Existence conditions for Feynman integrals

P. Muldowney

Abstract

For the Feynman path integral to exist in the sense of Henstock, the existence of certain finite-dimensional Henstock integrals involving the potential function V is sufficient.

1 Background

The integration system and other preliminaries required for this problem are given in [8]. We start with a review of some features of Henstock’s theory of integration in function spaces [1, 2, 3].

The primary space in which integration is required is R^{[τ′,τ[}. We integrate expressions f(x)Q(I), where f and Q are, respectively, functions of points and subsets of R^{[τ′,τ[}. In fact these subsets are cylindrical intervals I[N], restricted only in finite subsets N of dimensions in [τ′, τ[, and the integrand may depend explicitly on the elements of N, which are variable in the Riemann sums, just as x and I are. The integrand is given more generally as h(x, N, I), and the Riemann sums have the form ∑ h(x, N, I), where the cyindrical intervals I = I[N] in the sum form a disjoint cover for the domain of integration.

In Henstock’s theory of integration, the elements (x, I), (or (x, N, I) in the case of infinite dimensional integrals) may appear in a Riemann sum provided they are attached or associated. In the finite-dimensional situation involving only (x, I), this may mean that x must be a vertex of I. In the infinite-dimensional case, attachment or association of the components of (x, N, I) means that x(N), as an element of R^{N}, is a vertex of I(N), which is a parallelepiped in R^{N}.

Attachment or association may take other forms. For instance, when the Lebesgue integral in a one-dimensional space is constructed by the gauge integral method, the associated elements in the Riemann sums are (x, I), where x need not be contained in I, nor be a vertex of I, nor even be in the closure of I, but must only satisfy [4]

I ⊆ (x − δ(x), x + δ(x)).

Generally, the theory does not restrict us to two or three associated elements, as in (x, I), or (x, N, I). [5] gives an integration involving four associated elements (x, N, N^{∗}, I[N]).

A gauge γ is a rule which determines a set S_{γ} of associated elements (x, I) (or (x, N, I), (x, N, N^{∗}, I) and so on) from which Riemann sums may be formed.

Thus, in one dimension we may form Riemann sums ∑ f(x)Q(I) where each associated (x, I) belongs to the set S_{δ} satisfying |I| < δ. If the family A of all such S_{δ} is formed from the various choices of δ (a positive constant), the result is Riemann integration.

Thus Henstock’s theory of integration is a way of classifying systems of integration and generating new systems of integration. It does this by means of the family A which determines an integration by:

1. Specifying some rule of association between the integration elements x, I, …; and

2. Specifying the collections S_{γ} from which the terms of each γ-fine Riemann sum are produced.

In infinite-dimensional generalised Riemann integration, one of the most useful methods of integration is obtained by the following determination of A.

The integration elements x, N, I are associated or attached in the manner already described. A collection S_{γ} is specified as follows. Let A be any countable subset of [τ′, τ[. For each x in \overline{R}^{[τ′,τ[} let L_{A}(x) be a finite subset of A. For every finite subset N of [τ′, τ[ containing L_{A}(x) let

δ_{N} : \overline{R}^{N} ↦ R^{+}

be a gauge in R^{N} in the usual sense of finite-dimensional Henstock integration.^{1} These selections give us a gauge or rule γ in R^{[τ′,τ[} which determines membership of the set S_{γ}, so that (x, I[N]) is a member of S_{γ} (or (x, I[N]) is γ-fine) if and only if N contains L_{A}(x) and |I(N)| < δ_{N}(x(N). Each different selection of the countable set A and the mappings L_{A} and δ_{N} gives us a different gauge γ and a different set S_{γ}. The family A of all such S_{γ} determines the integration.^{2}

1In [7, 8] these finite-dimensional gauges were said to be determined by the same δ(x)

for all N. This was an error, in that the resulting system is not sufficient to enable us to integrate the functionals which arise in Brownian motion, quantum mechanics and other areas.

We define a relation ≺ on the gauges {γ} as follows. Suppose, for j = 1, 2, γ_{j} is defined by (A_{j}, L_{A_{j}}^{(j)}, δ_{N}^{(j)}). Then γ_{1} ≺ γ_{2} if and only if S_{γ_{1}} ⊇ S_{γ_{2}}; in other words, if and only if A_{1} ⊆ A_{2}, L_{A_{1}}^{(1)}(x) ⊆ L_{A_{2}}^{(2)}(x) for all x, and δ_{N}^{(1)}(x(N)) ≥ δ_{N}^{(2)}(x(N)) for all x and for all N ⊇ L_{A_{2}}^{(2)}(x) ⊇ L_{A_{1}}^{(1)}(x).

The family A is a net, in that the sets S_{γ} satisfy the conditions:

• S_{γ_{1}} ⊇ S_{γ_{2}}, S_{γ_{2}} ⊇ S_{γ_{3}} implies S_{γ_{1}} ⊇ S_{γ_{3}}; and,

• Given S_{γ_{1}} and S_{γ_{2}}, there exists S_{γ_{3}} with S_{γ_{1}} ⊇ S_{γ_{3}} and S_{γ_{2}} ⊇ S_{γ_{3}}.

Equivalently we can say that the set of rules or gauges γ form a net, so:

• γ_{1} ≺ γ_{2}, γ_{2} ≺ γ_{3} imply γ_{1} ≺ γ_{3}; and

• Given γ_{1} and γ_{2}, there exists γ_{3} with γ_{1} ≺ γ_{3}, γ_{2} ≺ γ_{3}.

To illustrate the reasoning involved in the latter step, let γ_{j} be defined by (A_{j}, L_{A_{j}}^{j}, δ_{N}^{j}) for j = 1, 2. We define a binary operation ∧ on gauges as follows. Let γ_{1} ∧ γ_{2} be γ_{3}, where γ_{3} is defined by (A_{3}, L_{A_{3}}^{3}, δ_{N}^{3}), with A_{3} = A_{1} ∪ A_{2}, L_{A_{3}}^{3}(x) = L_{A_{1}}^{1}(x) ∪ L_{A_{2}}^{2}(x), and δ_{N}^{3}(x) = min{δ_{N}^{1}(x), δ_{N}^{2}(x)}. This is equivalent to taking S_{γ_{3}} = S_{γ_{1}} ∩ S_{γ_{2}} in the alternative formulation.

2 Limits Relative To Nets

Let {c_{α}} be a set of real or complex numbers indexed by elements α of a net {α, ≻} ordered by a relation ≻. Then {c_{α}} converges (relative to {α, ≻}) to c if, given ε > 0, there exists α_{ε} so that α ≻ α_{ε} implies |c_{α} − c| < ε. Write c = lim_{α} c_{α} in this case.

Let h_{α} be a family of functions indexed by a net {α, ≻}. We say that {h_{α}} converges boundedly to h if there exists a positive function β(x, N, I) defined for associated (x, N, I), a gauge γ_{0} and an element α_{0} = α_{0}(x, N, I) of the net defined for each γ_{0}-fine pair (x, I[N]) so that • (E_{γ_{0}}) ∑ β(x, N, I) is bounded for all E_{γ_{0}}, and

2In [3], Henstock gives an elegant construction which does not use the countable set A.

• α ≻ α_{0} implies |h_{α}(x, N, I) − h(x, N, I)| < εβ(x, N, I) for all γ_{0}-fine (x, I[N]).

Henstock [2] gave criteria for taking limits of sequences of functions under the integral sign. We give the corresponding results and proofs here with nets instead of sequences. There is little difference in the wordings of either the statements or the proofs of Henstock’s results when they are re-written for nets instead of sequences.

Proposition 1 Suppose h_{α}(x, N, I) converges boundedly to h.

1. h is integrable in E if and only if there exist a ball B_{1} of arbitrarily small radius and, correspondingly, a gauge γ_{1}, and elements α_{1} = α_{1}(x, N, I), so that if α = α(x, N, I) satisfies α ≻ α_{1}(x, N, I) and E_{γ_{1}} is γ_{1}-fine, then

(E_{γ_{1}}) ∑ h_{α(x,N,I)}(x, N, I) ∈ B_{1}.

2. ∫_{E} h_{α} converges relative to the net {α, ≻} if and only if there exist a ball B_{2} of arbitrarily small radius, a corresponding element α_{2}, and, for each α with α ≻ α_{2}, a gauge γ_{α} so that for every γ_{α}-fine E_{γ_{α}} we have

(E_{γ_{α}}) ∑ h_{α}(x, N, I) ∈ B_{2}.

3. Given the existence of ∫_{E} h and lim_{α} ∫_{E} h_{α}, the two are equal if and only if B_{1} ∩ B_{2} ≠ ∅.

Proof

1. If h is integrable, with ∫_{E} h = a_{1}, then let ε > 0 be given, take γ_{1} = γ_{0}, take α_{1}(x, N, I) = α_{0}(x, N, I), and let K be an upper bound for (E_{γ}) ∑ β(x, N, I) for all E_{γ}, so that

(E_{γ}) ∑ h_{α(x,N,I}(x, N, I) ∈ B(a_{1}, Kε).

Conversely, if there exist B_{1}, γ and α_{1}(x, N, I) so that

(E_{γ}) ∑ h_{α(x,N,I)}(x, N, I) ∈ B_{1}

for all α satisfying α ≻ α_{1}(x, N, I) and all E_{γ}, then, with K as before, take γ′ = γ_{0} ∧ γ, and let α = α(x, N, I) satisfy α ≻ α_{j}(x, N, I) for j = 0, 1 and all γ′-fine (x, I[N]). Then, for every E_{γ′}, we have

(E_{γ′}) ∑ h(x, N, I) ∈ B_{1} + B(0, Kε),

giving the integrability of h.

2. If lim_{α} ∫_{E} h_{α}, = a_{2}, exists, then there exists α_{2} so that every α satisfying α ≻ α_{2} gives ∫_{E} h_{α} ∈ B(a_{2}, ε). In other words, there exists γ_{α} so that, for all E_{γ_{α}}, we have

(E_{γ_{α}}) ∑ h_{α}(x, N, I) ∈ B(a_{2}, 2ε).

Conversely, suppose there exist B_{2}, α_{2} and γ_{α}, with α_{2} ≻ α, so that every (E_{γ_{α}}) ∑ h_{α}(x, N, I) ∈ B_{2}. Then, since each h_{α} is integrable there exist gauges γ′_{α} so that for all E_{γ′_{α}},

|(E_{γ′_{α}}) ∑ h_{α} − ∫_{E} h_{α}| < ε.

Now take γ′′_{α} = γ_{α} ∧ γ′_{α}, and consider γ′′_{α}-fine divisions, so that for α satisfying α ≻ α_{2},

∫_{E} h_{α} ∈ B_{2} + B(0, ε).

Therefore lim_{α} ∫_{E} h_{α} exists.

3. If ε is an arbitrary positive number, then, given the existence of a_{1} = ∫_{E} h and a_{2} = lim_{α} ∫_{E} h_{α}, we have a_{1} = a_{2} if and only if B_{1} = B(a_{1}, Kε) and B_{2} = B(a_{2}, 2ε) have non-empty intersection.

It is sometimes useful to have Cauchy versions of these criteria. So we give details of a Cauchy version of the first one.

Proposition 2 h is integrable in E if and only if, for every ε > 0, there exists a gauge γ_{1}, and elements α_{1} = α_{1}(x, N, I) so that for α = α(x, N, I) ≻ α_{1}(x, N, I) and α′ ≻ α_{1}(x, N, I), and for each E_{γ_{1}}, we have

|(E_{γ_{1}})h_{α(x,N,I)}(x, N, I) − (E_{γ_{1}})h_{α′(x,N,I)}(x, N, I)| < ε.

Proof Assuming the latter inequality, let

S_{α_{1}} = {(E_{γ}) ∑ h_{α(x,N,I)}(x, N, I) : α ≻ α_{1}(x, N, I)}.

By the assumption, we can choose α_{1}, = α_{1}(ε), so that

diam(S_{α_{1}}) < ε.

For j = 1, 2, 3, …, choose α_{1} = α_{1j} so that diam(S_{α_{1j}}) < j^{−1}. Let R_{n} = ∩_{j=1}^{n}S_{α_{1j}}. For each n, choose s_{n} ∈ R_{n}. Then s_{n} is a Cauchy sequence, so there exists a_{1} with s_{n} → a_{1} as n → ∞. With ε > 0 given, choose n > ε^{−1}, so if s ∈ S_{α_{1}(ε)} we have

|s − a_{1}| ≤ |s − s_{n}| + |s_{n} − a_{1}|,

and s ∈ B(a_{1}, 2ε). The converse is straightforward.

3 Application to Feynman integrals

The notation is that of [8]. Let M denote a finite subset of [τ, τ[, and let M be the family of all M. M is a net under the relation of ⊆; because, given M_{1}, M_{2} ∈ M, there exists M_{3} ∈ M with M_{1} ⊆ M_{3}, M_{2} ⊆ M_{3}; and, given M_{1} ⊆ M_{2} and M_{2} ⊆ M_{3}, we have M_{1} ⊆ M_{3}.

Suppose E is a division of [τ′, τ[. Suppose the partition points of the division are τ′ = t_{0} < t_{1} < ⋯ < t_{n} = τ, and that (t, [t_{j−1}, t_{j})) is the jth term of E, with t = t_{j−1} or t_{j}. (If this is δ-fine then t_{j} −t_{j−1} < δ(t).) Now consider a division D^{(j)} of [t_{j−1}, t_{j}[. Let D_{E} := ∪_{j=1}^{n}D^{(j)}. Then D_{E} is a division of [τ′, τ[, and we call it a refinement of E.

If h(t, I) is defined on the points t and subintervals I of [τ′, τ[, we say that h is refinement-Henstock integrable in [τ′, τ[, with integral a, if, given ε > 0, there exists a gauge δ(t) > 0 defined for τ′ ≤ t ≤ τ, so that, for every δ-fine division E of [τ′, τ[, and every refinement D_{E}, we have

|(D_{E}) ∑ h(t, I) − a| < ε.

Since D_{E} need not be δ-fine, this is less than Henstock integrability, in the sense that there may be functions which are Henstock integrable, but not refinement-Hentock integrable.

Following the notation of [8], let

v(x, M) = exp (λ ∑_{j=1}^{n} V (x_{j−1})(t_{j} − t_{j−1})) , and

v(x) = exp (λ ∫_{τ′}^{τ} V (x(t))dt) , whenever the integral exists.

First we review some preliminary results.

Proposition 3 If c is a complex number with ℜ(c) ≤ 0, ℑ(c) ≥ 0, c ≠ 0, then ∫_{0}^{∞} exp(cy^{2})dy exists and equals \frac{1}{2} \sqrt{\frac{π}{−c}}.

This is Fresnel’s integral, and the evaluation is given in [5], Proposition 63, page 81.

Let D be the set of x in T which are discontinuous at some t in [τ′, τ], and let F_{0} be the “continuous modification” of G_{0} as described in [8], Proposition 4, page 135. We make use of the notion of the variation of a function in a set, as described in [5, 9].

Proposition 4 F_{0} has variation 0 in D.

This follows from [6], Proposition 4, page 135.

Proposition 5 If c is a complex number with ℜ(c) ≤ 0, ℑ(c) ≥ 0, c ≠ 0, then F_{0}(x, N, I) is integrable in every elementary set E of R^{[τ′,τ[}, and

∫_{R^{[τ′,τ[}} F_{0}(x, N, I) = (\frac{π}{−c}(τ − τ′))^{− \frac{1}{2}} exp \frac{c(ξ − ξ′)^{2}}{τ − τ′} .

Let E be a division of R^{[τ′,τ[}. Let M be the union of those dimension sets N for which (x, I[N]) ∈ E. For each such I[N] let I(M) denote the projection of I[N] into R^{M}, so the projection of E into R^{M} is a division E(M) = {(x(M), I(M))} of R^{M}. Then the Riemann sum (E) ∑ F_{0}(x, N, I) can be interpreted as

(E(M)) ∑ F_{0}(x(M), I(M)).

Except for the fact that we have excluded from the Riemann sums those terms for which x(t) = ±∞, t ∈ N, the result would now follow from Fresnel’s integral above. But, since these points form a subset of D, by the previous result we can choose a gauge γ so that, for all E_{γ}, we have |(E_{γ}) ∑ F_{0}(x, N, I)− ∫_{R^{M}} F_{0}(y(M)), I(M))dy(M)| < ε. So

|(E) ∑ F_{0}(x, N, I) − (\frac{π}{−c}(τ − τ′))^{− \frac{1}{2}} exp \frac{c(ξ − ξ′)^{2}}{τ − τ′} | < ε,

giving the required result.

Proposition 6 If V (x(t)) is refinement-Henstock integrable in [τ′, τ[, then lim_{M} v(x, M) exists and equals v(x).

Proof Let ε > 0 be given. Choose a δ-fine division E of [τ′, τ[ for which

|exp (λ(E) ∑ V (x_{j})(t_{j} − t_{j−1})) − exp (∫_{τ′}^{τ} λV (x(t))dt)| < ε.

Letting M_{E} be the partition points of E, this gives

|v(x, M_{E}) − v(x)| < ε.

Let M be any finite subset of [τ′, τ[ containing M_{E}. For every such M we can take a corresponding refinement division D_{E} of E, and

|exp (λ(D_{E}) ∑ V (x_{j})(t_{j} − t_{j−1})) − exp (∫_{τ′}^{τ} λV (x(t))dt)| < ε, or
|v(x, M) − v(x)| < ε,

giving the result.

Proposition 7 Suppose V satisfies the following pair of (finite-dimensional) integrability conditions. Firstly, V (x(t)) is refinement-Henstock integrable in [τ′, τ[ for each x ∈ C. Secondly, v(x(M), M) is Henstock integrable in R^{M} for each M. Then v(x) is F_{0}-integrable in R^{[τ′,τ[}.

Proof By [5], Proposition 69, page 87, assumption (2) implies the cylinder functional v(x, M) is F_{0}-integrable in R^{[τ′,τ[} for each fixed M. Let ε > 0 be given. Let γ_{1}, X_{k} and α_{k}, k = 1, 2, …, be as they are defined in [8], (14), page 15. Choose γ ≺ γ_{1} so that, for each x ∈ X_{k} and each L_{A}^{(γ)}(x), if we have M ⊇ L_{A}^{(γ)}(x), then

|v(x, M) − v(x)| < εα_{k}^{−1}2^{−k}.

Then for any γ-fine division E_{γ} of R^{[τ′,τ[}, and any choices M(x), M′(x) containing L_{A}^{(γ)}(x), we have

|(E_{γ}) ∑ v(x, M(x))F_{0}(I[N]) − (E_{γ}) ∑ v(x, M′(x))F_{0}(I[N])|
≤ (E_{γ}) ∑ |v(x, M(x)) − v(x, M′(x))||F_{0}(I[N])|
≤ (E_{γ}) ∑ |v(x, M(x) − v(x)||F_{0}(I[N])|
−(E_{γ}) ∑ |v(x) − v(x, M′(x)||F_{0}(I[N])|
< 2ε ∑_{k=1}^{∞} 2^{−k}
= 2ε.

The result follows from the Cauchy version, given above, of Henstock’s first criterion for limits under the integral sign.

References

[1] R. Henstock, Integration in product spaces, including Wiener and Feynman integration, Proc. London Math. Soc., 27 (1973) 317-344.

[2] R. Henstock, The General Theory Of Integration, Clarendon Press, 1991.

[3] R. Henstock, Math. Japonica, The construction of path integrals, Math. Japonica, 39(1) (1994) 15-18.

[4] E. J. McShane, Unified Integration, Academic Press, 1983.

[5] P. Muldowney, A General Theory Of Integration In Function Spaces, Pitman Research Notes in Mathematics 153, Longman, 1987.

[6] P. Muldowney, Infinite-dimensional generalised Riemann integrals, New Integrals, Springer-Verlag, Berlin, 1990.

[7] P. Muldowney, Topics in probability using generalised Riemann integration, Proc. Royal Irish Academy, A 99(1) (1999) 39-50.

[8] P. Muldowney, Feynman’s path integrals and Henstock’s non-absolute integration, Journal of Applied Analysis, 6(1) (2000) 1-24.

[9] P. Muldowney, The infinite dimensional Henstock integral and problems of Black-Scholes expectation, (pre-print).