corresponding left invariant Brownian motion on G as studied by McKean, see [McK], §4.7, and also [HL]. In particular, the Stratonovich holonomy can be represented as a “product integral” in the sense of McKean: (4.6.5)

Hol^{Str}(A) = ∏_{t∈[0,1]} exp(dB(t)) := lim_{q→∞} ∏_{ν=1}^{2^{q}} exp(B (\frac{ν}{2^{q}} ) − B (\frac{ν − 1}{2^{q}})),

see [HL], Thm.2. Here the product is taken in the order of increasing ν. In the sequel we will work with Hol^{Str}(A).

(4.7) The Malliavin calculus and the Feynman-Kac-Bismut formula. We now specialize (4.6) to the case when G = G_{n,d}(ℝ), 𝔤 = 𝔤_{n,d}(ℝ) and A = Ω^{(d)} is the constant 1-form Ω^{(d)} = ∑_{i=1}^{n} Z_{i,d}dy_{i}. We get the stochastic holonomy map

(4.7.1)  Hol^{Str}(Ω^{(d)}) : P_{n} → G_{n,d}(ℝ).

(4.7.2) Theorem. The probability measure θ_{d} on G_{n,d}(ℝ) is equal to

Hol^{Str}(Ω^{(d)})_{∗}(w),

the push-down of the Wiener measure under the holonomy map.

Proof: This is a fundamental property of (hypo)elliptic diffusions holding for any vector fields ξ_{1}, ..., ξ_{n} on a manifold M such that iterated commutators of the ξ_{i} span the tangent space at every point. In this case the operator Δ = ∑ Lie_{ξ_{i}}^{2} is hypoelliptic and has a uniquely defined, smooth heat kernel Θ(x, y), x, y ∈ M which is a function in x and a volume form in y and represents the operator exp(−Δ/2). Further, the heat equation

(4.7.3)  ∂u/∂t = −Δ(u)/2

is the “Kolmogoroff backward equation” for the M-valued stochastic process U(t) satisfying the Stratonovich differential equation

(4.7.4)  dU = ∑ L_{ξ_{i}}(U) ∘ db_{i}

with the b_{i}(t) being as before. This means that the fundamental solution of (4.7.3) is the pushforward of the Wiener measure under the process U(t). See [Ok], Th. 8.1. Our case is obtained by specializing to M = G_{n,d}(ℝ), ξ_{i} = Z_{i,d}.

⊓ ⊔

Further, let θ be the probability measure on G_{n}(ℝ) = lim_{⟵}_{d} G_{n,d}(ℝ) corresponding to the pro-measure (θ_{d}) by Theorem 3.3.5. Note that the maps Hol^{Str}(Ω^{(d)}) for various d unite into a map

(4.7.4)  Hol^{Str}(Ω) : P_{n} → G_{n}(ℝ), Ω = ∑ Z_{i}dy_{i}.

We get the following corollary.

(4.7.5) Corollary. The measure θ is the pushdown of the Wiener measure under Hol^{Str}(Ω).

(4.7.6) Theorem. (a) The support of the measure θ is contained in G_{n}(ℝ)^{ent}, the set of entire group-like power series.

(b) The convergent Fourier transform of θ is equal to Ξ(Z). In other words (taking into account part (a) and (4.7.5)), for any given Hermitian matrices Z_{1}^{0}, ..., Z_{n}^{0} of any given size N we have

exp(−\frac{1}{2} ∑_{j=1}^{n}(Z_{j}^{0})^{2}) = ∫_{γ∈P_{n}} Hol_{γ}^{Str}(A)(iZ_{1}^{0}, ..., iZ_{n}^{0})dw(γ).

(4.8) Stochastic iterated integrals and the proof of Theorem 4.7.6. In the situation of (4.5), assume given d smooth 1-forms ω_{1}, ..., ω_{d} on ℝ^{n}. We then define, following Fliess and Normand-Cyrot [FN], the iterated Stratonovich integral

(4.8.1)  ∫ ^{→Str}(ω_{1} · ... · ω_{d}) : P_{n} → ℂ

by the same iterative procedure as in (2.3). Like the ordinary Stratonovich integral, it is reparametrization invariant. This definition extends to the case when each ω_{i} takes values in a (pro-)finite-dimensional associative ℂ-algebra R (with unity). As before, we define the empty iterated integral (corresponding to d = 0) to be equal to 1. We will need some extensions of Proposition 2.3.1 to the stochastic case. The first statement deals with the nilpotent case.

(4.8.2) Proposition. Let I ⊂ R be a nilpotent ideal, i.e., I^{m} = 0 for some m. Let A be a smooth 1-form on ℝ^{n} with values in I. Consider A as a connection form with coefficients in R. Then

Hol^{Str}(A) = ∑_{d=0}^{∞} ∫ ^{→Str}(A · ... · A),

the series on the right being terminating.

Proof: This is a consequence of Theorem 2 of [FN].

(4.8.3) Corollary. The random variable Hol^{Str}(Ω) from (4.7.4), considered as a ℝ⟨⟨Z_{1}, ..., Z_{n}⟩⟩-valued random variable on P_{n}, has the form

Hol^{Str}(Ω) = ∑_{m=0}^{∞} ∑_{J=(j_{1},...,j_{m})} Z_{j_{1}}...Z_{j_{m}} ∫ ^{→Str}(dy_{j_{1}} · ... · dy_{j_{m}}).

Next, we look at convergence of the series in (4.8.3). Questions of this nature (“convergence of stochastic Taylor series”) were studied by Ben Arous [Be], and we recall some of his results. Denote by

(4.8.4)  B_{J} = ∫ ^{→Str}(dy_{j_{1}} · ... · dy_{j_{m}}) = ∫ db_{j_{1}} ∘ ... ∘ db_{j_{m}}

the coefficient in the series (4.8.3) corresponding to the multiindex J. Here the right hand side is the notation of [Be]. Let |J| = ∑ j_{ν} be the degree of the monomial corresponding to J.

(4.8.5) Theorem. Let (x_{J}) be a collection of real numbers given for each J = (j_{1}, ..., j_{m}), m ≥ 0, and satisfying the condition

|x_{J}| ≤ K^{J}, for some K > 0.

Then the series

∑_{m} ∑_{|J|=m} |x_{J}B_{J}|

of random variables on P_{n} converges almost surely.

This is Corollary 1 of [Be] (with the parameter α from loc. cit. taken to be 0).

We now deduce Theorem 4.7.6 from the above results. Let Z_{1}^{0}, ..., Z_{n}^{0} be fixed matrices of any given size N. For a matrix B denote by

∥B∥ = max_{v≠0} \frac{∥B(v)∥}{∥v∥}

the matrix norm of B. Let us apply Theorem 4.8.5 to

x_{J} = ∥(Z^{0})^{J}∥ = ∥Z_{j_{1}}^{0}...Z_{j_{m}}^{0}∥.

Take K = max(∥Z_{i}^{0}∥). Then |x_{J}| ≤ K^{J}, so (4.8.5) gives that the series ∑_{J} B_{J}(Z^{0})^{J} converges absolutely almost surely. This establishes part (a) of Theorem 4.7.6. Part (b) follows from (a) and from Theorem 4.2.6 about the formal Fourier transform.

5 Futher examples of NCFT.

(5.1) Near-Gaussians. In classical analysis, a near-Gaussian is a function of the form f(z)·e^{−∥z∥^{2}/2} where f(z), z = (z_{1}, ..., z_{n}), is a polynomial. In that setting, the Fourier transform of a near-Gaussian is another near-Gaussian.

A natural noncommutative analog of a near-Gaussian is a function of the form

(5.1.1)  F (Z) · Ξ(Z) · G(Z), F, G ∈ ℂ⟨Z_{1}, ..., Z_{n}⟩.

It can be represented as a (formal) Fourier transform using Proposition 3.2.3:

(5.1.2)  F (Z) · Ξ(Z) · G(Z) = F̂(θ_{•}L_{F} R_{G}),

where L_{F} is the system of left invariant differential operators on the G_{n,d}(ℝ), d ≥ 1, corresponding to F, while R_{G} is the system of right invariant differential operators corresponding to G.

It seems difficult to realize the measures θ_{d}L_{F} R_{G}, d ≥ 1, in terms of some transparent measures on the space P_{n}, as it requires using group translations on Π_{n}^{cont}, the group of continuous paths obtained by quotienting P_{n} by reparametrizations and cancellations.

(5.2) The Green pro-measure. Let g_{d} be the fundamental solution of the dth hypo-Laplacian on G_{n,d}(ℝ) centered at 1, the unit element, i.e.,

(5.2.1)  Δ_{d}(g_{d}) = δ_{1}.

By the general properties of hypoelliptic operators, g_{d} is a measure (volume form) on G_{n,d}(ℝ) smooth away from 1. In fact, if we denote by θ_{d,t} the kernel of exp(−tΔ_{d}/2), t > 0, i.e., the heat kernel measure at time t, then

(5.2.2)  g_{d} = ∫_{t=0}^{∞} θ_{d,t}dt.

This expresses the fact that the Green measure of a domain is equal to the amount of time a diffusion path spends in the domain. It is clear therefore that g_{•} = (g_{d}) is a pro-measure on G_{n}(ℝ).

(5.2.3) Examples. (a) For d = 1 we have the Green function of the usual Euclidean Laplacian in ℝ^{n} which has the form

g_{1}(y) = \frac{1}{4π} ln(y_{1}^{2} + y_{2}^{2})dy_{1}dy_{2}, n = 2,

g_{1}(y) = −\frac{((n/2) − 2)!}{4π^{n/2}} (∑ y_{i}^{2})^{1−n/2} dy_{1}...dy_{n}.

(b) Consider the case n = 2, d = 2 corresponding to the Heisenberg group, and let us use the exponential coordinates y_{1}, y_{2}, v as in Example 4.2.5(b). Then

g_{2}(y_{1}, y_{2}, v) = \frac{1}{π} \frac{1}{\sqrt{(y_{1}^{2} + y_{2}^{2}) + v^{2}}} dy_{1}dy_{2}dv,

as was found by Folland [Fo], see also [G], p. 101.

(5.3) The method of kernels. More generally, if F (Z_{1}, ..., Z_{n}) is a “noncommutative function” such that the operator F (L_{1,d}, ..., L_{n,d}) in functions on G_{n,d}(ℝ) makes sense and posesses a distribution kernel K_{d}(x, y)dy, then the distribution μ_{d} = K_{d}(1, y)dy is precisely the dth component of the prodistribution whose Fourier transform is F.

For example, hypoelliptic calculus allows us to consider F (Z) = ϕ (∑_{i=1}^{n} Z_{i}^{2}) where ϕ : ℝ → ℝ is any C^{∞} function decaying at infinity such as ϕ(u) = e^{−u^{2}/2} or ϕ(u) = 1/u, or 1/(u^{2} + 1). This leads to a considerable supply of prodistributions.

(5.4) Probabilistic meaning. An idea in probability theory very similar to our NCFT, viz. the idea of associating a noncommutative power series to a stochastic process, was proposed by Baudoin [Ba], who called this series “expectation of the signature” and emphasized its importance. From the general viewpoint of probability theory one can look at this series (the Fourier transform of a probability measure on the space of paths) as being rather an analog of the characteristic function of n random variables. Indeed, if x_{1}, ..., x_{n} are random variables, then their joint distribution is a probability measure on ℝ^{n}, and the characteristic function is the (usual) Fourier transform of this measure:

(5.4.1)  f(z_{1}, ..., z_{n}) = 𝔼[e^{i(z,x)}]

which is an entire function of n variables. Each time when we have a natural lifting of the characteristic function to the noncommutative domain, we can therefore expect some n-dimensional stochastic process lurking in the background.

6 Fourier transform of noncommutative measures.

(6.1) Nomcommutative measures. Following the general approach of Noncommutative Geometry [Con], we consider a possibly noncommutative ℂ-algebra R (with unit) as a replacement of a “space” (Spec(A)). A measure on R is then simply a linear functional (“integration map”) τ : I → ℂ defined on an appropriate subspace I ⊂ R whose elements have the meaning of integrable functions. We will call a measure τ finite, if I = R, and normalized, if it is finite and τ(1) = 1. If R has a structure of a ∗-algebra, then a finite measure τ is called positive if τ(aa^{∗}) ≥ 0 for any a ∈ R. A (noncommutative) probability measure on a ∗-algebra A is a normalized, positive measure.

(6.1.1) Examples. (a) Let R = Mat_{N}(ℂ) with the ∗-algebra structure given by the Hermitian conjugation. Then τ(a) = \frac{1}{N} Tr(a) is a probability measure.

(b) Let R = ℂ⟨Z_{1}, ..., Z_{n}⟩ with the ∗-algebra structure given by Z_{i}^{∗} = Z_{i}. Let Herm_{N} be the space of Hermitian N by N matrices. We denote by dZ = ∏_{i,j=1}^{N} dZ_{ij} the standard volume form on Herm_{N}. Let μ = μ_{N} be a volume form on (Herm_{N})^{n} of exponential decay at infinity. Then we have a finite measure on R given by

τ(f) = \frac{1}{N} Tr ∫_{Z_{1},...,Z_{n}∈Herm_{N}} f(Z_{1}, ..., Z_{n})dμ(Z_{1}, ..., Z_{n}).

If μ_{N} is a normalized (resp. probability) measure in the usual sense, then τ is a normalized (resp. probability) measure in the noncommutative sense. An important example is

μ_{N} = exp (−S(Z_{1}, ..., Z_{n})) dZ_{1}...dZ_{n},

where the “action” S(Z_{1}, ..., Z_{n}) ∈ ℂ⟨Z_{1}, ..., Z_{n}⟩ is a noncommutative polynomial with appropriate growth conditions at the infinity of Herm_{N}^{n}.

(c) Let R = ℂ⟨X_{1}^{±1}, ..., X_{n}^{±1}⟩ with the ∗-algebra structure given by X_{i}^{∗} = X_{i}^{−1}. If μ = μ_{N} is a finite measure on U(N)^{n}, then we have a finite measure τ on R given by

τ(f) = \frac{1}{N} Tr ∫_{X_{1},...,X_{n}∈U(N)} f(X_{1}, ..., X_{n})dμ(X_{1}, ..., X_{n}),

which is normalized (resp. probability) if μ_{N} is so in the usual sense.

(6.2) Free products. Let R_{1}, ..., R_{n} be algebras with unit. Then we have their free product R_{1} ⋆ .. ⋆ R_{n}. This is an algebra containing all the R_{i} and characterized by the following universal property: for any algebra B and any homomorphisms f_{i} : R_{i} → B there is a unique homomorphism f : R_{1} ⋆ ... ⋆ R_{n} → B restricting to f_{i} on R_{i} for each i. Explicitly, R_{1} ⋆ ... ⋆ R_{n} is obtained as the quotient of the free (tensor) algebra generated by the vector space R_{1} ⊕ ... ⊕ R_{n} by the relations saying the products of elements from each R_{i} are given by the existing multiplication in R_{i}. We will also use the notation ⋆_{i=1}^{n}R_{i}. (6.2.1) Example. If each R_{i} = ℂ[Z_{i}] is the polynomial algebra in one variable, then R_{1} ⋆ ... ⋆ R_{n} = ℂ⟨Z_{1}, ..., Z_{n}⟩ is the algebra of noncommutative polynomials. If each R_{i} = ℂ[X_{i}, X_{i}^{−1}] us the algebra of Laurent polynomials, then R_{1}⋆...⋆R_{n} = ℂ⟨X_{1}^{±1}, ..., X_{n}^{±1}⟩ is the algebra of noncommutative Laurent polynomials.

The following description of the free product follows easily from definition (see [VDN]).

(6.2.2) Proposition. Suppose that for each i we chose a subspace R_{i}^{∘} ⊂ R_{i} which is a complement to ℂ · 1. Then as a vector space

R_{1} ⋆ ... ⋆ R_{n} = ℂ · 1 ⊕ ⨁_{k>0} ⨁_{i_{1}≠i_{2}≠...≠i_{k}} R_{i_{1}}^{∘} ⊗ ... ⊗ R_{i_{k}}^{∘}.

The following definition of the free product of (noncommutative) measures is due to Voiculescu, see [VDN].

(6.2.3) Proposition-Definition 1. Let R_{i}, i = 1, ..., n be associative algebras with 1, and τ_{i} : R_{i} → ℂ be finite normalized measures. Then there exists a unique finite normalized measure τ = ⋆τ_{i} on ⋆_{i=1}^{n}R_{i} with the following properties:

(1) τ|_{R_{i}} = τ_{i}.

(2) If i_{1} ≠ ... ≠ i_{k} and a_{ν} ∈ R_{ν} are such that τ_{i_{ν}} (a_{ν}) = 0, then τ(a_{i_{1}}...a_{i_{k}}) = 0. If the R_{i} are ∗-algebras, and each τ_{i} is a probability measure, then so is τ.

Both the existence and the uniqueness of τ follow at once from (6.2.2), if we take R_{i}^{∘} = Ker(τ_{i}). The problem of finding τ(a_{1}...a_{k}) for arbitrary elements a_{ν} ∈ R_{i_{ν}} is clearly equivalent to that of writing a_{1}...a_{k} in the normal form (6.2.2). To do this, one writes

(6.2.4)  a_{ν} = τ_{i_{ν}} (a_{ν}) · 1 + a_{ν}^{∘},

with a_{ν}^{∘} defined so as to satisfy (6.2.4) and we have ϕ_{i_{ν}} (a_{ν}^{∘}) = 0. Then one uses the conditions (1) and (2) to distribute.

(6.2.5) Examples. Suppose we have two algebras A and B and normalized measures ϕ : A → ℂ and ψ : B → ℂ. Let χ : A ⋆ B → ℂ be the free product of ϕ and ψ. Then for a, a′ ∈ A and b, b′ ∈ B we have, after some calculations:

χ(ab) = ϕ(a)ψ(b), χ(aba′) = ϕ(aa′)ψ(b),
χ(aba′b′) = ϕ(aa′)ψ(b)ψ(b′) + ϕ(a)ϕ(a′)ψ(bb′) − ϕ(a)ϕ(a′)ψ(b)ψ(b′).

See [NS], Thm. 14.4, for a general formula for χ(a_{1}b_{1}...a_{m}b_{m}), a_{i} ∈ A, b_{i} ∈ B.

(6.2.6) Examples. (a) Let R_{i} = ℂ[x^{±1}], i = 1, ..., n and let τ_{i} be given by the integration over the normalized Haar measure d^{∗}x on the unit circle. Thus

τ_{i} (f(x) = ∑_{m} a_{m}x^{m}) = ∫_{|x|=1} f(x)d^{∗}x = a_{0}.

The free product of these measures is the functional on ℂ⟨X_{1}^{±1}, ..., X_{n}^{±1}⟩ given by

τ (f(X) = ∑_{γ∈F_{n}} a_{γ}X^{γ} ) = a_{0},

the constant term of a noncommutative Laurent polynomial. The asymptotic freedom theorem for unitary matrices (1.4.7) says that this functional is the limit, as N → ∞, of the functionals from Example 6.1.1(c) with μ_{N}, for each N, being the normalized Haar measure on U(N)^{n}.

(b) Let R_{i} = ℂ[z], i = 1, ..., n, and let τ_{i} = δ(z − a_{i}) be the Dirac deltafunction situated at a point a_{i} ∈ ℂ, i.e., τ_{i}(f) = f(a_{i}). Then the free product τ = τ_{1} ⋆ ... ⋆ τ_{n} is given by:

τ(f(Z_{1}, ..., Z_{n})) = f(a_{1} · 1, ..., a_{n} · 1),

in other words, it depends only on the image of f in the ring of commutative polynomials ℂ[z_{1}, ..., z_{n}]. This can be seen from the procedure (6.2.4) using the fact that each τ_{i} : ℂ[z] → ℂ is a ring homomorphism.

(c) Let R_{i} = ℂ[z], i = 1, ..., n, and let τ_{i} be the integration over the standard Gaussian probability measure

τ_{i}(f) = \frac{1}{\sqrt{2π}} ∫_{−∞}^{∞} f(z)e^{−z^{2}/2}dz.

Their free product is a probability measure on ℂ⟨Z_{1}, ..., Z_{n}⟩ denoted by ξ_{n} and called the free Gaussian measure. The asymptotic freedom for Hermitian Gaussian ensembles [V] can be formulated as follows.

(6.2.7) Theorem. The measure ξ_{n} is the limit, as N → ∞, of the measures from Example 6.1.1(b) where, for each N, we take for μ_{N} the Gaussian probability measure on the vector space (Herm_{N})^{n} corresponding to the scalar product ∑ Tr(A_{i}B_{i}) on this vector space:

μ = μ_{N} = \frac{1}{(2π)^{nN^{2}/2}} exp (−\frac{1}{2} ∑_{i=1}^{n} Z_{i}^{2}) dZ_{1}...dZ_{n}.

(6.3) The Fourier transform of noncommutative measures. In classical analysis, Fourier transform is defined for measures on ℝ^{n}, not on an arbitrary curved manifold. We will call a measure on ℝ_{NC}^{n} (“noncommutative ℝ^{n}”) a datum consisting of a ∗-algebra R, a ∗-homomorphism ℂ⟨Z_{1}, ..., Z_{n}⟩ → R (i.e., a choice of self-adjoint elements in R which we will still denote Z_{i}), and a measure τ on R. Elements of R for which τ is defined, will be thought of as functions integrable with respect to the measure. This concept is thus very similar to that of n noncommutative random variables in noncommutative probability theory, except we do not require any positivity or normalization.

Let τ be a measure on ℝ_{NC}^{n}. Its Fourier transform is the complex valued function 𝔉(τ) on the group Π_{n} of piecewise smooth paths in ℝ^{n} defined as follows:

(6.3.1)  𝔉(τ)(γ) = τ(E_{γ}(iZ_{1}, ..., iZ_{n})), γ ∈ Π_{n}.

Here we assume that the “entire function” E_{γ}(iZ_{1}, ..., iZ_{n}) lies in the domain of definition of τ. In physical terminology, 𝔉(τ)(γ) is the “Wilson loop functional” (defined here for non-closed paths as well).

(6.3.2) Example: delta-functions. (a) For every J = (j_{1}, ..., j_{m}) we have the measure δ^{(J)} on ℂ⟨⟨Z_{1}, ..., Z_{n}⟩⟩ given by

δ^{(J)} (∑_{m} ∑_{I=(i_{1},...,i_{m})} a_{I}Z_{i_{1}}...Z_{i_{m}} ) = a_{J}.

The Fourier transform of δ^{(J)} is the function W_{J} : Π_{n} → ℂ which associates to a path γ the iterated integral along γ labelled by J:

W_{J}(γ) = ∫\overrightarrow{γ} dy_{j_{1}}...dy_{j_{m}}.

We will call these functions monomial functions on Π_{n}

(b) If we take for τ the free product of (underived) delta-functions δ_{a_{1}} ⋆ ... ⋆ δ_{a_{n}}, as in Example 6.2.6(b), then

𝔉(τ)(γ) = exp(i(e(γ), a)),

where e(γ) ∈ ℝ^{n} is the endpoint of γ. This follows from the fact that τ is supported on the commutative locus, i.e., τ(E_{γ}(iZ)) depends only on the image of E_{γ}(iZ) in the commutative power series ring, which is exp(ie(γ), z)).

(6.4) Convolution and product. Let τ, σ be two measures on ℝ_{NC}^{n}, so we have homomorphisms

α : ℂ⟨Z_{1}, ..., Z_{n}⟩ → R, β : ℂ⟨Z_{1}, ..., Z_{n}⟩ → S

and τ is a linear functional on R while σ is a linear functional on S. Their (tensor) convolution is the measure τ ∗ σ which corresponds to the homomorphism

(6.4.1)  ℂ⟨Z_{1}, ..., Z_{n}⟩ → R ⊗ S, Z_{i} ↦ α(Z_{i}) ⊗ 1 + 1 ⊗ β(Z_{i}),

and the linear functional

τ ∗ σ : R ⊗ S → ℂ, r ⊗ s ↦ τ(r) ⊗ σ(s).

For commutative algebras this corresponds to the usual convolution of measures with respect to the group structure on ℝ^{n}.

(6.4.2) Proposition. The Fourier transform of the convolution of measures is the product of their Fourier transforms:

𝔉(τ ∗ σ) = 𝔉(τ) · 𝔉(σ).

Proof: This is a consequence of the fact that the elements E_{γ}(iZ_{1}, ..., iZ_{n}) of ℂ⟨⟨Z_{1}, ..., Z_{n}⟩⟩ are group-like, see the exponential property (2.4.7). ⊓ ⊔

(6.5) Formal Fourier transform of noncommutative measures. The product of two monomial functions on Π_{n} is a linear combination of monomial functions. This expresses Chen’s shuffle relations among iterated integrals:

(6.5.1)  W_{j_{1},...,j_{m}}W_{j_{m+1},...,j_{m+p}} = ∑_{s} W_{j_{s(1)},...,j_{s(m+p)}},

the sum being over the set of (m, p)-shuffles. An identical formula holds for the convolution of the measures δ^{(j_{1},...,j_{m})} and δ^{(j_{m+1},...,j_{m+p})}, as both formulas describe the Hopf algebra structure on ℂ⟨⟨Z_{1}, ..., Z_{n}⟩⟩.

The ℂ-algebra with basis W_{J} = W_{j_{1},...,j_{m}} and multiplication law (6.5.1) is nothing but the algebra

(6.5.2)  ℂ[G_{n}] = lim_{⟶} ℂ[G_{n,d}]

of regular functions on the group scheme G_{n} = lim_{⟵} G_{n,d}. The multiplication in G_{n} corresponds to the Hopf algebra structure given by

(6.5.3)  Δ(W_{j_{1},...,j_{m}}) = ∑_{ν=0}^{m+1} W_{j_{1}...j_{ν}} ⊗ W_{j_{ν+1},...,j_{m}}.

Elements of ℂ[G_{n}] can be called polynomial functions on Π_{n}.

Note that formal infinite linear combinations (series) ∑_{J} c_{J}W_{J} still form a well defined algebra via (6.5.1), which we denote ℂ[[G_{n}]]. This is the algebra of functions on the formal completion of G_{n} at 1. The rule (6.5.3) makes ℂ[[G_{n}]] into a topological Hopf algebra.

Let τ be a measure on ℝ_{NC}^{n}. We will call the formal Fourier transform of τ the series

(6.5.4)  𝔉̂(τ) = ∑_{J=(j_{1},...,j_{m})} τ(Z_{i_{1}}...Z_{j_{m}}) · W_{J} ∈ ℂ[[G_{n}]].

As before, we see that convolution of measures is taken into the product in ℂ[[G_{n}]].

7 Towards the inverse NCFT.

(7.0) In this section we sketch a possible approach to the problem of finding the inverse to the NCFT F from (2.2). In other words, given a “noncommutative function” f = f(Z_{1}, ..., Z_{n}), how to find a measure μ on (possibly some completion of) Π_{n} such that F(μ) = f? Note that unlike in classical analysis, the dual Fourier transform 𝔉 (from noncommutative measures to functions on Π_{n}) does not provide even a conjectural answer, since there is no natural identification of functions and measures.

So we take as our starting point the case of discrete NCFT (1.4.5) where Theorem 1.4.6 provides a neat inversion formula.

(7.1) Fourier series and Fourier integrals. We recall the classical procedure expressing Fourier integrals as scaling limits of Fourier series, see [W], §5. Let f(x) be a piecewise continuous, ℂ-valued function on ℝ of sufficiently rapid decay. We can restrict f to the interval [−π, π] which is a fundamental domain for the exponential map z ↦ exp(iz), ℝ → S^{1}, and then represent f on this interval as a Fourier series in e^{inz}, n ∈ ℤ.

Next, let us scale the interval to [−A, A] instead. Then the orthonormal basis of functions is formed by

(7.1.1)  \frac{1}{\sqrt{2A}} exp (\frac{nπiz}{A} ) , n ∈ ℤ,

so on the new interval we have

(7.1.2)  f(z) = \frac{1}{2A} ∑_{n∈ℤ} exp (\frac{nπiz}{A} ) ∫_{−A}^{A} f(w) exp (\frac{−nπiw}{A} ) dw.

If we associate the Fourier coefficients to the scaled lattice points, putting

(7.1.3)  g (\frac{nπ}{A} ) = \frac{1}{\sqrt{2π}} ∫_{−A}^{A} f(z) exp (\frac{−nπiz}{A} ) dz,

then

(7.1.4) f(z) = \frac{1}{\sqrt{2π}} ∑_{n∈ℤ} g (\frac{nπ}{A} ) exp (\frac{nπiz}{A} ) Δ (\frac{nπ}{A} ) , z ∈ [−A, A],

where Δ (\frac{nπ}{A} ) = \frac{π}{A} is the step of the dual lattice. So when A → ∞, the formulas (7.1.3-4) “tend to” the formulas for two mutually inverse Fourier transforms for functions on ℝ. In other words, the measures on ℝ (with coordinate y) given by infinite combinations of shifted Dirac delta functions:

(7.1.5)  \frac{1}{\sqrt{2π}} \frac{π}{A} ∑_{n∈ℤ} g (\frac{nπ}{A} ) δ (y − \frac{nπ}{A} ) ,

converge, as A → ∞, to a measure whose Fourier transform is f.

(7.2) Matrix fundamental domains. We now consider the analog of the above formalism for Hermitian matrices instead of elements of ℝ and unitary matrices instead of those of S^{1}. Let Herm_{N}^{≤A} be the set of Hermitian N by N matrices whose eigenvalues all lie in [−A, A]. Then Herm_{N}^{≤π} is a fundamental domain for the exponential map

(7.2.1)  Z ↦ X = exp(iZ), Herm_{N} → U(N).

Note that the Jacobian of the map (7.2.1) is given by (7.2.2)

J(Z) = det_{N^{2}×N^{2}} \frac{e^{ad(Z)} − 1}{Z} = ∏_{j,k} \frac{e^{i(λ_{j}−λ_{k})} − 1}{λ_{j} − λ_{k}} = ∏_{j<k} 2\frac{1 − cos(λ_{j} − λ_{k})}{(λ_{k} − λ_{k})^{2}}.

Here λ_{1}, ..., λ_{N} are the eigenvalues of Z, see [Hel], p. 255. Using the formula for the volume of U(N), see, e.g., [Mac], we can write the normalized Haar measure on U(N) transferred into Herm_{N}^{≤π}, as

(7.2.3)  d^{∗}X = \frac{J(Z)dZ}{V_{N}} , V_{N} = ∏_{m=0}^{N−1} \frac{2π^{m+1}}{m!} .

Let f(Z_{1}, ..., Z_{n}) be a “good” noncommutative function (for example an entire function or a rational function defined for all Hermitian Z_{1}, ..., Z_{n} and having good decay at infinity). Then we can restrict f to (Herm_{N}^{≤π})^{n} and transfer it, via the map (7.2.1), to a matrix function on U(N)^{n}. This matrix function is clearly nothing but

(7.2.4)  f(−i log(X_{1}), ..., −i log(X_{n})),

where −i log : U(N) → Herm_{N}^{≤π} is the branch of the logarithm defined using our choice of the fundamental domain. Although (7.2.4) is far from being a noncommutative Laurent polynomial (indeed, it is typically discontinuous), one can hope to use the procedure of Theorem 1.4.6 to expand it into a noncommutative Fourier series. In other words, assuming that for each γ ∈ F_{n} the limit (7.2.5)

a_{γ} = lim_{N→∞} \frac{1}{N} Tr ∫_{X_{1},...,X_{n}∈U(N)} f(−i log(X_{1}), ..., −i log(X_{n}))X^{−γ} ∏_{j=1}^{n} d^{∗}X_{j} =

= lim_{N→∞} \frac{1}{N} Tr ∫_{Z_{1},...,Z_{n}∈Herm_{N}^{≤π}} f(Z_{1}, ..., Z_{n})E_{γ^{−1}}(iZ_{1}, ..., iZ_{n})) ∏_{j=1}^{n} \frac{J(Z_{j})dZ_{j}}{V_{N}}

exists, we can form the series

(7.2.6)  ∑_{γ} a_{γ}X^{γ} = ∑_{γ} a_{γ} E_{γ}(iZ_{1}, ..., iZ_{n}), Z_{j} ∈ Herm_{N}^{≤π} .

By analogy with the classical case one can expect that this series converges to f|_{(Herm_{N}^{≤π})^{n}} away from the boundary.

(7.3) Scaling the period. In the situation of (7.2) let us choose A > 0 and restrict f to (Herm_{N}^{≤A})^{n}. The same procedure would then expand the restriction into a series in

(7.3.1)  X_{j}^{π/A} = exp (i\frac{π}{A}Z_{j}) , j = 1, ..., n.

Let F_{n}^{π/A} ⊂ G_{n}(ℝ) be the group generated by the X_{j}^{π/A}. We can think of elements of F_{n}^{π/A} as rectangular paths in ℝ^{n} with increments being integer multiples of π/A. The coefficients of the series for the restriction given then a function

g_{A} : F_{n}^{π/A} → ℂ,

so the series will have the form

f(Z) = ∑_{γ∈F_{n}^{π/A}} g_{A}(γ)E_{γ}(iZ), Z = (Z_{1}, ..., Z_{n}), Z_{j} ∈ Herm_{N}^{≤A} .

Now, as A → ∞, we would like to say that the g_{A}, considered as linear combinations of Dirac measures on Π_{n} (or some completion) tend to a limit measure. Although Π_{n} is not a manifold, we can pass to finite dimensional approximations

Π_{n} ⊂ G_{n}(ℝ) ⟶^{p_{d}} G_{n,d}(ℝ).

Let F_{n,d}^{π/A} = p_{d}(F_{n}^{π/A}). This is a free nilpotent group of degree d on generators

p_{d}(X_{j}^{π/A}), and is a discrete subgroup (“lattice”) in G_{n,d}(ℝ). As A → ∞, these lattices are getting dense in G_{n,d}(ℝ). Supposing that the direct image (summation over the fibers) p_{d∗}(g_{A}) exists as a function on F_{n,d}^{π/A} or, what is

the same, a measure on G_{n,d}(ℝ) supported on the discrete subgroup F_{n,d}^{π/A}, we can then ask for the existence of the limit

μ_{d} = lim_{A→∞} p_{d∗}(g_{A}) ∈ Meas(G_{n,d}(ℝ)).

These measures, if they exist, would then form a pro-measure μ_{•} which is the natural candidate for the inverse Fourier transform of f. The author hopes to address these issues in a future paper.

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