A Framework for Non-Gaussian Functional Integrals with Applications

Home , Gaussian integral

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And yet there is still no widely accepted mathematically rigorous definition encompassing all types of functional integrals. There are, of course, already some rigorous constructions of functional integrals of limited type that have been developed [7]–[19].2 An excellent up-to-date synopsis of various approaches and a good source of authoritative references is [20]. But it is fair to say these constructions are not generally viewed as definitive; perhaps because they are restricted to subclasses of functions that have limited applicability [7, 9, 11, 15, 17, 18] or perhaps because they abandon the notion of integration with respect to an orthodox measure [8, 10, 12, 19]. It would be satisfying to have a definition of functional integrals that promises the possi- bility of mathematical rigor and broad applicability while maintaining the pragmatic heuris- tics that is their hallmark. Needless to say, such a definition would need to be consistent with existing heuristic and rigorous constructions. If one is searching for such a definition, taking stock of shared characteristics among various approaches is a good place to start. In doing so, we can immediately make two rather obvious observations about the various approaches: i) functional integrals are typically defined in terms of a limiting sequence of finite dimensional objects and/or by some Fourier-type duality; and ii) evaluating integrals invariably involves some kind of reduction/localization in the integration domain that eventually leads to finite-dimensional, or at least measurable, subspaces. Indeed, from a rigorous mathematical perspective, it has proven impossible (so far) to devise a consistent and general scheme any other way. In essence, the observations seem to be telling us that a single functional integral associated with a function space is realized by a whole family of bona fide integrals corresponding to different ‘questions’3 one may ask — much like the case of general versus particular solu- tions in the theory of differential equations: The entire family represents a tool to probe the function space, and measure-theoretic aspects come into play only after a specific ‘question’ has been posed. This idea is not new. It (seemingly) is always implicit in any functional integral evaluation, and it is often even explicitly stated in the form of localization principles (cf. [21, 23]). Our aim is to identify the mathematical structure that captures this essence. With these observations in mind, we propose to define functional integrals in terms of data (G, B,GΛ) where G is a topological group, B is a Banach space, and GΛ := {Gλ,λ ∈ Λ} is a countable family of locally compact topological groups indexed by group epimorphisms λ : G → Gλ. With this data we define integral operators intλ on a suitable space of integrable functionals4 F(G) ∋ F: G → B by

intλ(F) ≡ F(g)Dλg := f(gλ) dν(gλ) (1.1) ZG ZGλ 2The cited references are meant to be representative: They are deﬁnitely not exhaustive. Apologies to all excluded authors. 3By ‘question’ we mean some kind of restriction or constraint pertinent to one’s problem that singles out a subclass of measurable functions in subspaces of the integration domain. 4We use the term functional (imprecisely) to refer to a Banach-valued function on some topological space. Strictly, functional refers to a scalar-valued function on some vector space. However, in the context where Banach-valued functions on topological spaces are integrands of functional integrals, we will continue to use the term imprecisely.

2 1 with f denoting the restriction F|Gλ such that f ∈ L (Gλ, B) for all λ ∈ Λ. We call intΛ a family of integral operators on F(G) and DΛg its associated family of integrators. The right-hand side of (1.1) is clearly well defined once a choice of Haar measure ν(Gλ) is made. So the definition will be meaningful if the set of group epimorphisms Λ and integrable F ∈ F(g) can be quantified. In the sequel we will give some examples of well- known embodiments of Λ, and we will see that intΛ is really nothing more than a shorthand notation for the two observations discussed above. Nevertheless, the shorthand intΛ has value. First, it serves as a vehicle to transfer algebraic structure between B and F(G). Second, it simultaneously incorporates both measure- theoretic and Fourier-duality approaches. And third, it begs for the introduction of non- Gaussian integrators that have useful potential. The plan of this paper is to give evidence to these three attributes of intΛ. We first state the proposed definition of functional integral and investigate its algebraic properties when they exist. In fact, the definition is actually rather trivial — almost. As already discussed, it does not attempt to formulate a rigorous measure theory in infinite dimensions, but instead it incorporates the notion that measure theory should apply only after suitable localization is effected in some topological group. Again, this notion is not new and in a sense the definition is just bookkeeping, but it does provide a framework to realize F(G) as a Banach algebra through its relation to B; and this is important, particularly for physics. Next, some examples of well-known successful approaches to functional integrals are shown to be included in the framework. These approaches are based on the prototypic Gaussian integrator family so it is appropriate that we spend some time developing these integrators in detail. We consider integrators induced by symmetric and skew-symmetric quadratic forms. It is significant that the latter are characterized by Pfaffians; thus enabling the construction of functional integrals with Grassmann-like change of variables without having to introduce Berezin functional integrals. To illustrate, we construct a functional integral representation of the Mathai-Quillen Thom class representative in the spirit of [21]. Taken together, the Gaussian and skew-Gaussian can be combined into what we call a Li- ouville integrator. It localizes via the Berline-Vergne and Duistermaat-Heckman theorems, and its heuristic counterpart (developed in [22]) has found application in topological and cohomological QFT (for a review see e.g. [23]). Under suitable restrictions Liouvelle-type functional integrals can mimic supersymmetric functional integrals but with a notably different physical interpretation of degrees of freedom: they (the degrees of freedom) can be viewed as currents and the corresponding space of integrable functionals can be viewed as an exterior algebra of current-dual differential forms. Finally, we develop a non-Gaussian integrator family closely allied with gamma statistics and then use it to define what can be viewed as ‘distributionals’ on a certain dual topological group. We formulate the gamma-type integrator also for non-abelian topological groups; rendering certain matrix functional integrals accessible. For certain parameters, the gamma-type integrator family can be interpreted as Poisson-type; and, to present a novel application, we use it to give functional integral representations of some average counting functions involving both single primes and prime k-tuples. The accuracy of these counting functions suggests the physical model underlying these functional integral representations may provide a useful perspective in number theory. The approach should also prove use-

3 ful in counting average prime cycles and average correlated prime cycles in the context of spectral determinants and dynamical zeta functions. We want to stress this point: The goal here is not to rigorously define a measure on general infinite-dimensional spaces of functions. Instead, we claim that meaningful questions of measure (e.g. associated with a specific quantum system) necessarily distinguish specific locally compact subspaces thereof. Our notion of functional integral is not “a number- generating machine” but rather “a generator of number-generating machines”: There is no single rigorous functional integral per se but a collection of realizable Haar integrals. As such, there is no new measure-theoretic value in our definition. The real value lies in the algebraic structure of F(G) inherited from Λ and B, and a point of view that leads to the construction of new integral types that should profit mathematical physics.

2 Functional integral

2.1 Ingredients The key components of the proposed definition are topological groups and Banach-valued integration, so it is appropriate to first recall relevant definitions and theorems. (Proofs of theorems in this subsection will be omitted since they are not particularly germane to our development. The proofs, and much more, can be found in [24],[25].)

Deﬁnition 2.1 A Hausdorﬀ topological group G is a group endowed with a topology such that; (i) multiplication G × G → G by (g, h) 7→ gh and inversion G → G by g 7→ g−1 are continuous maps, and (ii) {e} is closed.

Deﬁnition 2.2 G is locally compact if every g ∈ G has a neighborhood basis5 comprised of compact sets.

Deﬁnition 2.3 G is a Lie group if there exists a neighborhood U of {e} such that, for every subgroup H, if H ⊆ U then H = {e}.

Remark that the closure hypothesis on {e} together with the topology and group structure allow the closure property to be ‘transported’ to every element in G. That is, G is Hausdorff iff {e} is closed. Moreover, since G is Hausdorff, it is locally compact iff every g ∈ G possesses a compact neighborhood. We will need the exponential function on topological groups. The motivation for the following definition comes from analogy with the exponential map for finite Lie groups.

Deﬁnition 2.4 ([24, ch. 5]) A one-parameter subgroup φ : R → G of a topological group 6 is the unique extension of a continuous homomorphism f ∈ HomC (I ⊆ R,G) such that f(t + s)= f(t)f(s) and f(0) = e ∈ G. Let L(G) denote the set of one-parameter subgroups

5A neighborhood basis at g ∈ G is a family N of neighborhoods such that given any neighborhood U of g there exists an N ∈ N such that N ⊂ U. 6 Denote continuous homomorphisms G → H by HomC (G, H).

4 HomC (R,G) endowed with the uniform convergence topology on compact sets in R. The exponential function is deﬁned by

expG : L(G) → G by expG(g) := φg(1) (2.1)

−1 where g ∈ L(G) is identiﬁed with dφg(d/dt) ∈ Te(G). Similarly, its inverse logG := expG is a well-deﬁned function.

In particular, if G is an abelian topological group, then L(G) is a topological vector space with the uniform convergence topology on compact sets. Let B be the set of invertible elements (units) of some unital Banach algebra B. Then B is a topological group and there exists a homeomorphism η : B → L(B) such that B φη(b)(1) = expB(η(b)) =: expB(b). Moreover, for any subgroup G ⊆ B, there is an induced B homeomorphism ηGB : B → L(G ). Consequently, the exponential function extends to the algebra level expB : B → B by b 7→ expB(tb)= φb(t), and it enjoys the standard properties if B is endowed with a Lie bracket.

Deﬁnition 2.5 ([24, def. 5.1]) Let B1(1) be the unit ball about the identity element 1 ∈ B. The exponential, expB : B → B, and logarithm, logB : B1(1) → B, are deﬁned by

∞ 1 exp (b) := bn ∀ b ∈ B, B n! 0 X ∞ (−1)n+1 log (1 + b) := bn for kbk < 1 . (2.2) B n 1 X The two functions are absolutely convergent and analytic for the indicated b ∈ B.

Proposition 2.1 ([24, prop. 5.3]) −1 Let N0 be the connected component of the 0-neighborhood of expB B1(1). Then

i) logB(expBb)= b ∀ b ∈ N0 .

ii) expB(logB b)= b ∀ b ∈ B1(1) . iii) exp | : N → B (1) is an analytic homeomorphism with B N0 0 1 analytic inverse logB : B1(1) → N0 . (2.3)

Deﬁnition 2.6 ([24, def. 5.32]) Let BL ⊆ (B, [·, ·]) be a closed Lie subalgebra of some Banach algebra (B, [·, ·]) equipped with a Lie bracket. Let Bl be a subgroup of the multiplica- B tive group of units of such that expBL is a homeomorphism mapping a neighborhood of {0} ∈ BL into a neighborhood of {e} ∈ Bl. A topological group is a linear Lie group if it is isomorphic to Bl.

Proposition 2.2 ([24, th. 5.41]) If G is a linear Lie group, then the set L(G) is a completely

normable topological real Lie algebra and expG is a homeomorphism from a 0-neighborhood of L(G) to an e-neighborhood in G.

5 There are many other interesting and useful results regarding topological group structures (e.g. for subgroups, quotients, product groups, etc.) that will only be recorded as needed. The most important for our application is the following well-known result: Theorem 2.1 If G is locally compact, then there exists a unique (up to positive scalar multiplication) Haar measure. If G is compact, then it is unimodular. If G is a locally compact linear Lie group then it is a dim-L(G) manifold. Evidently, locally compact topological groups can be used as a footing on which to ground functional integration: They supply measure spaces on which to model functional integrals and their associated integrators. With this in mind, recall Banach-valued integration on locally compact topological groups:

Proposition 2.3 ([25, prop. B.34]) Let Gλ be a locally compact topological group, µ its associated Haar measure, and B a Banach space possibly with an algebraic structure. Then the set 1 of integrable functions L (Gλ, B) ∋ f, consisting of equivalence classes of measurable functions equal almost everywhere with norm kfk1 := kf(gλ)kdµ(gλ) ≤kfk∞ µ(supp f) < ∞, Gλ is a Banach space. Moreover, f 7→ f(gλ)dµ(gλ) is a linear map such that Gλ R R k f(gλ)dµ(gλ)k≤kfk∞ µ(supp f) (2.4) ZGλ 1 for all f ∈ L (Gλ, B),

ϕ f(gλ)dµ(gλ) = ϕ (f(gλ)) dµ(gλ) (2.5) ZGλ ZGλ for all ϕ ∈ B′, and

LB f(gλ)dµ(gλ) = LB (f(gλ)) dµ(gλ) (2.6) ZGλ ZGλ for bounded linear maps LB : B → B2. Moreover, Fubini’s theorem holds for all equivalence 1 classes f ∈ L (G1 × G2, B).

∗ ∗ ∗ Corollary 2.1 Let B be a C -algebra and π : B → LB(H) a representation with LB(H) the algebra of bounded linear operators on Hilbert space H. Then

π f(gλ)dµ(gλ) v|w = hπ (f(gλ)) v|wi dµ(gλ) , (2.7) ZGλ ZGλ ∗ ∗ f(gλ)dµ(gλ) = f(gλ) dµ(gλ) , (2.8) ZGλ ZGλ and

a f(gλ)dµ(gλ)b = af(gλ)bdµ(gλ) (2.9) ZGλ ZGλ where v,w ∈ H and a, b ∈ M(B∗) with M(B∗) the multiplier algebra7 of B∗.

7 ∗ Recall the multiplier algebra can be characterized as the set of adjointable linear operators L∗(BB) on B∗ viewed as a right Hilbert module over itself. If B∗ is unital then M(B∗)= B∗.

6 1 ∗ It can be shown ([25, appd. B]) that L (Gλ, B ) is a Banach ∗-algebra when equipped with: i) the k·k1 norm, ii) the convolution

−1 f1 ∗ f2(gλ) := f1(hλ)f2(hλ gλ)dµ(hλ) , (2.10) ZGλ and iii) the involution ∗ −1 ∗ −1 f (gλ) := f(gλ ) ∆(gλ ) (2.11) where ∆ is the modular function on Gλ.

2.2 Definition With this mathematics backdrop, our proposed functional integral story starts by specifying data (G, B,GΛ). The physical or mathematical model of interest determines G and B. But the more difficult work of quantifying Λ and integrable functionals F : G → B remains. This task is application specific. Consequently we will indicate pertinent aspects as they occur in the examples. Suppose we are given (G, B,GΛ) where G is a Hausdorff topological group, B is a Banach space that may have additional associative algebraic structure, and GΛ := {Gλ,λ ∈ Λ} is a countable family of locally compact topological groups indexed by group epimorphisms λ : G → Gλ. Insofar as λ is surjective and G contains at least one point such that every open neighborhood does not contain a compact neighborhood, we can think of Gλ as a subset Gλ ⊂ G, and loosely refer to λ as a ‘topological localization’. The idea is to use the rigorous B-valued integration theory associated with {Gλ,λ ∈ Λ} to define and characterize functional integration on G.8

Deﬁnition 2.7 Let F(G) represent a space of functionals F: G → B, and denote the 9 restriction of F to Gλ by f := F|Gλ . Let ν be a left Haar measure on Gλ. A family (indexed by Λ) of integral operators intΛ : F(G) → B is deﬁned by

intλ(F) ≡ F(g)Dλg := f(gλ) dν(gλ) (2.12) ZG ZGλ 1 if f ∈ L (Gλ, B) for all λ ∈ Λ. We say that F is integrable with respect to the integrator family DΛg, and F(G) ⊆ F(G) is the subspace of all such integrable functionals. Further, if B is an algebra, deﬁne the functional ∗-convolution on F(G) by

−1 (F1 ∗ F2)λ (g) := F1(˜g)F2(˜g g)Dλg˜ (2.13) ZG for each λ ∈ Λ.

8 It is probably fruitful to consider GΛ as a locally compact topological groupoid, but this would add a layer of complexity that is better left as a separate investigation. 9The Haar measure ν does not necessarily have unit normalization. Also, recall that if ν and µ are left −1 −1 −1 and right Haar measures respectively, then ν(Gλ) = µ(Gλ ) and dν(gλ) = ∆(gλ )dµ(gλ ) where ∆ is the modular function on Gλ.

7 For any given λ, the integral operator is linear and bounded according to

kintλ(F)k ≤ kf(gλ)k dν(gλ) =: kfk1 < ∞ (2.14) ZGλ which follows from the Cauchy-Schwarz inequality and proposition 2.3. This suggests to

deﬁne the norm kFkF := supλ kintλ(F)k. It is not hard to see k·kF is a norm using the fact that k·k1 is a norm. But there may be more suitable norms depending on the context, and we don’t insist on this particular choice. Since F(G) is a normed space, its completion (which will be denoted by the same symbol) is a Banach space. The ∗-convolution then implies

intλ(F1 ∗ F2) = (F1 ∗ F2)(g)Dλg ZG −1 = f1(˜gλ)f2(˜gλ gλ) dν(˜gλ,gλ) ZGλ ZGλ

= f1(˜gλ)f2(gλ) dν(˜gλ)dν(gλ) ZGλ ZGλ

= f1(˜gλ)f2(gλ) dν(˜gλ)dν(gλ) ZGλ ZGλ = intλ(F1) intλ(F2) (2.15) where the second line follows from left-invariance of the Haar measure and the last line follows from Fubini. A similar computation (using left-invariance and Fubini) establishes associativity (F1 ∗ F2) ∗ F3 = F1 ∗ (F2 ∗ F3). Moreover, Banach B, eq. (2.15), and the multiplicative property of sup imply kF1 ∗F2kF ≤kF1kF kF2kF. Consequently, F(G) inherits the algebraic structure of B and we have shown:

Proposition 2.4 F(G) equipped with the ∗-convolution is a Banach algebra when completed with respect to the norm kFkF := supλ kintλ(F)k (or other suitable norm).

With an eye toward applications to quantum systems, we want to add the additional structure of ∗-algebras. To that end,

Proposition 2.5 Suppose now B is a Banach ∗-algebra. Then intλ is a ∗-homomorphism, and F(G) — endowed with a suitable topology and involution F∗(g) := F(g−1)∗∆(g−1) and completed with respect to the norm k·kF — is a Banach ∗-algebra.

Corollary 2.2 If B is a C∗-algebra, then F(G) is C∗-algebra.

The corollary is important. It means the as-deﬁned functional integral implements a duality between two C∗-algebras. In particular, if B happens to be the space of bounded, linear operators on a Hilbert space LB(H), this duality provides a link between operator and functional integral methods in quantum physics.

8 Proof : First,

∗ ∗ intλ(F ) = F (g)Dλg ZG ∗ = f (gλ) dν(gλ) ZGλ −1 ∗ −1 = f(gλ ) ∆(gλ ) dν(gλ) ZGλ ∗ = f(gλ) dν(gλ) Gλ Z ∗

= f(gλ) dν(gλ) Gλ Z ∗ = intλ(F) , (2.16) where the fourth line follows by virtue of the Haar measure. Together with (2.15), this shows that the integral operators are ∗-homomorphisms. From the previous proposition, F(G) is a Banach algebra so it remains to verify the ∗- algebra axioms. The ∗-operation is continuous for a suitable choice of topology, and linearity is obvious. Next,

(F∗)∗(g) := F∗(g−1)∗∆(g−1) = (F(g)∗)∗∆(g)∆(g−1)=F(g) (2.17) and

∗ ∗ ∗ ∗ −1 (F1 ∗ F2)λ (g) := f1 (˜gλ)f2 (˜gλ gλ) dν(˜gλ) ZGλ −1 −1 −1 −1 ∗ = f2(gλ g˜λ)∆(gλ g˜λ)f1(˜gλ )∆(˜gλ ) dν(˜gλ) ZGλ ∗ −1 −1 −1 = f2(gλ g˜λ)f1(˜gλ )∆(gλ ) dν(˜gλ) ZGλ −1 ∗ −1 = (F2 ∗ F1)λ(g ) ∆(g ) ∗ = (F2 ∗ F1)λ (g) (2.18) where we used the deﬁnition of involution, left-invariance of the Haar measure, and the fact that the modular function ∆ is a homomorphism. For the norm, Banach B and eq. (2.16) ∗ ∗ ∗ imply kintλ(F )k = kintλ(F) k = kintλ(F)k which implies kF kF = kFkF. Conclude that F(G) is a ∗-algebra. If B is a C∗-algebra, the corollary follows from (2.15) and (2.16) since now

∗ ∗ ∗ 2 kintλ(F ∗ F )k = kintλ(F) intλ(F) k = kintλ(F)kkintλ(F) k = kintλ(F)k (2.19)

∗ 2 implies kF ∗ F kF = kFkF.

9 1 An immediate observation: although the products in F(G) and L (Gλ, B) are trivially equivalent by definition, their respective norms are not. Our choice of norm on F(G) 1 (together with the fact that the involution is only defined within each L (Gλ, B)) renders the restriction of F(G) to GΛ a direct sum if the cardinality of GΛ is finite; namely 1 F(G)|GΛ = λ∈Λ L (Gλ, B) ([26, pg. 16–17]). In this sense, a ‘question’ — which (by definition) corresponds to a measurable element O ∈ F(G) and a particular choice of λ — induces a projection.L

Remark 2.1 In the context of quantum mechanics, this observation and the topological framework provide a Heisenberg-picture interpretation of the measurement process. Sup- pose that F(G) is the C∗-algebra characterizing some quantum system, G governs the system dynamics, and O ∈ F(G) is an observable. But the very notion of observation/measurement requires a restriction to locally compact GΛ. Further, restricting to closed quantum systems, insist that the set Λmeas ⊆ Λ yields a finite family GΛmeas that can be represented by unitaries in B. We have seen that, as far as measure is concerned, the functional O (represented as a functional integral) is realized as an entire family of functions. It is easy to imagine that the possible physical quantum states of the macroscopic measuring device corresponds to this family of functions and, hence, are indexed by the set Λmeas. Which member λ ∈ Λmeas is realized in a measurement of course can’t be known. Alternatively, one can imagine that a particular ‘topological localization’ G → Gλ represents a subjective “choice” made by the observer. Either way we have a topological interpretation: performing a measurement10 and thereby F 1 B 11 actualizing an observable corresponds to a particular projection of (G)|GΛmeas onto L (Gλ, ). Precisely which projection is effected by the measurement cannot be known. Subsequent mea- 1 surement will of course be referred to L (Gλ, B) ⊂ F(G) unless interaction dynamics takes the system out of this subspace. Evidently, evolution of a prepared closed quantum system 1 would be modeled by L (Gλ, B) where λ ∈ Λmeas (corresponding to the preparation) encodes its initial state. In effect, the topological model supplies a family of isomorphic (under measurement) Hilbert spaces. The family represents indeterminacy; not of the system but of the measuring “ruler”. Once a measurement has been made, it is given only comparative meaning (that is, it can be compared to subsequent measurements) through a specific representation of the associated observable carried by the Hilbert space based on a specific, locally compact Gλ. Barring external interaction, this representation will continue to be descriptive; otherwise the representation will no longer be valid in subsequent measurements. Alternatively, since we have stipulated that Gλ is represented by unitaries in B, we can imagine a single Hilbert space with unitarily-equivalent undetermined bases. An initial measurement allows identification

10By the phrase “performing a measurement” we mean not the system/measurement device interaction, but the query of the device’s readout which takes place at some ﬁnite space-time interval away from the system/measurement device interaction. 11We do not mean to imply that this projection has any causal eﬀect on physical reality: We are in the Heisenberg picture so the system’s wave function can be ontological (if one insists) while the observable representing a measurement is both deterministic and epistemic.

10 of a ﬁducial basis by which comparisons with subsequent measurement can be made — if no external interactions interfere. If there are external interactions, the dynamics are generically governed by G so the initial basis may be no longer relevant and measurement may settle on a new Gλ′ .

Before leaving this section it is appropriate to stress that intΛ is an algebra homomorphism. In consequence, one can anticipate applications in quantum physics where B ≡ LB(H). But instead of viewing the algebraic properties of F(G) as proceeding from B, turn it around and posit an abstract C∗-algebra F(G) that is realized by B through functional integrals.

3 Subsumed approaches

The proposed deﬁnition of functional integrals is only useful to the extent that it includes known successful approaches. So it is important to check that this is the case.

Example 3.1 The Wiener and Feynman path integral via time slicing: m Consider P0C , the infinite dimensional vector space of piece-wise continuous pointed m paths x : [ta, tb] ⊂ R → C with x(ta)=0, and take G to be its underlying abelian group n equipped with a suitable topology. Choose B ≡ C, and let Λ = {λn : G → Hλ ∀ n ∈ N+} n by x 7→ (x(t1), x(t2),...,x(tn)) where t1 < t2 ... < tn and the Hilbert space Hλ comprises states characterized by a mean and covariance.12 n ′ ′ Evidently the pair (Hλ,Pnn ) is a projective system for G with maps Pnn determined by 1 n ′ ′ F Pnn ◦ λn = λn. It achieves the reduction (G)|GΛ = λn L (Hλ, C) for finite time slices. m Under the restrictions B ≡ R and x : T → R with T := [ta, tb], the projective system can be used to define a promeasure with Gaussian weight (theL Wiener measure). But in the generic case, with B ≡ C and x : T → Cm, to get the Feynman path integral one must work harder and either: i) analytically continue the restricted case; ii) use the projective system to construct time-sliced integrals defined using the Trotter product formula; or iii) use the topological dual space, its associated projective system, and Fourier duality to define projective distributions according to [8, 11]. Of course one must still determine 1 n the class of integrable functions f ∈ L (Hλ, C) allowed by each approach through functional analysis. Remark that one can contemplate more general projective systems that are not based m on time slicing. This was rigorously achieved in [12] in the case where P0C carries the structure of an infinite-dimensional real, separable Hilbert space. The projections are or- dered according to their finite dimension. Coupled with the theory of oscillatory integrals, the projective system for G gives rigorous access to Feynman-type path integrals and their localization by stationary phase. In the context of gauge field theory, projective systems have been constructed [13, 14] based on the holonomy of a connection. Once again, in these more general projective systems the most difficult work is quantifying integrable functions.

12 n Technically, λn maps G to the abelian group underlying Hλ, but this distinction in not necessary here and it is better to use familiar notation.

11 Unfortunately, projective systems derived from time slicing run into issues if non-cartesian coordinates are used on Cm (see e.g. [18, ch. 8]). Complications arise because there are consistency conditions that must be obeyed by the projective system, and it may be difficult to find a suitable projective system and/or integrable functions. Even more troublesome; in the case the target manifold is more general than Cm, the space of pointed paths is no longer a Banach space and the projective method cannot be applied directly. In either case, one must be careful to pay close attention to delicate mathematical issues — undermining the intuitive and formal appeal of path integrals. For Feynman path integrals at least, the shortcomings of the projective method can be sidestepped by utilizing dual topological abelian groups in the framework of Fourier/Pontryagin duality as exemplified in [10, 19]. In this approach, one no longer attempts to define a rigorous measure on the integration domain. Instead the path integral is related through Fourier duality to a bona fide integral. There is, however, no topological localization in this context since the dual space is assumed measurable from the beginning. The next example is a brief outline of the Cartier/DeWitt-Morette (CDM) approach which illustrates this idea.

Example 3.2 Cartier/DeWitt-Morette functional integration scheme:[19] The CDM scheme for functional integration corresponds to the particular case of B ≡ C and (as above) G an abelian topological group underlying an infinite dimensional Banach space. More precisely, G is the abelian (additive under point-wise addition) group of contin- m m uous pointed maps x :(T, ta) → (C , 0) equipped with a suitable topology, and X0 := P0C is its associated Banach space over C.13 To be consistent with the notation of CDM, we will abuse notation and write G ≡ X0 keeping in mind that scalar multiplication is strictly not allowed. Consequently, any question regarding scale must ultimately be referred to scalar multiplication in F(X0) through the definition of intΛ. Since X0 is abelian, the space of one-parameter subgroups L(X0) is a topological vector ′ m ′ space. The abelian group X0 underlying the topological dual (P0C ) is assumed to be a locally ′ compact Polish space when equipped with a suitable topology. Hence, X0 can be equipped with ′ 1 ′ a positive measure µ, L(X0) is a locally compact Banach space, and L (X0, C) is a Banach algebra under convolution. 14 The space of integrable functionals F(X0) is the set of functionals defined by

′ ′ Fµ(x) := Θ(x, x ) dµ(x ) (3.1) X′ Z 0 ′ ′ where Θ(x, x ): X0 × X0 → C is continuous, bounded and integrable with respect to µ. Then F(X0) is a Banach space with an induced norm deﬁned as the total variation of µ. Bounded

13 CDM uses X0 as their domain of integration. But the field structure is not relevant to the integration in the sense that their integrators are not invariant under scalar multiplication. Of course scale is an important issue, but it is better handled within the algebraic structure of F(X0). We accomplish this by including a scale factor s ∈ C+ in the definition of Dλx (where C+ is the right-hand complex plane). Otherwise said; the scale s is part of the specification of Λ. 14 Since Haar measures can only differ by a scalar multiple, the µ designation of Fµ can be dropped if we agree to use the normalized Haar measure.

12 linear integral operators X Dλx with k X FµDλxk≤kFµk on F(X0) are defined by R R ′ ′ Fµ(x)Dλx := Fλ(x ) dµ(x ) (3.2) X X′ Z 0 Z 0 where b ′ ′ Θ(x, x )Dλx := Fλ(x ) (3.3) ZX0 1 ′ defines the integrator family DΛx (with Fλ ∈ L (X b, C)). In particular, with the standard ′ 0 choice Θ(x, x′) = e−2πihx ,xi, then (3.1) can be interpreted as the Fourier transform of the measure µ. b Note that an affine transformation x 7→ x + xa along with the translation invariance m Dλ(x+xa)= Dλx yields integration on Xa, the space of pointed maps x :(T, ta) → (C , xa). ′ It is evident that a choice of λ corresponds to a class of functions Fλ(x ) integrable with respect to µ. Conversely, a choice of µ corresponds to a class of functions Fµ(x) integrable with respect to λ. In this sense, λ and µ are Fourier dual. For the archetypicalb Gaussian ′ case with Θ(x, x′) = e−2πihx ,xi, parameter λ characterizes the mean and covariance of the Gaussian paths of interest. The functional integral on the left-hand side of (3.2) is exact (in the sense it is also specified by the same λ), because there is a one-to-one correspondence between the two sides for Gaussian paths by duality. That is, the Fourier transform of a Gaussian is still a Gaussian. So here we have an explicit determination of Λ: it is the ′ Fourier dual to the set {µ} of measures on X0

To handle spaces (which are not topological groups) of pointed maps Pma M where now m : T → U ⊆ M with m(ta) = ma and U ⊆ M an open neighborhood of a smooth dim(M)= m Riemannian manifold, CDM uses the left-invariant vector ﬁeld Lie algebra Ga at a point ma to identify the non-abelian linear Lie group G underlying Pma M. In this case, a one- parameter subgroup morphism p : L(X0) → L(G) induces a morphism e Exp : L(X0) → PmaeM

x 7→ Exp(x)= expGe ◦ p (x) . (3.4) Given Exp and the fact that Pma M is contractible since it is a pointed space, the parametrization

P : X0 → Pma M

x 7→ Exp(logX0 (x)) (3.5)

allows the integral on Pma M to be deﬁned by

′ F(m)Dλm := Fµ(P (x))Dλ(P (x)) := Fµ(P (x)) |DetλP(x)|Dλx . (3.6) ZPma M ZX0 ZX0 The left-hand side furnishes the path integral route to QM. Note that it has limited applicability if M is not geodesically complete.

13 Meanwhile, if it happens that M = G is a Lie group manifold, then (3.6) can be readily used since the Lie algebra of left-invariant vector ﬁelds is automatically available at each point of the group manifold;

F(m)Dλm := Fµ(P (x))Dλ(P (x)) = f(gλ) dν(gλ) . (3.7) ZPga G ZX0 ZG In particular, (as is well-known) this means that the free point-to-point propagator on a group manifold is ‘exact’ in the sense that it can be expressed as a sum over relevant λ of ﬁnite dimensional integrals. Again, the left-hand side is exact due to Pontryagin duality. Alternatively, CDM can use the soldering form θ on the frame bundle F (M) equipped with a connection to construct the development map parametrization. The explicit construction n of the development map uses the identiﬁcation θ(hor(vp)) =z ˙ where z˙ ∈ C and vector hor(vp) ∈ Tp(F (M)) is tangent to the horizontal lift m(T) of m(T). The tangent to the horizontal lift m(T) induces the morphism Dev : P˙ M → P M e e ma ma m˙ 7→ Dev(m ˙ )=(π ◦ m) (3.8) where P˙ M is the abelian topological group of pointed maps m˙ : T = [t , t ] → U ⊆ T (M) ma e a b ma with m˙ (ta)=0 and π is the projection on the frame bundle. Then

F(m)Dλm := Fµ(Dev(m ˙ ))Dλ(Dev(m ˙ )) . (3.9) ˙ ZPma M ZPma M When M = G and the connection is Riemannian, Dev and Exp amount to the same thing.

It should be noted that [19] already suggested generalizing the space of pointed paths in their scheme to include locally compact abelian groups. And we stress that the localization/projective system in the CDM framework is eﬀected indirectly through Pontryagin duality.

Example 3.3 Loop groups: The previous two examples can readily be applied to continuous loops x : S1 → G yielding 1 functional integrals whose domains are the free loop group LG = HomC (S , G) or based loop 1 group Lga G = HomC ((S ,sa), (G,ga)) of some Lie group G. Utilizing a suitable parametrization `ala the CDM scheme, these can be extended to loop spaces LM and Lma M of a Rie- mannian manifold. We are, of course, glossing over symmetry issues regarding invariance

of the initial point for paths in LM. (We explicate the CDM parametrization for Lma M in the non-trivial example presented in §4.2.)

The next obvious generalization is to promote paths to ﬁelds with suitable analytic properties; x : D → M where D is a smooth Riemannian manifold with dim(D) = d ≤ m. The CDM scheme can be used to construct functional integrals for ﬁelds x : D → M according to

the definition, but we will not verify that specifying Λ and F|GΛ captures the intricacies of renormalization. Consequently, we can only claim that the proposed definition of functional integrals includes free fields since in this case they are characterized as Gaussian.

14 Example 3.4 CDM for fields:[27] Let FCm be the Sobolev space W k,p(U) of Lp maps x : U ⊆ D → Cm with U open and D a compact (with or without boundary) or open Riemannian manifold. If D has boundary ∂D, m k,p k,p ∞ let F0C = W0 (U) be the closure in W (U) of the vector space of C maps with compact k,p k,p support in U. Recall that W (U) and W0 (U) are Banach. Continue to take B ≡ C, m m and again abuse notation by writing FC ≡ X or F0C ≡ X. Of course, particular applications require consideration of some type of boundary conditions or functional constraints implemented through Λ. Let FM denote the space of fields x : D → M for both open and compact D for notational simplicity. Introduce the exterior differential system

{ωI =0} , ωI ∈ Λ(FM) (3.10)

with I ∈{1,...,N} and N ≤ m. This system deﬁnes a parametrization P : X →FM by

∗ P ωI =0 ∀ I. (3.11)

As with the previous case of paths, two particularly prevalent parametrizations arise from Pfaﬀ exterior diﬀerential systems associated with the exponential map Exp : Tx(FM) →FM

and the development map Dev : FTma (M) →FM. Finally, deﬁne

′ F(m)Dλm := Fµ(P (x))Dλ(P (x)) := Fµ(P (x)) |DetλP(x)|Dλx (3.12) ZFM ZX ZX

where Fµ ∈ F(X) is deﬁned by

′ ′ Fµ(x) := Θ(x , x) dµ(x ) (3.13) ′ ZX with µ the Haar measure on the dual group X′ underlying the topological dual (FCm)′. (As in CDM for paths, X′ is assumed locally compact Polish.) Structurally, functional integrals for fields are not much different from path integrals. But there are complications lurking in FM concerning the localization associated with Λ. Borrowing from the path integral case, one approach is to specify Λ by means of finite projections. In the context of QFT, the convention is to construct a projective system based on causal ordering in R3,1 and account for the spatial dependence through the formal device m limm→∞ C . That is, one considers a field on space-time to be a path with an infinite number of components indexed by some space-like surface (including any other discrete quantum numbers carried by the field). More generally, in the context of FQFT one decomposes15 D = Σ × T and constructs the projective system Σ ×{t} with t ∈ T. The sewing axiom is a consequence of the composition of projections, but unitarity and locality are extra conditions imposed by the physics. As

15Of course there is nothing forcing the decomposition D = Σ × T, and there are on-going attempts at making sense of the general case in the guise of “extended” FQFT. However, now one loses grounding in canonical quantization so physics can no longer supply direct guidance or motivation.

15 in the case of paths, Fourier duality and dual projective systems allow for the definition of projective Sobolev distributions. And the CDM scheme allows to effectively transfer Λ from FM to the set of measures on the space of integrable functionals on the dual X′. The projective approach for fields is a direct generalization of the original d = 0+1 path version, and it gains legitimacy through comparison with canonical quantization and operator methods. In favorable circumstances, one is able to find a fairly simple description

of Λ and integrable functions F|GΛ leading to exactly solvable models. Examples include partition functions and local n-point functions in free-ﬁeld QFT, rational CFT, and TQFT.

Conversely, specifying Λ and F|GΛ in the context of perturbative QFT is far more involved.

For one thing, one must find integrable functions F|GΛ at each order of perturbation. In addition, one typically requires invariance of various objects under some kind of infinite dimensional symmetry, and this redundancy must be accounted for consistently at each level of perturbation. Resolution of these two issues requires the programs of renormalization and gauge theory. For QFT applications, it is important to implement these programs within this framework, but the study is expected to be lengthy and so better presented elsewhere. We do want to make two brief remarks, however. First, one cannot interpret renormalization as a rescaling of fields in this framework because scalar multiplication in X is not supported — strictly (despite notation) X is an abelian group. So interactions that necessitate rescaling through renormalization require modified integrators Dλ(m) at each order of perturbation. It is tempt- ing to speculate that physical considerations might lead (through appropriate choices of λ) to some topology on the dual space X′ that would effectively act as a cut-off for continuous fields and hence regulate unruly integrals. In this sense, the renormalization group appears to

fit into the framework via Λ and F|GΛ . Second, the idea of effective field theories seems to be captured by the notion of topologically induced localization if the topological cover somehow corresponds to energy scale. (Note, the cover refinement characterizes continuity in F(X), so it implies a minimum time resolution for continuous dynamical evolution.)

Although Examples 3.1–3.4 ﬁt into the proposed deﬁnition, for the most part they are rather elementary functional integrals that have been well studied and understood. There are however more sophisticated, formal functional integrals developed in the context of TQFT and equivariant cohomological QFT that need to be included. For that we will need to assemble some tools in the next section.

4 Quadratic-type integrators

This section develops integrators based on sesquilinear forms on complex abelian topological groups. This is a substantial simplification of the general non-abelian case. However, such forms give rise to the ubiquitous Gaussian integrator family as well as the skew-Gaussian integrator family to be introduced below. In this section we restrict to ‘path integrals’ because this simpler context allows attention to be focused on the properties of the integrators without the distractions and issues fields bring to the story. Nevertheless, extending these quadratic-type integrator families to include fields should be straightforward; regularization

16 and local symmetry notwithstanding. Similarly, the development can be extended to non- abelian topological groups GC by considering sesquilinear forms on T ∗(GC). m Our starting point is the vector space PaC of piece-wise continuous, pointed paths m z : (T, ta) → (C , za). Let Za denote the underlying complexiﬁed abelian group (under ′ point-wise addition) of this vector space. The dual group Za is the space of linear forms hz′, zi→ C. The involution and complex structure on Cm induce an involution and complex structure on Za according to the prescriptions z∗(t) = z(t)∗ Jz(t) = iz(t) Jz∗(t) = −iz(t)∗ . (4.1)

′ ′ ′ ′ By duality, these structures can be transferred to Za. For example J z (t) = iz (t) where the ′ ′ ′ ′ transpose J is determined by hJ z , zi = hz , Jzi. Remind that J induces a Z2 grading on Za through the projection P± := 1/2(Id ∓ iJ); likewise for J′. ′ Let Za be endowed with a continuous, sesquilinear form ′ ′ ′ F : Z a × Za → C ′ ′ ′ ′ ′ ′ ′ (z 1, z2) 7→ F (z 1, z2) := hz 1, Gz2i (4.2) ′ ′ where the (linear) covariance G : Za → Za is nondegenerate with domain DG = Za. On the dual space Za, construct an associated sesquilinear form

F: Za × Za → C

(z1, z2) 7→ F(z1, z2) − B(¯z1, z¯2)= −hDz1, z2i =: FB(z1, z2) (4.3)

′ where D : Za → Za is linear and B(¯z1, z¯2) is a sesquilinear boundary form depending on a set of mean paths {z¯ : Dz¯ =0} (4.4) endowed with suitable boundary conditions. (Because the argument of a form or map already indicates its domain, from now on we will not distinguish between say F′ and F or D and D′ in the sequel unless to avoid confusion.)

Let Zz¯a := Za\Ker(D). Then, restricting to this factor space, we require

DG = Id ′ Zz¯a

GD = IdZz¯a , (4.5) ′ and so in this sense F and F are inverse modulo a boundary form on Zz¯a . Further, any ′ ′ ′ z ∈ Za can be reached from a givenz ¯ by z =z ¯ + Gz for all z ∈ Zz¯a . Consequently, each mean path spawns a copy of Zz¯a in Za. Decompose FB into Hermitian and skew-Hermitian parts according to FB = QB + ΩB where 1 1 Q (z , z )= − {hDz , z i + hDz , z i} = − (D + D†)z , z B 1 2 2 1 2 2 1 2 1 2 1 =: − h Qz , z i (4.6) 2 1 2

17 and 1 1 Ω (z , z )= − {hDz , z i−hDz , z i} = − (D − D†)z , z B 1 2 2 1 2 2 1 2 1 2 1 =: − h Ωz , z i . (4.7) 2 1 2

∗ ∗ Note that QB(z1, z2)=QB(z2, z1) but ΩB(z1, z2) = −ΩB(z2, z1) . To make contact with

QM, use QB(z, z) (resp. iΩB(z, z)) to deﬁne a norm on Zz¯a in the usual way, then complete

Zz¯a with respect to this norm to get the Hilbert space HQB (resp. HΩB ) of paths possessing boundary conditions encoded in B(·, ·). Remind that this structure induces a canonical ′ isomorphism between Za and Za. Clearly the Hilbert space based on QB can be very diﬀerent from that based on ΩB.

4.1 Gaussian integrators

Gaussian integrators are constructed from quadratic forms QB. Let Za be the underlying m ′ complex abelian group of the space of pointed paths PaC . Equip its dual group Za with a ′ suitable topology and denote the Pontryagin duality by h·, ·i : Za × Za → C.

16 Deﬁnition 4.1 A family of Gaussian integrators DΛωz,¯ QB (z) is characterized by

′ ′ 2πihz ,(z−z¯)i−(π/s)QB(z−z¯) Θz,¯ QB (z, z )= e ′ ′ −(π/s)B(¯z) 1/2 −πsWB(z ) Zz,¯ WB (z )= e Det(sWB) e (4.8)

where WB is inverse to QB in the sense that

δ2Q δ2W B δjk B = δ(t − t′)δ (4.9) δz(t)iδz(t)j δz′(t′)kδz′(t′)l il ∼ with i,j,k,l ∈{1,...,m}. The parameter s ∈ C+ = R+ ×iR, and the functional determinant is assumed to be well-deﬁned/regularized so that Detλ(·) coincides with det(·) (up to a possible phase). The Gaussian integrator family is deﬁned in terms of the primitive integrator family DΛz;

−(π/s)QB(z−z¯) Dλωz,¯ QB (z) := e Dλz (4.10)

where primitive integrator family DΛz is characterized by

′ 2πihz′,zi−(π/s)Id(z) Θ0,Id(z, z )= e ′ 1/2 −πs Id(z′) Z0,Id(z ) = Det(sId) e (4.11)

with Id(z)= hIdz,zi = |z|2.

16This definition uses a different normalization from the usual Gaussian integrator in the CDM scheme. Both definitions are valid: we choose this normalization because it seems more consistent with definitions of other integrator families and it highlights the role of the functional determinant.

18 −1 ′ Remark there is the obvious restriction (s Q(z)|Gλ ) ∈ C+ yielding integrable Θz,¯ QB (z, z ) for suitable QB. Loosely, the primitive integrator Dz (which is characterized by zero mean and trivial covariance) is the inﬁnite dimensional analog of the Lebesgue measure on Cn. Note that WB (and hence also Det WB) inherits the boundary conditions imposed on z, and note the normalizations −(π/s)Id(z) 1/2 Dλω0,Id(z)= e Dλz = Detλ(sId) , (4.12) ZZ0 ZZ0 and 1/2 −(π/s)B(¯z) Dλωz,¯ QB (z)= Dλωz,¯ QB (z)= Detλ(sWB) e . (4.13) Za Zz¯a Z XZ{z¯} Z XZ{z¯} Three points to emphasize: The ﬁducial Gaussian integrator Dλω0,Id(z) is associated with

the bona ﬁde Banach space Z0 = Zz¯0 where the primitive integrator is translation invariant, i.e. Dλ(z1 − z2)= Dλ(z1). For any givenz ¯ ∈{z¯}, the middle integral in (4.13) can therefore be written as an integral over Z0 by a change of integration variable z −z¯ 7→ z with z(ta)=0 since the primitive integrator is translation invariant. Finally, since there is a copy of Zz¯a for each non-trivial zero mode, we see clearly why an integral over the fulle spaceeZa must include a sum/integral over allz ¯.

Remark 4.1 The definition of functional integral allows to take limits of Gaussian integral operators with respect to the parameter s when the limits exist for the finite-dimensional integrals. Accordingly, one can define an integrator analog of the Dirac measure;

1 ′ 2πihz′,(z−z¯)i lim Θz,¯ QB (z, z )Dλz =: e δ(z − z¯)Dλz |s|→0 Zz,¯ W (0) B ZZz¯a ZZz¯a 2πihz′,zi =: e Dλδz¯(z) Zz¯a Z ′ = lim e−πsWB(z ) |s|→0 = 1 . (4.14)

−1 This definition makes sense because: i) the ratio Θz,¯ QB (z, 0)/Zz,¯ WB (0) = Θz,¯ QB (z, 0)/Z (0) z,¯ QB localizes to a Gaussian distribution which tends to a delta function as |s|→ 0; and ii) it ob- viously is consistent with the finite-dimensional definition. It is not surprising that the Dirac integrator behaves as one expects under a linear map ′ M: Za → Za with Det M =6 0 ([28, pg. 387]);

′ −1 F (z)Dλδz¯(M(z)) = F (z)Dλδ(M(z)) = DetλMz0 F (z0) (4.15) Zz¯ Z Z a Z 0 z0 X ′ 17 given that M(z0)(t) = 0, and DetλMz0 =06 for all z0 ∈ ker(M).

17The reduction to the middle integral comes from D(M(z)) = M′z¯ + M(Dz¯) = M′z¯ = 0 which implies ′ z¯ = 0 for DetλMz0 6= 0.

19 On the other hand, for |s|→∞,

′ 2πihz′,(z−z¯)i lim Θz,¯ QB (z, z )Dλz =: e Dλz |s|→∞ Zz¯a Zz¯a Z Z ′ −(π/s)B(¯z) 1/2 −πsWB(z ) = lim e Detλ(sWB) e |s|→∞ − −1/2 −(π/˜s)Q 1(z′) = lim Detλ(˜sQB) e B |˜s|→0 =: δ(z′) . (4.16)

Again, the deﬁnition makes sense for the same reasons. But notice the mismatch in normalization between the two cases.

4.2 Skew-Gaussian integrators Use the Hermitian form to deﬁne an inner product

(z1|z2)Za := QB(z1, z2) . (4.17)

Similarly, use ΩB to deﬁne a pre-symplectic form Ω on Za by 1 (Ωz |z ) := − hΩz , z i = Ω (z , z ) . (4.18) 1 2 Za 2 1 2 B 1 2 ′ Employ the complex structure J on Za and the topological pairing hz1, z2i to deﬁne the ′ associated map J : Za → Za by hJz1, z2i := (Jz1|z2)Za . Skew-Gaussian integrators are constructed from ΩB provided that the pre-symplectic form is J-skew, that is ΩJ + J†Ω = [Ω, J]=0 . (4.19)

To emphasize the skew nature of ΩB, we will change notation z → η in this subsection but stress that η is not Grassmann: The notation η is only meant to remind of the underlying skew symmetry.

Deﬁnition 4.2 Given a J-skew pre-symplectic form Ω on Za, a family of skew-Gaussian integrators Dωη,¯ ΩB (η) is characterized by

′ ′ 2πihη ,(η−η¯)i−πsΩB(η−η¯) Θη,¯ ΩB (η, η )= e ′ ′ −πsB(¯η) −(π/s)MB(η ) Zη,¯ MB (η )= e Pf(MB/s)e (4.20) where

ΩB(η) := iΩB(η, η)=(Ωη|iη)Za = (Ωη|Jη)Za (4.21) and ′ ′ ′ ′ MB(η ) := (Mη |Jη )Za (4.22) such that M = S†Ω S (4.23)

20 where S is a symplectomorphism on Za. Again the parameter s ∈ C+, the restriction

(sΩ(η)|Gλ ) ∈ C+ ensures integrability for suitable ΩB, and the functional Pfaffian is assumed well defined so that Pfλ(·) = pf(·) (up to a possible phase). This integrator family is defined in terms of the primitive skew-Gaussian integrator Dη according to −πsΩB(η−η¯) Dωη,¯ ΩB (η) := e Dη (4.24) where Dη is characterized by

′ Θ(η, η′)= e2πihη ,ηi−πsId(η) ′ Z(η′) = Pf(Id/s) e−(π/s) Id(η ) . (4.25)

Here Id(η) := ihJη,ηi = hJη, Jηi = hJJη|ηi =: (η|η)Za.

Clearly, given λ, the ‘localized’ integrator Dλη cannot correspond to a measure on the target space under a class of pointed maps. Instead it corresponds to a class of (formal) top/volume forms on (Za)λ. In essence, Za and Θ(η)Dη are the inﬁnite dimensional analog of current/form duality with respect to integration. From a mathematics perspective, one could argue this characterization is merely a change of notation from the Berezin representation. But, physically, it transfers the burden of anti-symmetry from the dynamical variables to the operators acting on them; which is a substantial re-interpretation. In contrast to the Gaussian case, the opposite extreme values of s lead to

1 ′ 2πihη′,(η−η¯)i lim Θη,¯ ΩB (η, η )Dλη =: e δ(η − η¯)Dλη |s|→∞ Zη,¯ MB (0) Za Za Z Z ′ = lim e−(π/s)MB(η ) |s|→∞ = 1 , (4.26) and

′ 2πihη′,(η−η¯)i lim Θη,¯ ΩB (η, η )Dλη = e Dλη |s|→0 Za Za Z Z ′ −πsB(¯η) −(π/s)MB(η ) = lim e Pfλ(MB/s)e |s|→0 =: δ(η′) . (4.27)

Skew-Gaussian integrators enable functional integration on Banach spaces of p-forms as exempliﬁed by the following.

Example 4.1 Mathai-Quillen Thom class representative: ′ † Suppose Za is the vector space of all p-forms on some given manifold and Ω = D JD where D: Za → Za is a first-order linear differential operator. Its transpose is determined by hD′η′, ηi := hη′, Dηi. Typically, JD is a covariant derivative associated with some chosen connection on a principal fiber bundle of interest. Below we will not distinguish between D′ and D and simply write D (unless to avoid confusion).

21 ′ Recall J : Za → Za is canonically associated with the complex structure J such that ′ † † ′ † † J J = IdZa and J = −J. Now, M = S ΩS implies M ≡ DJ D with D = S DS and 1 M (η′ , η′ ) = − hDJ ′D†η′ , η′ i−heDJe′†D†η′ , η′ei B 1 2 2 1 2 1 2 ′ n† ′ † ′ o = hJ D ηe1, Deη2i e e =: MB(ψ1, ψ2) (4.28) e e ′ where we have deﬁned ψ := Gη . e ′ ′ ′ ′ Restricting to Zz¯a := Za\Ker(D ) yields a sesquilinear form on Zz¯a given by

′ 2 M (η )| ′ = M (ψ)| ′ := iM (ψ, ψ)| ′ =(ψ|ψ) ′ = |ψ| . (4.29) B Zz¯a B Zz¯a B Zz¯a Zz¯a

To simplify, choose boundarye conditionse for η¯ so that B(¯η)=0. The corresponding symplectic functional integral encodes a useful Fourier duality between η and ψ that generates a Thom class representative;

2 −2πihDψ,ηi−πsΩB(η) −(π/s)|ψ| e Dλη = Pfλ(MB/s)e . (4.30) Za Z XZ{η¯} Consider the action functional contained in (4.30),

2 S(η, ψ) := 2πihDψ, ηi + πsΩB(η) − (π/s)|ψ| . (4.31)

Using hD′ψ, ηi = hψ, Dηi, which follows from duality (which is Poincar´eduality in this case), it is straightforward to show that δS(η, ψ)=0 for the variations

δη = iψ δψ = sJ ′Dψ . (4.32)

′ Notice that δη ∈ Zza while δψ ∈ Za, i.e. the variations are not quite in dual spaces. This BRST-like symmetry owes its existence to the skew-symmetry of ΩB, the Z2 grading induced ′ by the complex structure possessed by Za, and the topological duality between Za and Za: ′ ′ ′ ′ ′ ′ They induce isomorphisms (Ω, Ω ): Za × Za → Za ⊗ Za and (J, J ): Za × Za → Za ⊗ Za satisfying [Ω,J]=0 and [Ω ′,J ′]=0 that will feature prominently later. To verify that exp[S(η, ψ)]Dη is indeed a Thom class representative when JD is a Za covariant derivative, note that: i) it is clearly horizontal and invariant under the rotational R′ symmetry of (·|· )Za, ii) the action functional is δ-closed, but introducing an auxiliary map that is Gaussian-dual to ψ in the usual way renders it δ-exact,18 and iii) normalization is enforced by (4.47). The topological and cohomological aspects of (4.30) are well studied and understood — usually in the context of supersymmetric QM/QFT and Berezin functional

2 18This boils down to re-writing e−(π/s)|ψ| as a Gaussian functional integral for the dual pair hψ,Bi over the dual space Zza .

22 integrals (see e.g. [21]). It is noteworthy that complex pointed paths and their skew-Gaussian integrators are also germane in this context.19 In order to illustrate how all this works in detail, we follow a standard example and use it to express the Euler characteristic of a smooth compact Riemannian manifold M. Its 2,1 1 based loop space Lma M consists of smooth L loops γ : S → M with γ(ta) = ma for each C ∗ C ma ∈ M. It is convenient to parametrize (Lma M) by the Banach space Zma ≡ Tγ (Lma M) ∗ C C via the exponential map Exp : Tγ (Lma M) → (Lma M) . ∗ C ∗ C ∼ ∗ ∗ C Since Zma ∋ ηγ : T → Tγ (M) , with Tγ (M) = γ (T (M) ) being the complexiﬁed co-

tanget bundle over γ(t), the parametrizing space Zma is isomorphic to the Banach space of sections Γ(γ∗(T ∗(M)C)). It follows that the domain of the skew-Gaussian functional integral ∼ ∗ ∗ C ∗ C is the Banach space of sections ηγ ∈ Zma = Γ(γ (T (M) )) such that ηγ (t) ∈ Tγ(t)(M) . ∗ C The complex structure J induces a complex structure on Tγ (M) allowing the decomposition ∗ C ∗ + ∗ − Tγ(t)(M) = Tγ(t)(M) ⊕ Tγ(t)(M) for all t ∈ T. This in turn induces the decomposition of + − Zma = Zma ⊕Zma into holomophic and anti-holomorphic 1-forms which allows to fix boundary + + conditions such that ηγ (ta) = ηγ(tb)=(ςγ , 0). Think of ςγ as an infinitesimal holomorphic variation of loop γ at the point ma. Choose the linear first-order operator D = J †∇ where ∇ is the Levi-Civita connection on the complex frame bundle F (M)C. Evidently, JD is an exterior covariant derivative on ∗ C ∗ C β β T (M) . The action of JD on a holomorphic frame in T (M) is ∇eα = eβ ⊗ ω α where ω α are the connection (1, 0)-forms. Likewise,

† † † β α αβ J ∇ eα = −J e ⊗ ω β = ieβ ⊗ ω (4.33) which implies † † α αβ ∇J ∇ e = ieβ ⊗ R (4.34)

with Rαβ the curvature (1, 1)-forms. Hence, in a local trivialization of F (M)C with

∗ + ηγ (t) = (γ(t), v γ(t)(t)) ∈ Tγ(t)(M) , (4.35)

the Hermitian form ΩB(η) is represented by the matrix of the (anti-symmetric) curvature components; that is ∗ αβ ΩB(η)= vαR vβ . (4.36) The explicit action functional in this case is

∗ α ∗ αβ 2πi vα∇tψ + πs vαR vβ − π/s(ψ|ψ) γ(t) dt (4.37) ZT where ∇t denotes the pull-back of ∇ by γ.

19 Of course this is no accident. A Cliﬀord algebra obtains if one deﬁnes an inner product on Zz¯a by polarization of ΩB(η) according to η1 · η2 := η1η2 + η2η1 := ΩB(η1 + η2) − ΩB(η1) − ΩB(η2); which leads to the Grassmann/Berezin picture for any Lagrangian subspace of Za ×Za since then ΩB(·) vanishes. Also, note that variation of S(η, ψ) with respect to η yields Dψ = −isD†JDη so on shell {δ†,δ}η = |s|2DJD†η = |s|2Ωη. ′ Consequently δ certainly ‘looks’ like a supersymmetry charge, but in general ψ ∈ Zza and η ∈ Za are not topological duals so we’ll call δ a ‘BRST-like’ symmetry.

23 Now, since Dη¯ = JD¯η = ∇η¯ = 0 (4.38) by deﬁnition, the boundary conditions on η imply

η¯(t) = (ma, 0) ∀t ∈ T . (4.39)

† † ∗ C That is, (∇J ∇ )¯η = 0 implies that [¯η(t), η(t)] is horizontal for all η(t) ∈ Tγ(t)(M) (with [·, ·] the Lie bracket) and therefore

v¯γ(t)(t) = 0 ∀t ∈ T . (4.40)

This holds for every based loop, and so the set {η¯ : JD¯η = 0} coincides with the zero ∗ + ∼ section Γ0(T (M) ) = M. It follows that Lma M can be parametrized by the disjoint union + Xa := ma Zma . + Finally, ﬁx s=1 and momentarily restrict to a single Zma . Thereby obtain the Euler class representativeF `ala Mathai-Quillen

−2πihη,Dψi−πΩB(η)+π(ψ|ψ) e Dλη = Pfλ(MB) + ZZma = Pfλ(ΩB)

= pf((Rαβ)ma ) (4.41)

∗ + where ψ : M → Tγ (M) , and a normalization/regularization for Pfλ has been ﬁxed by suitable choice of λ. In particular, for the zero section given by ψ(γ(t)) = 0 ∀t ∈ T, the entire set {η¯ : JD¯η =0} yields an integration over the full space Xa producing the Euler characteristic

−πΩB(η) −πΩB(η) e Dλη = e Dλη = pf(Rma ) . (4.42) + Xa Zma M Z XZ{η¯} Z Z Alternatively, re-instate the parameter s and take the limit |s| → 0 along the imaginary axis. In this case, if the section ψ is generic, the left-hand side of (4.41) localizes onto the zero-locus of ∇ψ according to (4.27).20 Recall that we took B(¯η)=0 to simplify the exposition, but note that we can arrange ′ for dZη,¯ MB (η )/ds=0 with a suitable choice of the boundary form B(¯η); in which case s- independence with s=1 , ψ =0 versus |s|→ 0 yields the well-known equivalence between the Gauss-Bonnet versus Poincaré-Hopf representations of the Euler characteristic χ(M). To finish the story, recall (3.6). Then, by definition

−πΩB(η) −πΩB(η) F(m)Dm := e Dλη = e Dλη + ZLma M ZXa ZM ZZma

= pf(Rma ) (4.43) ZM 20 ′ The delta functional δ(η ) represents a complex quantity whose phase at an isolated zero m0 in this case is normalized/taken to be iInd(∇ψ,m0).

24 as long as F and Dm are chosen to satisfy F(P (η)) D(P (η)) = e−πΩB(η) Dη. In essence, the −πΩB(η) functional integral is realized by an entire family of integrals + e Dλη := pf(Rma ) Zma indexed by the points of M — which exhibits yet again the localization idea. R Before moving on, we digress for a moment to indicate why characterizing a skew- Gaussian integrator so that Pf(MB) ∼ Pf(ΩB) makes sense (compare to the Gaussian case 1/2 −1/2 † where Det(WB) ∼ Det(QB) ) — after all, it seems a bit contrived to insist M = S ΩS. Suppose Za is endowed with Hermitian form QB and complex structure J. Let ΩB be ′ the associated K¨ahler form. From the topological duality, QB, J, and ΩB extend to Za ⊗ Za which is Z2-graded and decomposes as

′ ± ′± ± ′∓ + − Za ⊗ Za = Za ⊗ Za ⊕ Za ⊗ Za =: Wa ⊕ Wa =: Wa . (4.44) ± M + − ′ We can characterize an invariant integrator D(w ,w ) on Za ⊗ Za by

+ − + − + − −(π/s)B+(w ¯ )−πsB−(w ¯ ) + − Θ(w ¯ ,w¯ ),FB (w ,w ) Dλ(w ,w ) := e (4.45) ′ ZZa⊗Za ± ± with w ∈ Wa , and a sesquilinear form on Wa × Wa

+ − + + − − FB(w ,w )=QB(w1 ,w2 ) + ΩB(w1 ,w2 ) (4.46)

− − − − where ΩB(w1 ,w2 )=QB(w1 , Jw2 ). As long as w+ and w− are independent we have D(w+,w−) = Dw+Dw−, and, to sim- ± plify, we ﬁx boundary conditions B±(¯w ) = 0. Then the characterization of skew-Gaussian integrators follows from the prescribed normalization (and functional Fubini);

+ − + − 1 ≡ Θ(w ,w ) Dλ(w ,w ) ′ ZZa⊗Za + − + − = Θ(w ,w ) Dλw Dλw − + ZWa ZWa − − −1/2 −πsΩB(w −w¯ ) − = Det(QB/s) e Dλw . (4.47) − ZWa Under a change of integration variable, the measure is invariant (by assumption) implying + − + − Dλw Dλw = Dλw Dλw . Meanwhile, suppose the change of variable renders QB 7→ IdB, + 1/2 + − −1 − 21 then Dλw ∼ Detλ(QB/s) Dλw . So we need Dλw ∼ Pfλ(ΩB/s) Dλw . ePhysically,e this construction can be interpreted to mean that Gaussian integrators represent conﬁguration-typee (Fourier pairs with like parity)e degrees of freedom characterized by symmetric bilinear forms (hence by covariance — in the statistical sense). The Fourier duals in this case must be related through inverse symmetric bilinear forms. On the other

21 For a general change of variable, say QB 7→ QB, the two integrators are related through the ratio 1/2 e e QB −ran(QB)/2−ind(QB)+ind(QB) 1/2 1/2 Detλ e /s := s det(QB) /[det(QB) ] . The same conclusion follows. QB e e

25 hand, skew-Gaussian integrators represent phase-space-type (‘skew-Fourier’ pairs with opposite parity) degrees of freedom characterized by skew-symmetric bilinear forms (hence by conjugation — in the dynamical sense). So it is natural to impose a symplectic relationship between the corresponding duals, and therefore the condition M = S†ΩS, as this expresses (dynamical) invariance between Lagrangian subspaces in phase space; which is the hallmark of a dynamical system. Reversing integration order allows to deﬁne a functional Liouville integrator (since ΩB is a K¨ahler form in the case at hand)

+ + + − − + −(π/s)QB(w −w¯ ) + Θ(w ,w ) Dλw Dλw = Pf(MB/s)e Dλw + − + ZWa ZWa ZWa + + −(π/s)QB(w −w¯ ) + = Pf(ΩB/s)e Dλw + ZWa + + =: Dλℓw¯ , ΩB (w ) . (4.48) + ZWa This Liouville integrator clearly has application in the context of phase-space path integrals since Za models a symplectic manifold. Notice that a symplectic transformation to canonical coordinates on Za implements the analogue of the Nicolai map — albeit for a quadratic action and (generally) position-dependent J. Skew-Gaussian integrators possess two key properties: 1) they provide access to Pfaffian- type partition functions without invoking Berezin integration, and 2) like the Gaussian case, ′ they allow to define a delta functional on Za but in the inverse s scaling limit. In consequence, ′ a delta functional localization cannot be achieved on Za ⊗ Za simultaneously using Gaussian and skew-Gaussian integrators since their respective delta functionals require inverse limits + for |s|. On the other hand, the limit |s|→∞ does yield delta functional localization on Wa − while |s|→ 0 localizes on Wa . These properties, along with the physical interpretation of the integrators discussed ′ above, suggests that in the context of quantum physics we can interpret Za and Za as furnishing two (often Fourier) dual representations of a quantum Hilbert space while the de- + − composition Wa and Wa can be interpreted as manifesting a probabilistic and a dynamical nature (respectively) simultaneously possessed by the degrees of freedom.22 From a mathematical perspective, this entire section can be viewed as simply a re-wording of standard physics Gaussian and Berezin functional integrals. However, the exercise is not without value: For one thing, the skew-Gaussian integrator connotes dynamical relation- ships among variables whereas the Gaussian integrator connotes correlations. Mixing these ′ attributes together on Za ⊗ Za = Wa when ΩB is Kähler and η, ψ contain equal degrees of freedom leads to behavior that mimics supersymmetry — but without the need to intro-

22We do not mean to imply that the dichotomy inherent in the decomposition actually reflects a physical reality. The two aspects are different sides of the same coin. That is to say, the manifested nature depends on how one chooses to look at it. This contrasts with the usual construction and interpretation of boson/fermion symmetry. The difference stems from the conception of the skew symmetry: in our setup, it is attributed to a skew-symmetric bilinear form (which characterizes an integrator) while the standard SUSY approach attributes it to fermionic degrees of freedom (which characterize particle states).

26 duce superpartners into the physical arena.23 In light of current experiment, this may prove useful. For another thing, it points to the importance of starting with sesquilinear forms on complex Banach spaces. At any rate, nice mathematical properties normally supplied by supersymmetry are still available in the functional integral framework as we have deﬁned it. Moreover, everything in this section can be promoted to complex non-abelian topological ′ groups. In this case, one would identify Za and Za with the Lie algebra and Lie co-algebra respectively of a non-abelian group. The non-trivial Lie bracket would allow development of Lie derivatives, exterior and interior products, etc.; enabling the investigation of symmetry properties of integrators.

5 Non-Gaussian integrators

One of the main themes we want to emphasize is that the proposed deﬁnition of functional integral — though not new in the sense that it depends on a family of Haar measures — gives rise to a perspective that spurs new types of functional integrals beyond the familiar Gaussian case. This section highlights three non-Gaussian integrators.

5.1 Liouville integrators This subsection further develops the Liouville integrator introduced in the previous section beyond the Kähler case. We give the geometric setting, characterize the integrator, and indicate some of its general features. However, there are substantial symmetry and boundary aspects pertinent to physics/mathematics applications that will be addressed in future work. e C π Start with a pointed principal bundle Ge → Pp• → Zz• . The pointed base space Zz• is a complex Lie group of dimC(Z)= m equipped with a Hermitian metric g. We further suppose C ∼ C that Pp• is endowed with a metric connection χ : Pp• → G = TeG and a complex one-form ∼ ω : Pp• → Z = TeZz• (often taken to coincide with a connection on the frame bundle). ∗ ∗ π Construct an associated pointed cotangent bundle Z → T Z(z•,0) → Zz• with structure C ∗ x group G and typical fiber Z the co-Lie algebra of Zz• . Let ε : (X, •) ֒→ (Z, z•) be a based embedding of a sufficiently regular, compact, complex manifold X with dimC(X) = ∗ n ≤ m. We will restrict to ∂X = ∅. The embedding ε induces pull-back bundles ε Pp• and ∗ ∗ ∗ ε T Z(z•,0) =: T Xx• now over the pointed base space Xx• . ∗ 0 1 Let Λ(0,1)(ε Pp• ) ∋ (α ,α ) denote the (direct sum) vector space of compactly supported, ∗ ∗ ∗ 24 based 0-forms on ε Pp• and 1-forms on ε Pp• with values in Z . Given a family of local

trivializations {Ui, ϕi} of Pp• with its associated canonical sections, we can canonically iden- ∗ ∗ 0 1 tify Λ(0,1)(ε Pp• ) ≡ Λ(0,1)(T Xx• ) and focus on based forms (also denoted by (α ,α )) on

23 e Clearly one coulde interpret the constituents of Wa as superpartners. However, to the extent that Za models already a symplectic manifold and the Liouville integrators characterize dynamical quantum systems; only one side of the symmetric/skew-symmetric split contributes to the physical Hilbert space. It goes counter to experiment (at least up to now) to build a realistic physical model whose transition amplitudes include both at once — quite analogous to the need for a polarization in geometric quantization. This of course does not imply mathematical models of this sort are not valuable. 24There is no reason not to allow for equivariant p-forms in the construction, but for a ﬁrst pass it helps to keep it simple.

27 ∗ Xx• with values in T Xx• . The identification of 1-forms via canonical sections includes χ ∗ and ω as well, so we also use the same symbols for their counterparts on T Xx• . In local µ ∗I ∗ 0 1 coordinates (x , z ) on Ui × Z with Ui ⊂ X, a representative of (α ,α )(x) is expressed as 0 x 1 x I xµ ∗ α ( ), (α ( ))µ d ⊗ eI . ∗ Given a local trivialization {Ui,ϕi} of the pointed cotangent bundle T Xx• with a based 0 1 local section α|Ui = (αi ,αi ), the complex 1-forms χ and ω are employed to construct two ∗ fiducial functionals SB, ΩB : Λ(0,1)(T Xx• ) → C+ defined by

0 1 1 0 1 x µ¯ 0 x I SB(αi ,αi ) := ≪ αi |Dαi ≫ dτ = (α ( ))I (Dα ( ))µ dτ (5.1) ZUi ZUi and

1 1 1 ΩB(αi ) := ≪ αi |dω(αi ) ≫ dτ ZUi 1 1 = ≪ αi |[dω,αi ] ≫ dτ ZUi 1 x I ∗ D x K 1 x J ∗ = ≪ α ( ) ⊗ eI |cKJ (dω( ) ∧ α ( ) ) ⊗ eD ≫ dτ ZUi 1 x µ¯ x IJ 1 x ν = (α ( ))I (dω( )µν¯) (α ( ))J dτ (5.2) ZUi ∗ ∗ where ≪·|·≫ denotes the inner product on T Xx• coming from the pull-back metric ε g on X and the Hilbert-Schmidt inner product on Z∗, the diﬀerential operator D = d − χ is D the covariant derivative, τ is the volume form on X, and cKJ are the structure coeﬃcients of the Lie algebra. ∗ Endow Λ(0,1)(T Xx• ) with a suitable topology to arrive at an abelian topological group ′ Zx• of based 0-forms and 1-forms along with its dual Zx• . As in the previous section, Zx• and ′ ′ Zx• inherit complex structures from Z. The induced complex structure on Zx• × Zx• then ′ ′ ∼ ′ ′ leads to a Z2-grading of the tangent space Tz Zx• ⊗ TzZx• = T(z ,z)(Zx• ⊗ Zx• ). Evidently, we ′ + − can view T (Zx• ⊗ Zx• ) =: Wx• ⊕ Wx• =: Wx• as a Z2-graded Whitney sum bundle. This will be the domain of the Liouville integrator family.

Deﬁnition 5.1 Given the geometric constructions above, the Liouville family of integrators + DΛlw,¯ SB, ΩB (ˇw) on the topological group Wx• is characterized by

−(π/ˇs)SB(w ˇ−w¯) Θw,¯ SB, ΩB (ˇw) = Pf(ΩB/ˆs)e − + Zw,¯ SB, ΩB = Vol Wx• / Wx• (5.3)

0 1 + + where wˇ := (w ,w ) ∈ Wx• is an even-grade 0-form and 1-form on X, and Vol(·) is interpreted (formally) as a normalized volume functional that localizes (via λ) to a volume ratio.

28 The Liouville family is deﬁned in terms of the primitive integrators Dλwˇ and Dλ(ˇw, wˆ) ≡ 25 DλwˇDλwˆ such that

−(π/ˇs)SB(w ˇ−wˇ¯) Dλlw,¯ SB, ΩB (ˇw) := Pf(ΩB/ˆs) e Dλwˇ (5.4) with

−πˆsΩB(w ˆ−wˆ¯) Pfλ(ΩB/ˆs) := e Dλwˆ (5.5) − ZWx• 0 1 − − wherew ˆ := (w ,w ) ∈ Wx• is an odd-grade 0-form and 1-form. Restoring the Pfaﬃan to integral form gives (puttingw ¯ = 0 for simplicity)

−(π/ˇs)SB(w ˇ)−πˆsΩB(w ˆ) − + Dλl0,SB, ΩB (ˇw)= e Dλ(ˇw, wˆ) = Volλ Wx• / Wx• (5.6) + ZWx• ZWx• which helps explain the normalized volume interpretation since integration against X yields a volume ratio. Actually, we are cheating a bit by characterizing the Liouville integrator in terms of

Zw,¯ SB, ΩB since the volume functional, generically, is only a formal object. However, when SB − + is quadratic, we can be more explicit since then Vol(Wx• / Wx• ) → Pf(ΩB/ˆs) / Det(SB/ˇs). In particular, as we have seen before, for the K¨ahler case ΩB ◦J = SB with equivalent scaling − + p ˇs ≡ ˆswe have Volλ(Wx• / Wx• ) ∼ pf(Id); indicating even-odd symmetry. It is instructive to apply the Liouville integrator to a familiar example.

Example 5.1 Berline-Vergne and Duistermaat-Heckman localization: By taking diﬀerent limits of ˇs, the Liouville family will be shown to be consistent with the Berline-Vergne theorem and, with suitable specialization, the Duistermaat-Heckman theorem. Not having explored the symmetry aspects of the Liouville family, we cannot claim that the considered integrals do not depend on the scaling parameters ˇs and ˆs. So this example should be viewed not as a proof but as a demonstration of manipulations of the Liouville integrator that are consistent with the two localization theorems. hor ∗ For Berline-Vergne, we restrict to the vector space Λ(0,1)(T Xx• ) of horizontal forms + + and construct the space W := x• Wx• . Consider the integral F 1/2 0 Det(SB/ˇs) Dλlw,¯ SB, ΩB (w ) (5.7) + ZWx• 26 where SB is now quadratic and given by

0 0 + 0 0 µ¯ 0 0 ρ 0 0 SB(w ,Dw ) = ≪ Dw |Dw ≫ dτ = D w (x)ρ Dµw (x) dτ = tr Dw ∧∗Dw ZX ZX ZX (5.8) 25Sincew ˇ andw ˆ are supposed independent, they can be scaled diﬀerently; hence the two scaling factors ˇs and ˆs. 26This can be implemented by inserting a delta functional δ(w1 − Dw0)+ into (5.6), using the original 0 1 + 1 0 1 + SB(w , w ) = X ≪ w |Dw ≫ dτ , and then integrating over (w ) . R

29 and the pre-symplectic form is

1 − 1 1 1 x µ¯ x ρσ 1 x ν 1 1 ΩB(w ) = ≪ w |Ωw ≫ dτ = w ( )ρ (Ω( )µν¯) w ( )σ dτ = tr w ∧ Ω ∧∗w . ZX ZX ZX (5.9) Take the limit |ˇs|→ 0 with ˆs= 1 to get

1/2 0 0 0 0 lim Det(SB/ˇs) Dλlw,¯ SB, ΩB (w ) = Pf(ΩB)δ(w − w¯ ) Dλw |ˇs|→0 + + ZWx• ZWx• 0 = Pf(ΩB) Dλδw¯0 (w ) + ZWx• 0 0 =: Dλlw,¯ SB, ΩB (δw¯ (w )) (5.10) + ZWx• Analogous to translation of the Gaussian integrator in the abelian case, we can arrange for C x• = e by group action of G to get

0 Dλl0,SB, ΩB (δ(w )) = pf((Ωρσ)e) =pf(Ωe) . (5.11) + ZWe 0 Then since w¯ is determined by Dw¯ =0, it represents a horizontal lift of Ui ⊂ X with initial point (x•, 0). Therefore

0 0 0 Dλlw,¯ SB, ΩB (δw¯ (w )) = Dλl0,SB, ΩB (δ(w )) = pf(Ωx) (5.12) + + W We X Z {Xw¯x• } Z Z + ∼ which expresses the localization W |w0={w¯0} = X. More generally, one can localize onto zero sections of wˇ under an endomorphism L: + + x 0 x Wx• → Wx• in which case (4.15) yields (assuming isolated zeros of (Lˇw)( ) := Lλ(w ( )) ′ such that Lλ is an isometry with non-degenerate derivative map Lλ)

0 0 x Pfλ(ΩB) F (w )Dλl0,Se , Ω (δ(Lw )) = F (ˇwo( )) (5.13) + B B ′ We x DetλLwˇo {Xw¯x• } Z {wXˇo( )} where SB := SB ◦ L, and wˇo(x) ∈ ker(Lλ). In particular, restricting to an equivalence class ′ of isometries such that Lλ ∼ Ω ◦ J with equivalence relation relative to symplectic transfor- mations,e (5.13) reduces to

′ pf(L x ) 0 0 wˇo( ) F (w )Dλl e (δ(Lw )) = F (ˇwo(x)) (5.14) + 0,SB 2 ′ We x |pf (Lwˇ (x))| {Xw¯x• } Z {wXˇo( )} o This can be interpreted as the functional integral analog of Berline-Vergne localization. From here, one could use the Berline-Vergne theorem to show that, for a suitable choice of λ, the

left-hand side is a (topologically) localized Liouville functional integral given by X F τ where F τ is an equivariantly closed top-form on X. R

30 Alternatively, return to (5.7) and instead take the limit |ˇs|→∞ to get (still with ˆs= 1)

−π ΩB(w ˆ−w¯) e Dλwˆ = Pfλ(ΩB) . (5.15) − ZWx• 0 Multiply both sides by F (w ) and sum over {w¯x• } to get

0 −π ΩB(w ˆ) 0 0 F (w )e Dλwˆ = F (w )(x)pf(Ωx) =: F (w ) . (5.16) − We X X {Xw¯x• } Z Z Z From here, if one can show that F (w0) is an equivariantly closed top-form and that (5.7) does not depend on ˇs and ˆs, then the Berline-Vergne theorem for Liouville functional integrals follows. But, as we have said, showing this requires the study (which we won’t initiate here) of symmetry aspects of the Liouville integrator. Duistermaat-Heckman is just a few short steps away. Assume that GC furnishes a Hamil- tonian action on Z, and let Xg ∈ T Z represent the vector ﬁeld induced by the action of some C − g ∈ G . Choose a ﬁxed element µ ∈ Wx• that generates a momentum map by x x x HXg ( ) := hµ( ),Xg( )i (5.17) x so that HXg ( ) is a generalized Hamiltonian. Given the 2-form Ω = dω = Dω (since ω is horizontal) and its associated equivalence class of isometries Lλ, choose (if it exists) a suitable µ satisfying x 0 ′ x ′ dhµ( ), (Lw ) ( )i = i(Lw0) (x)Ωx . (5.18) Restore the variable scaling ˆs and take

0 ′ 0 ′ F (w0)= e−πˆs hµ,(Lw ) i = e−πhµ,ˆs(Lw ) i . (5.19)

Then it turns out[29] that F (w0) is indeed an equivariantly closed top-form so we verify Duistermaat-Heckman (assuming ˇs,ˆs-independence, isolated zeros of Lˇw, and using shorthand notation for the Pfaﬃan ratio)

′ ′ e−πhµ,ˆs(Lw ˇo) i e−πˆs hµ, (Lw ˇo) i F (w0)= = ˆs−n (5.20) pf(ˆsL′ ) pf(L′ ) X x wˇo(x) x wˇo(x) Z {wXˇo( )} {wXˇo( )} since dimR(X)=2n. Evidently, the localization displayed by the Duistermaat-Heckman theorem hinges on three conditions: i) the integrals do not depend on the scaling parameters ˇs and ˆs; ii) up to sym- ′ plectomorphisms, Lλ ∼ Ω ◦ J; and iii) there exists a momentum map such that all three ′ x objects Lλ, Ω, and µ are related through (5.18) for all ∈ X.

Although the Liouville integrator was presented for ∂X = ∅, it can be generalized to the important case when X is a bordism. Like the symmetry aspects, the bordism case requires a more lengthy study than we can include here.

31 5.2 Gamma integrators

Definition 5.2 Let T0 be the space of continuous pointed maps τ :(T+, ta) → (C+, 0) where T+ ⊆ R+ and C+ := R+ × iR is the right-half complex plane. T0 is an abelian topological group under point-wise multiplication in the first component and point-wise addition in the ′ ′ second component. Let β be a fixed element in the dual group T0 of characters HomC (T0, C). A gamma family of integrators DΛγα,β′ (τ) on T0 is characterized by ′ ihτ ′,τi−hβ′,τi α Θα,β′ (τ, τ )= e τ ′ ′ ′ −α Zα,β′ (τ ) = Det (β − iτ ) (5.21) α where α ∈ C+, τ is defined point-wise by τ α(t) := eα log τ(t) (5.22) with the principal value prescription for log τ(t), and the functional determinant Det(·) is assumed to be well defined (through specification of λ). The gamma integrator family is defined in terms of the primitive integrator Dλτ by −hβ′,τi α Dλγα,β′ (τ) := e τ Dλτ (5.23)

where Dλτ is characterized by ′ ′ ′ Θ0,Id′ (τ, τ ) = exp{ihτ , τi−hId , τi ′ Z0,Id′ (τ )λ =Γλ(0) , (5.24) and implicit in Γλ(0) is a regularization.

In applications, one often imposes a bound on τ(t); say |τ(t)|≤|c| for all t ∈ [ta, tb] and for some finite constant c ∈ C+. The obvious tool to enforce this constraint is the functional analog of Heaviside; yielding a ‘cut-off’ gamma family that generalizes the previous definition but reduces to it as the cutoff |c|→∞:

′ Definition 5.3 Let T0 be the space of continuous pointed maps τ :(T+, ta) → (C+, 0). Let β ′ ′ be a fixed element in the dual group T0 and fix a fiducial τo ∈ T0 such that hβ , τoi = c ∈ C+. A lower gamma family of integrators DΛγα,β′,c(τ) on T0 is characterized by ′ ihτ ′,τi−hβ′,τi α Θα,β′ (τ, τ )= e τ ′ ′ ′ −α Zα,β′,c(τ )= γ (α,c)Det (β − iτ ) (5.25) where γ (α,c) is the lower incomplete gamma function given by ∞ (c)α+n γ (α,c)=Γ(α)e−c , (5.26) Γ(α + n + 1) n=0 X and again the functional determinant Det(·) is assumed to be well defined by suitable choice of λ. An upper gamma family of integrators DΛΓα,β′,c(τ) is defined similarly where Γ(α,c)=Γ(α) − γ (α,c) (5.27) is the upper incomplete gamma function.

32 Using this notion, the ﬁducial gamma integrator Dλτ is Dλγ0,Id′,∞(τ). By deﬁnition it is normalized according to 1 1 D γ ′ (τ) := 1 =: D Γ ′ (τ) , (5.28) Γ(0) λ 0,Id ,∞ Γ(0) λ 0,Id ,0 ZT0 ZT0 but the other family members yield

1 ′−α 1 D γ ′ (τ) = Det β = D Γ ′ (τ) . (5.29) Γ(α) λ α,β ,∞ λ Γ(α) λ α,β ,0 ZT0 ZT0 ′−α ′−α ′ −α Remark that Detλ(β ) is an element of B = C here, and in general Detλ(β ) =6 det(β ) .

5.2.1 Distributionals As we have repeatedly stressed, an important aspect of the proposed scheme is localization in function spaces. The aim of this subsection is to develop some tools to eﬀect localization ′ on the dual group T0. As motivation, consider the lower gamma integrator family, and restrict to τ ‘imaginary’, n i.e. τ :(T+, ta) → ({0} × iR, 0). Put α = 1, and consider the linear map Ln : T0 → iR and ˜ n ′ ˜ n its transpose Ln : R → T0 such that hω, Ln(τ)i = hLn(ω), τi = 2πω · iu with u, ω ∈ R and ω = ω∗. Then, Ln −2πiω·u Dλγ1,β′ (τ) −→ e du = δ(ω) (5.30) n ZT0 ZR with the integral over Rn understood as an inverse Fourier transform. On the other hand,

′−1 −c ′−1 Dλγ1,β′,c(τ) := γ(1,c)Detλ(β )=(1 − e )Detλ(β ) , (5.31) ZT0 and so the integrator Dλγ1,β′,∞(τ) can be understood as a limit;

Dλγ1,β′,∞(τ) := lim Dλγ1,β′,c(τ) . (5.32) |c|→∞ ZT0 ZT0 Consequently, when c is strictly imaginary, Dλγ1,β′,∞(τ) can be interpreted as the functional analog of a two-sided Laplace transform implying

c −c ′−1 Dλγ1,β′,∞(τ) = lim (e − e )Detλ(β ) ; (5.33) |c|→∞ ZT0 ′−1 which formally vanishes except when Detλ(β ) diverges. This can be interpreted as the functional analog of a delta function. In particular, this integrator can be used to localize onto the kernel of β′. Conversely, if τ is ‘real’, i.e. τ :(T+, ta) → (R+ ×{0}, 0), then

′−1 −c ′−1 Dλγ1,β′,∞(τ) := lim γ(1,c)Detλ(β ) = lim (1 − e )Detλ(β ) , (5.34) c→∞ c→∞ ZT0 which we interpret as a principal value. These observations suggest the deﬁnition:

33 ′ ′−1 ′ Deﬁnition 5.4 Suppose hβ , τi ∈ iR and Detλ(β ) diverges. A delta functional on T0 is deﬁned by ′ 1 δ (β ) := D γ ′ (τ) . (5.35) λ Γ(1) λ 1,β ZT0 ′ If instead hβ , τi ∈ R+ , ′ 1 Pv (β ) := D γ ′ (τ) . (5.36) λ Γ(1) λ 1,β ZT0 Together these indicate the linear functional heuristic

Det(β′−1) = Pv(β′)+ iδ(β′) (5.37)

′ with hβ , τi ∈ C+. That is, given a choice of λ, we have

′−1 Det(β ): T0 → C

τ 7→ Dλγ1,β′ (τ) . (5.38) ZT0 Remark that Deﬁnition 5.4 suggests the characterization

′ α−1 (α−1) ′ i δ (β )= D γ ′ (τ) (5.39) λ Γ(α) λ α,β ZT0 ′ ′−1 when hβ , τi ∈ iR and Detλ(β ) diverges. The characterization is “good” in the sense that ′ n δλ(β ) reduces to the usual Dirac delta function under linear maps Ln : T0 → iR for any n. So for α = m ≥ 1 with m ∈ N we have

′ m (m−1) ′ m−1 Γ(m − 1) δ δ (β )(t) = i Dγ ′ (τ) . (5.40) λ Γ(m) δβ′(t)m 0,β ZT0 It appears that gamma integrators and their associated functional integrals might be used as a basis for a theory of what might be called ‘distributionals’, but of course much work is required to develop and verify such a concept.

Remark 5.1 Delta functionals defined in terms of gamma-type integrators are important for imposing constraints that lead to certain types of localization. Notice that they can be interpreted as the functional analog of “the two-sided Laplace transform of the identity map”, and the duality allows them to be transferred to T0. On the other hand, as remarked in the previous section, the notion of delta functionals can also be formulated using Gaussian- type integrators [30]. Again, duality — but this time Fourier duality — allows them to be transferred between dual spaces. But there is a big difference between the two: Whereas the gamma family integrator is invariant under ‘rescaling’ by a linear map, the Gaussian family picks up a functional determinant àla Faddeev-Popov. How is one to know which type of delta functional is appropriate in a given application? The answer proposed in [31] is based on analogy with Bayesian inference in probability theory. In essence, the type of delta functional depends on the integrator family characterizing the function space of interest. For example, if the function space is T0, then the

34 gamma-type delta functional is indicated. However, if the function space is a Banach space characterized by a Gaussian integrator family, both types of delta functional are required in general. Specifically, one uses a gamma-type to localize the mean and a Gaussian-type to localize the covariance. The latter corresponds to the Faddeev-Popov method successfully utilized in QFT. However, it is known that the Faddeev-Popov method is not appropriate for all types of localization:27 In particular, it is not applicable to fixed energy path integrals or paths with fixed boundary conditions. But these types of constraints localize the mean of a Gaussian and should therefore be implemented with gamma-type delta functionals [31, 28].

5.3 Poisson integrators Restrict gamma to integers, i.e. α = n ∈ N. Take the lower gamma integrator and regularize by replacing γ(n, c) with the regularized lower incomplete gamma function

P (n, c) := γ(n, c)/Γ(n) . (5.41)

Likewise for the upper incomplete gamma function

P (n, c) := Γ(n, c)/Γ(n) . (5.42)

Note that, for N ∈ Pois(c) a Poissonb random variable, we have (c)k Pr(N < n)= e−c = P (n, c) . (5.43) k! Xk

∞ k 1 −c (c) Dλγ0,Id′,c(τ)= e . (5.45) Γ(0) T0 k! Z Xk=0 On the other hand, (c)k e−c c c e−c = ··· dτ ,...,dτ . (5.46) k! k! 1 k Z0 Z0 Evidently, the Poisson distribution is closely related to the restricted gamma integrator which motivates the following deﬁnition:

27The failure of the Faddeev-Popov method in some instances is sometimes attributed to the failure of quantization and constraints to commute. Our view is that functional integrals (should) perform both oper- ations simultaneously, and so a proper formulation must include constraints and provide tools to implement them. From our perspective, failure of Faddeev-Popov in certain cases reﬂects the wrong choice of delta functional.

35 Deﬁnition 5.5 Let T0 be the space of continuous pointed maps τ : (T+, ta) → (C+, 0) ′ endowed with a lower gamma family of integrators. Let α = n ∈ N and hβ , τoi = c with c ∈ C+. The Poisson integrator family DΛπn,β′,c(τ) is characterized by

′ ihτ ′,τi−hβ′,τi n Θn,β′ (τ, τ )= e τ ′ ′ ′ −n Zn,β′,c(τ )= P (n, c)Det (β − iτ ) . (5.47) h i The Poisson family is deﬁned in terms of the primitive integrator Dλτ by

−hβ′,τi n Dλπn,β′ (τ) := e τ Dλτ . (5.48) Note the normalization of the ﬁducial Poisson integrator

Dλπ0,Id′,c(τ) := 1 , (5.49) ZT0 and the rest of the family

Dλπn,Id′,c(τ)= P (n, c) . (5.50) ZT0 For quantum evolution, one can restrict τ : (T+, ta) → (iR, 0) and ℜ(c) = 0. In this restricted case, T0 becomes a Banach space over R, and it is useful to define the ‘shifted’ primitive integrator by −hβ′,τi n Dλπn,β′ (τ) := e (τ − τ0) Dλτ . (5.51) Use the shifted Poisson integrator to define; b ′ Definition 5.6 Suppose τ : (T+, ta) → (iR, 0). Then the Poisson integration of β with respect to some fiducial τ0 is defined by

′ −hβ′,τ i hβ i := D π ′ ′ (τ)= e 0 λ . (5.52) λ τ0 λ 0,β ,hβ ,τ0i ZT0 For example, if we take hβ′, τ i = − tb βb′ (t) dt, then 0 λ ta λ ∞ 1 tbR tb hβ′ i = β′ (t ) ··· β′ (t ) dt ,...,dt λ τ0 n! λ 1 λ n 1 n n=0 ta ta X Z Z ∞ tb tb tb ′ ′ ′ = βλ(t1) βλ(t2) ··· βλ(tn) dt1,...,dtn (5.53) n=0 ta t1 tn X Z Z Z where ta ≤ t1 < ··· < tn ≤ tb. This implies Lemma 5.1 The Poisson integration of β′ solves the ﬁrst-order evolution equation

∂ ′ ′ ′ hβλiτ0 = −βλ(tb)hβλiτ0 (5.54) ∂tb Of course, using the shifted Poisson integrator to obtain the exponential function seems rather contrived; but no more so than adding an external source to a Gaussian integral.

36 5.4 Matrix gamma integrators

The target manifold for gamma integrators can be readily generalized from C+ ⊂ C to n positive cones C+ ⊂ C . The same can be done for Gaussian integrators. This brings matrix functional integrals into the fold — complex Wishart for gamma and random matrix models for Gaussian. We’ll do it here for gamma integrators.

Definition 5.7 Let T 0 denote the space of continuous, matrix-valued pointed maps τ : (T+, ta) → (VH , 0) where T+ ⊆ R+ and VH is the vector space of rank (n, n) positive-definite ′ Hermitian matrices. T 0 is an abelian topological group. Let β be an invertible fixed element ′ ′ ′ in the dual group T 0 of characters HomC (T 0, C) with duality given by hβ , τ i = tr(β (τ )) ′ where β : T 0 → VH . Explicitly;

tb hβ′, τ i = tr(β′(τ (t))) dt (5.55) Zta

A (matrix) gamma family of integrators DΛγα,β′ (τ ) on T 0 is characterized by

′ ihτ ′,τ i−hβ′,τ i α Θα,β′ (τ , τ )= e det(τ ) ′ ′ ′ −α Zα,β′ (τ ) = Det (β − iτ ) (5.56) where α ∈ C, τ α is deﬁned point-wise by

τ α(t) := eα log τ (t) . (5.57)

Note that log τ (t) is real analytic here since τ (t) is Hermitian. As before, the functional determinant Det(β′ − iτ ′) is assumed to be well deﬁned. The gamma integrator family is deﬁned in terms of the primitive integrator Dλτ by

−hβ′,τ i α Dλγα,β′ (τ ) := e det(τ ) Dλτ (5.58) where Dλτ is characterized by

′ ′ ′ Θ0,Id ′ (τ , τ ) = exp{ihτ , τ i−hId , τ i} ′ Z0,Id ′ (τ )λ =Γn(0)λ , (5.59) where −tr(τ ) α Γn(α)λ := e det(τ ) Dλτ (5.60) ZT 0 and implicit in Γn(0)λ is a regularization. ∼ In applications, the archetypical ‘localizing’ epimorphism turns out to be λ : T 0 → C+ = VH , the positive cone in Cn(n+1)/2; yielding, in particular, the standard Wishart integral for the special case α = d/2 with d > n − 1. Analogous to the gamma integrator, impose a cut-oﬀ τ o, and interpret it as a bound on k 1/k the spectral radius ρ(τ (t)) ≤ kτ o(t) k for all t ∈ [ta, tb] and all k ∈ N. This leads to a lower gamma integrator;

37 Definition 5.8 Let T 0 denote the space of continuous, matrix-valued pointed maps τ : ′ ′ (T+, ta) → VH . Let β be an invertible fixed element in the dual group T 0 and fix a fiducial ′ ′ τ o ∈ T 0 such that hβ , τ oi = r ∈ R+. A lower gamma family of integrators DΛγα,β ,τ o (τ ) on T 0 is characterized by

′ ihτ ′,τ i−hβ′,τ i α Θα,β′ (τ , τ )= e det(τ )

′ ′ ′ ′ ′ −α Zα,β ,τ o (τ )= γn (α, hβ , τ oi)Det (β − iτ ) (5.61) ′ where γn (α, hβ , τ oi) is the lower incomplete gamma function associated with Γn(α) defined in (5.60), and again the functional determinant Det(β′ − iτ ′) is assumed to be well defined ′ by suitable choice of λ. The upper gamma family DΛΓα,β ,τ o (τ ) is defined similarly.

The next two examples use gamma and matrix gamma integrators to formulate functional integral representations of certain average prime and prime k-tuple counting functions. Al- though we focus on prime numbers, the approach can be adapted to physical models that count average prime and correlated prime cycles in the context of spectral determinants and dynamical zeta functions. We formulate the counting functions in the spirit of quantum mechanical expectation values in the sense that they represent a sum over all ‘paths’ with certain attributes. This approach leads to average single-prime counting functions that agree with their known number theory counterparts — albeit from a quite diﬀerent perspective. However, the average prime k-tuple counting functions, though in agreement asymptotically with the Hardy-Littlewood prime k-tuple conjecture, actually yield more accurate counts (see appendix A).

Example 5.2 Counting primes: If one is willing to compare the supposedly pseudo-random28 occurrence of prime numbers and their predictable averages to quantum evolution, then analogy with quantum systems suggests that counting prime powers can be formulated as some integral kernel associated with a gamma functional integral. That is, the very nature of the counting process of a random prime power event dictates a Poisson process which can be represented by a gamma functional integral. All we need do is determine the relevant, perhaps non-homogeneous, scaling factor that parametrizes the process. It may be useful to have a physical picture29 in mind: Consider a quantum system of two-state (integer/not-integer) ‘entities’ on the positive reals R+. The observables of interest are projections onto either of the two possible states. Observation at a random point via a projector gives integer or not integer. Fortunately, once a metric has been established one knows precisely where to observe the integers. Enumeration of the integer observations within an interval (0, x] then gives a correspondence between a subset of the natural numbers N+

28The distribution of prime numbers is not random since even numbers beyond 2 can’t be prime (although there is evidently pseudo-random behavior). However, the toy model to be presented suggests that prime powers are random variables — or at least can be modeled as such. To be precise, the physical model posits that counting prime powers is a random process following a constrained gamma distribution. 29This toy model describes the mathematical procedure of assigning ordinals to the set of positive integers, and it suggests that gamma functional integrals may be useful in studying the assignment of ordinals to sets more general than the positive integers.

38 and integer states in the lattice Z+ contained within the interval. This correspondence can be used to characterize/label an integer eigenstate located in R+ by its associated natural number — thus yielding a model of Z+ in terms of N+ as the cut-off x → ∞. We will show below that the projector onto integers follows the trivial gamma distribution, and enumeration of the integers is given by a certain trace of the associated propagator over integer states. Now consider a quantum system of two-state (prime-power/not prime-power) ‘entities’ localized on the lattice of positive integers Z+. Counting ‘prime-power events’ is postulated to be a constrained dynamical random process. As in the case of integers, we use the quantum model on R+ given a suitable projector. Unfortunately, in this case we have no metric to tell where the next prime power event occurs: Having localized onto some integer, we then must test whether its natural-number label has non-trivial divisors. This is where randomness enters the process.30 Counting prime powers in an interval corresponds to the expectation of a suitable non-trivial evolution operator, (i.e. an integral kernel which follows a non-trivial gamma distribution), generated by the projection onto primes. And the integral kernel can be represented as a constrained gamma functional integral. To see how to proceed, let’s first calculate the expected number of integers occurring up to some cut-off integer x ∈ R+ by defining a suitable α-trace applied to the simple case of an homogenous process. That is, we take β′ = −Id′ in the lower gamma integral and fix λ by restricting the domain of paths via the linear map L : T0 → R+. In this case, the functional integral can be explicitly evaluated and we get

Γ(1 − α) α N(x) := tr Dγ ′ x(τ(t)) := [(−1) γ(α, x)] dα α α,−Id , 2πi ZR+ ZC ∞ (−1)2n = γ(n, x) (n − 1)! n=1 X∞ = P (n, x)= x (5.62) n=1 X where the contour encircles the positive real axis and the (−1)α comes from the phase of the 31 determinant. This motivates and justiﬁes the deﬁnition of trα. Now postulate that the prime-power counting function is the expectation of a gamma process with unknown scale parameter due to the constraint associated with counting only primes. According to [31], the constrained functional integral that represents the expectation value can be written as a constrained functional that is integrable with respect to two marginal gamma integrators. And the functional integral that enforces the constraint must be a gamma integral because the conjugate prior of a gamma distribution with unknown scaling parameter

30Fine, but once the prime powers are located in any given interval how can we say they are randomly distributed? Well, as long as our hypothetical quantum system that models the interval is a closed system, the observed state eigenvalues and natural-number labels remain valid and the system is deterministic. But if an external agent were to act on the system, for example by some unknown re-assignment of ordinals or re-arrangement of the points on Z+, then we would no longer have a correspondence between N+ and Z+ and the location of prime powers in the interval would have to be re-established. 31 ′ ′ We could, of course, take β =+Id and adjust the deﬁnition of trα.

39 is again a gamma distribution. Accordingly, let us posit the average number of prime powers up to some cut-oﬀ integer x is given by

k x ′ x ′ Np ( )λ = trα+1 Dλγα,−Id , (τ) Dλγ1,ic (τ),∞(c) ZT0×C

= trα+1 Dλγα,−Id′,s(x)(τ) (5.63) ZT0 where c′(τ) represents the constraint (see [31]) and s(x) represents an unknown possibly non- homogenous scaling factor. The α-trace has been shifted by one because the counting should begin with the second event (since primes start with p =2). Note that the constraint imposes the non-homogeneous scaling factor on the cut-oﬀ. Under L : T0 → R+, this evaluates to x α x Npk ( ) = trα+1 [(−1) γ(α,s( ))] 1 π csc(π(α + 1)) = [(−1)αγ(α,s(x))] dα 2πi Γ(α + 1) ZC+1 ∞ (−1)2n+1 = γ(n, s(x)) n! n=1 X ∞ Γ(n) = − P (n, s(x)) (5.64) Γ(n + 1) n=1 X where the new contour begins at ∞ above the real axis, circles the point {1} counter-clockwise, and returns to ∞ below the real axis. Roughly speaking, this calculation simply sums the positive integers appropriately adjusted with a non-homogenous scaling factor and weighted by Γ(n)/Γ(n +1)=1/n. The series converges absolutely since n! |γ(n +1,s(x))| 1 lim = lim s(x)=0 . (5.65) n→∞ (n + 1)! |γ(n, s(x))| n→∞ (n + 1)

And notice that

∞ s(x+1) x x 1 −t n−1 −1 Npk ( + 1) − Npk ( )= − e t dt ∼ (5.66) n! x s(x) n=1 s( ) X Z is supposed to represent the average density of prime powers at x (in this section the symbol ∼ means asymptotically). Accordingly, a good and obvious choice for the scaling factor is s(x)= − log(x) yielding the Poisson functional integral representation

∞ 1 N k (x) = − D π ′ x (τ) . (5.67) p λ n λ n,−Id ,− log( ) n=1 T0 X Z Now |dγ(n, − log(x))/dx| dx = γ(n, − log(x)) together with the absolute convergence of γ(n, − log(x))/n! implies that (under L : T → R ) the sum (5.64) can be expressed as R 0 + x x x P Npk ( ) = li( ) − log(log( )). (5.68)

40 x x x x It is reasonable, therefore, to view Npk ( ) as an approximation to J( ) − w( ) where J( ) is Riemann’s counting function 1 Λ(n) J(x) := = , (5.69) k log(n) k x n≤x pX≤ X and w(x) is the weighted sum of prime power divisors of x 1 Λ(n) w(x) := = . (5.70) k log(n) k x x Xp | Xn | x x x This is justified since Npk ( )= J( ) − w( ) where Λ(n) J(x) := li(x) ≈ (5.71) log(n) n≤x X and Λ(n) w(x) := log(log(x)) ≈ . (5.72) x log(n) Xn | What if we had started with β′ = +Id′ instead? Repeating the calculation for this case yields ∞ ∞ n 1 (−1) Γ(n) 2 − Dπ ′ x (τ(t)) = − P (n, − log(x)) = li(x) − li(x ) . (5.73) n n,Id ,− log( ) Γ(n + 1) n=1 R+ n=1 X Z X Up to a minus sign this is k k 2 p p nΛ(n) σ k (x) = li(x ) − li(x) ≈ − = . (5.74) p k k log(n) k x k x pX≤ Xp | Xn≤x n ∤ x So β′ = −Id′ counts the number while β′ = Id′ counts the sum of prime powers weighted by 1/k that don’t divide the cut-off x. It seems that Dγα,β′,− log(x)(τ) knows something about the distribution of prime powers. Let’s put α → α +1 in the integrator. For β′ = −Id′ we get a Poisson functional integral representation of an average Chebyshev function for prime powers up to cut-off x given by ∞ ∞

x ′ x Chpk ( )= Dπn+1,−Id ,− log(x)(τ(t)) = P (n +1, − log( )) n=1 R+ n=1 X Z X = ψ(x) − dΛ(x) = x − log(x) 1 1 ≈ log(pk) − log(pk) k k k x k x pX≤ Xp | = Λ(n) . (5.75) Xn≤x n ∤ x

41 For β′ = Id′ we get what can be called the ‘weighted entropy of prime powers’:

∞ ∞

x ′ n x Hpk ( )= Dπn+1,Id ,− log(x)(τ(t)) = (−1) P (n +1, − log( )) n=1 ZR+ n=1 X X 1 1 ≈ pk log(pk) − pk log(pk) k k k x k x pX≤ Xp | = nΛ(n) . (5.76) Xn≤x n ∤ x

x There is a clear pattern emerging, and it starts with the dominant term of Npk ( ) which is a sum of lower incomplete gamma functions that happens to equal li(x). In the spirit of the physical toy model, this average can be interpreted as an expectation value of a counting operator, say hx|J(x)|0i ≡ J(x) = li(x). When extended to the complex plane as J(xs) with s ∈ C and integrated along a suitable contour with respect to log(ζ(s)), the average converges x to Riemann’s countingb function J( ). Likewise for the other averages in the pattern. They can be combined into a compact expression:

where C ∈ C+ is some appropriate contour,br, i ∈ Z+ and log(ζ(s)) with s ∈ C\{1} is the log-zeta function. Evidently exact counting operators are sums of operators dictated by the pole and zeros of zeta. The pole gives the average while the zeros conspire to “morph” the smooth average into a step function. x Λ(n) For example, r, i =0 gives the exact Riemann prime counting function J( )= n≤x log(n) , and r =0, i =1 gives the exact second Chebyshev function ψ(x)= Λ(n). Meanwhile, n≤x P for r =1, i =0 we get the exact weighted sum of prime powers P pk nΛ(n) 1 ∞ = = li(x2) − li(x1+ρ) − C − − li(x−2k+1) , (5.78) k log(n) 2 k x n≤x ρ pX≤ X X Xk=0 and for r =1, i =1 we get the exact weighted entropy of prime powers

1 1 x1+ρ pk log(pk)= nΛ(n)= x2 − − C + arctan(x−1) (5.79) k 2 1+ ρ k x n≤x ρ pX≤ X X

42 where ρ represents a zeta-zero and C = 12 log(A)−1 being A the Glaisher-Kinkelin constant. If we deﬁne ∞ µ(n)Λ(n) log (z(s)) := − , ℜ(s) > 1 , (5.80) log(n)ns n=1 X which (up to a minus sign) is the prime zeta function in disguise, we get explicit integrals on the other side of the Moebius inversion representing exact prime (as opposed to prime-power) counting functions:

′ π(x; r, i) := hx| J (i) (xs+r) d log(z(s)) |0i c > r . (5.81) ZC These explicit integrals for exact countingb functions are already famously known in number theory — although they are expressed and interpreted diﬀerently than (5.77) and (5.81). Our purpose here is to highlight a perspective that will be proﬁtable for counting prime k-tuples explicitly.

Example 5.3 Counting prime k-tuples: The functional integral approach to counting single primes provides four clues: average counting follows gamma statistics, the random variables are prime powers, the maps τ are logarithmic, and expectations do not include ordinals that divide the cut-off x. In addition, the functional integral framework requires Gλ to be a locally compact topological group with a Banach target space. These clues suggest that the cut-off exerts a strong influence on the counting process and perhaps we were too hasty to construct the toy model on R+; after all, counting always necessarily includes a finite cut-off as is true of any observation of a quantum system. In consequence, we will replace the space T0 in the definition of the gamma integrator with the 32 x x space of continuous pointed maps τ : (T+, ta) → (B , 2) where B is the Banach space with module (Z/xZ)× (the group of units of (Z/xZ)) over the field Z/pZ (for some prime x x × x ω( ) ri p) and suppose now that L : T0 → (Z/ Z) . Since = i pi by prime decomposition, the prime-power k-tuple quantum states in our model correspond to elements in the direct product group Q ω(x) x × ∼ ri × (Z/ Z) = (Z/pi Z) (5.82) i Y with ω(x) the number of distinct prime factors. We maintain the postulate of a gamma distribution for prime powers, and rerun the single prime counting example to get (once again) ∞ 1 N k (x) = − D π ′ x (τ) . (5.83) p λ n λ n,−Id ,− log( ) n=1 T0 X Z x x x x × However, here we can’t say Npk ( ) = li( ) − log(log( )) since our group (Z/ Z) is finite. x x Nevertheless, we learned the expectation Npk ( ) includes no divisors of so numerically at

32 Note that we are now taking τ(ta) = 2. This avoids singularities, and we don’t start counting before 2 for primes anyway.

43 x x x x × least we can expect Npk ( ) ≈ li( ) − log(log( )) and (under the map L : T0 → (Z/ Z) ) we have ∞ 1 x ′ Npk ( ) = − Dπn,−Id ,− log(x)(τ(t)) n x × n=1 (Z/ Z) X Z ∞ 1 = − Dπn,−Id′,− log(x)(τ(t)) x × n Z(Z/ Z) n=1 X1 − x = − x log(x) x∈(Z/xZ)× xX x − 1 ≈ C dx (5.84) x log(x) Z2 × for some normalization constant C that expresses the ratio of measures on (Z/xZ) v.s. R+. x (Interchanging the sum and integral is justified since Npk ( ) < ∞.) With these preliminaries, we build a model to count admissible prime k-tuples (to be defined shortly). Let Hk := {0,...,hk} be a set with |Hk| = k and distinct hi ∈ Z+ such × that Hk does not cover the residue classes associated with (Z/pZ) for any prime p. Let x x k k k x ( ,..., + hk) ∈ N+ ⊂ Z+ where N+ is the pair-wise coprime k-lattice and ≥ 2. Define 1/k the geometric mean of (x,..., x + hk) by x(k) := [x(x + h2) ... (x + hk)] . An admissible k prime k-tuple is a point (p,...,p + hk) ∈ N+ for any prime p. Note that, since we are on xk ∼ k x the pair-wise coprime lattice, Z/ (k)Z = i=1 Z/( + hi)Z, and so counting prime k-tuples will require the matrix gamma integrator. xk xk Q (k) (k) Let T 0 be the space of continuous pointed maps τ :(T+, ta) → (B ,Id) where B is xk × the Banach space with module (Z/ (k)Z) over the field Z/pZ. We follow the single-prime ′ ′ ′ case and choose β = −Id with cut-off hβ , τ oi = x(k). The points (2, 2+ h2,..., 2+ hk) and x x x k ( , +h2,..., +hk) define a ray (of discrete points) Rk := (r, r +h1,...,r +hk) ⊂ N+ along xk × ′ which we will count prime k-tuples. We take L : T 0 → (Z/ (k)Z) such that β (τ (t)) = Rk, xk ∼ k x and we generalize the α-trace to Z/ (k)Z = i=1 Z/( + hi)Z in the obvious way. With the hypothesis that prime powers in Nk are independent random events, the counting Q+ xk × ∼ k x × factors into a product since (Z/ (k)Z) = i=1(Z/( + hi)Z) . Hence, the expected number of admissible prime k-tuples along R up to cut-off x is (under L) k Q

x ′ N k ( ) = trα+1 δR Dγα,−Id ,− log(x )(τ (t)) p(k) k (k) (Z/xk Z)× "Z (k) # k

′ = trα+1 δr+hi Dγα,−Id ,− log(x+hi)(τ(t)) x × "i=1 Z(Z/ Z) # Y k k α α−1 = trα+1 (−1) δr+hi (−1) (− log(x + hi)) dx . (5.85) x × " (Z/ Z) i=1 # Z Y xk ∼ k x The second line uses Z/ (k)Z = i=1 Z/( + hi)Z. Since the prime powers are assumed independent, an independent α-trace should be taken over each component of τ (t) separately. Q

44 This yields ∞ k (−1)2ni x k ni−1 N k ( ) = (−1) δr+hi (− log(x + hi)) dx p(k) (Z/xZ)× ni! Z (n1,...,nk)=1 i=1 x X Y k Q(x, Hk) = (−1) k x log(k)(x) x∈Rk (k) X x Q(x, H ) ≈ (−1)k C k dx (5.86) k xk log (x) Z2 (k) (k) where Ck is a measure normalization, log(k)(x) := log(x) log(x + h2) ··· log(x + hk) and Q(x, Hk) is a polynomial. For example, Q(x, H1)=(1−x), Q(x, H2)=(1−x)(1−(x+h2))) 33 and Q(x, H3)=(1 − x)(1 − (x + h2))(1 − (x + h3))). By analogy with the single-prime case, deﬁne x x k 1 k 1 J(k)(x) := (−1) ≈ (−1) Ck dx . (5.87) R log(k)(x) 2 log(k)(x) xX∈ k Z xk We don’t know how to explicitly sum counting functions on B (k) (for k > 1) so it is crucial to have the approximate R+ integral representation. The normalization Ck allows to k compare the measure on Rk ⊂ N+ with the measure on R+. We claim the normalization is given by (the Hardy-Littlewood singular series) ν (H ) 1 −k C = 1 − p k 1 − (5.88) k p p p Y where νp(Hk) is the number of distinct congruence classes mod p covered by the admissible set Hk. To see this, we appeal to a theorem by T´oth: Let k,m,u ≥ 1 and (u) Pk (m)= 1 (5.89)

1≤a1X,...,ak≤m (ai,aj )=1,i6=j

(ai,u)=1

be the number of k-tuples (a1,...,ak) on the pair-wise coprime lattice with 1 ≤ ai ≤ m and (ai,u)=1 for all i ∈{1,..., k}. Theorem 5.1 ([6]) For a ﬁxed k ≥ 1, we have uniformly for m, u ≥ 1, (u) k k−1 k−1 Pk (m)= Akfk(u)m + O θ(u)m log (m) (5.90) where θ(u) is the number of squarefree divisors of u and 1 k−1 k − 1 Ak = 1 − ′ 1+ ′ ′ p p Yp k fk(u) = 1 − ′ . ′ p + k − 1 Yp |u 33 k k xk Apparently Q(x, Hk)/x(k) somehow excludes, on average, the divisors n(k)| (k).

45 To apply the theorem along Rk, restrict to u = p for some prime p and choose m> x + hk. Then the density of points that are coprime to a given prime p (or prime power) is

(p) (p) k k−1 Dk (m) := Pk (m)/m = Akfk(p)+ O log (m)/m . (5.91)

Of course m is automatically coprime to all primes p > m. Now, in our case the k-tuples (p) (a1,...,ak) are restricted to the admissible ray Rk so Dk (m) must be divided by the density of points with GCD(x + hi,p)=1 for each hi and each p. This density is given by 1 p (p − νp(Hk)). Hence, the total density of prime powers along an admissible ray Rk is

D(p)(m) 1 k ν (H ) −1 lim k = 1 − 1 − p k . (5.92) m→∞ νp(Hk) p p p≤m 1 − p Y p Y With the prime-power states properly normalized on R+, we can now count them and assign ordinals.34 Appendix A compares average counts of some prime 2-tuples predicted by (5.86) with the Hardy-Littlewood conjecture. We are ﬁnally in a position to make use of the interpretation aﬀorded by (5.77) and xk (5.81): As already mentioned, we don’t know how to sum counting functions on B (k) — except for k=1 since then we can use the Riemann zeta function which has the nice interpretation as a kind of spectral measure on (Z/xZ)×. Evidently, explicit counting for k > 1 xk × requires a generalized zeta function to play the part of a spectral measure on (Z/ (k)Z) . Let’s try to discover one. First, note that an exact counting function for prime k-tuples given Hk is

Λ(n) Λ(n + h ) Λ (n) π (x)=(−1)k µ(n) ··· µ(n + h ) ··· k =(−1)k µ (n) (k) . (k) k log(n) log(n + h ) (k) log (n) n≤x k n≤x (k) X X (5.93) This suggests to deﬁne a generalized zeta function implicitly by

∞ Λ (n) log ζ (s) := (k) ℜ(s) > 1 . (5.94) (k) log (n) ns n=1 (k) (k) X Then ∞ ′ Λ (n) log(k−1) ζ (s) =(−1)k−1 (k) logk−1(n ) . (5.95) (k) log (n)ns (k) n=1 (k) (k) X Now deﬁne Λ(k)(n) k−1 φ(k)(x) := log (n(k)) . (5.96) log(k)(n) Xn≤x n ∤ x

34 Note that C1 = 1 which explains the success of our naive model for counting single primes on R+.

46 Its average is x x Q(x, H ) Q(x, H ) φ (x)= k logk−1(x ) ≈ C k logk−1(x ) dx (5.97) (k) xk log (x) (k) k xk log (x) (k) x∈R (k) (k) 2 (k) (k) Xk Z k xk with dominant term (which ostensibly includes divisors n(k)| (k)) x x 1 k−1 1 k−1 ϕ(k)(x) := log (x(k)) ≈ Ck log (x(k)) dx R log(k)(x) 2 log(k)(x) xX∈ k Z =: Ck Li(k)(x) . (5.98)

Note that asymptotically ϕ(k)(x) ∼ Ck li(x) so this situation is very much like the single prime case. Together with the interpretation of (5.77) and (5.81), this motivates s s s s k Proposition 5.1 Deﬁne log(k)(x ) := log(x ) log((x + h1) ) ... log((x + hk) )= s log(k)(x). Put x(k) := x(k) +ǫ with x ∈ N+ and 0 <ǫ< 1. Let σa be the abscissa of absolute convergence k−1 ∞ λ(k)(n) log (n(k)) of n=1 ns . Then, for c > σa, f (k) P (k−1) x c+iT (−1) log(k)( ) s (k−1)′ ϕ(k)(x) = lim lim Li(k)(x(k) ) d log (ζ(k)(s)) , c>σa ǫ→0 T →∞ 2πi logk(x ) (k) Zc−iT Λ (n) = (k) logk−1(n ) . f (5.99) log (n) (k) n≤x (k) X The proof is presented in appendix B.

With hx|J(k)(x)|0i ≡ J(k)(x) := Li(k)(x), the heuristic (5.77) from the single-prime case becomes

x ′ b log(k)( ) (i) s+r J(k)(x; r, i)) = hx| J (x ) d log(ζ(k)(s)) |0i c>k + r . (5.100) logk(x ) (k) (k) ZC x To count weighted prime-power k-tuples, useb J(k)( ;0, 0) which clearly gives Λ (n) J (x)= (k) (5.101) (k) log (n) n≤x (k) X after integration by parts. Unfortunately, we don’t have an explicit expression for ζ(k)(s) so we can’t use the integral to do exact counting via residues — in contrast to the single-prime case. However, we do have a guess: Conjecture 5.1

1 −k −k ζ (s) =? =? 1 − p−s =? 1 − ν (H ) p−s 1 − p−s . (5.102) (k) ns (k) p k n∈R (k) p∈R p Xk Yk Y If it turns out that ζ(k)(s) can be well deﬁned according to this conjecture or otherwise, then knowledge of its zeros and poles would allow evaluation of the explicit integral (5.99) and potentially have important implications in number theory. In particular, it might allow to extend the Prime Number Theorem to prime k-tuples.

47 6 Conclusion

A common property seemingly possessed by useful functional integrals is that they eventually reduce to a set of bona fide integrals due to some form of localization in the domain of integration. In other words, an induced localization in the infinite-dimensional functional- integral domain leads to a quantifiable/measurable object or objects that can be associated with the original functional integral. Promeasures in particular exemplify this observation. It is natural therefore to view integral operators on an infinite-dimensional function space in much the same way as differential operators: For a given function space, they represent a family that can be formally manipulated but whose individual members are well defined. Such a view suggests defining functional integrals on topological groups in terms of families of well-defined Haar measures on locally-compact topological groups. Further, the rigorous integration theory on locally compact groups allows to consider Banach-valued integrable functions. So, we construct a space of Banach-valued functionals on a topological group that inherits any algebraic structure possessed by the target Banach space. In this approach, functional integrals serve a dual purpose: i) they are a tool to formally manipulate a Banach algebra of integrable functionals on a topological group; and ii) given a ‘topological localization’, they become bona fide measure-theoretic objects. The second point in particular offers an interpretation of the measurement process in quantum theory (recall Remark 2.1). Of course the nature of the Banach algebra of functionals F(G) and its image under integral operators depend on the underlying topological group and its associated family of Haar measures. This immediately leads one to contemplate functional integrals beyond the familiar Gaussian type. Being obvious infinite-dimensional counterparts to finite-dimensional integrals, one can develop functional integrators for useful finite-dimensional integrals. In this paper we detailed skew-Gaussian, Liouville, gamma, and Poisson functional integrals as specific examples and gave some particular applications of each relevant to physics and mathematics. Significantly, the skew-Gaussian and Liouville integrators can generate exterior algebras of functionals. Their construction exposes inherent BRST/SUSY in the context of sesquilinear forms on complex topological groups, but it points to an unorthodox physical interpretation reminiscent of wave/particle duality. One aspect of these integrators that we ignored (with the exception of translation invariance of the Gaussian integrator) is their invariance under relevant symmetries of the domains of integration. This is of particular importance and will be explored further.35 Also, for the most part we merely offered functional integral representations of some pertinent objects: The more difficult task of solving non-trivial integrals remains, but hopefully the framework proposed here, augmented with a study of symmetry properties of integrators, will enable reliable manipulations and perhaps suggest new solution strategies. Although our pragmatic definition of functional integrals could be fairly characterized as mere bookkeeping, it does provide an uncluttered landscape by which to envision and construct new functional integral tools that should be of use in mathematical physics. More importantly, it provides a bridge between concrete B and abstract F(G) Banach ∗-algebras.

35 Having skew-Gaussian and Liouville integrators in hand, the strategy is to consider topological homo- topy groups and deﬁne forms, invariant forms, etc. through their ﬁnite-dimensional counterparts based on form/chain duality.

48 As such, it combines operator and functional integral methods under one roof. In particular one can characterize a C∗-algebra representing a quantum system through a family of integrators taking values in LB(H). In this regard, beyond what has been presented here, we have in mind generalizing gamma integrators in the context of functional Mellin transforms. Functional Mellin transforms are expected to be useful for both constructing and representing (quantum) C∗-algebras and will be developed in subsequent work.

A Comparing gamma and Hardy-Littlewood

Let’s compare the counts of prime doubles between (5.86) and Hardy-Littlewood. First, note that 1 1 h h2 ∼ − 2i + O 2i . (A.1) log(x) log(x + h ) log(x)2 x log(x)3 x2 log(x)3 2i So, for given cut-off x and off-set h2i, we don’t expect much difference when x ≫ h2i. But for the other way around h2i > x, there may be. The table below contains the exact number of prime doubles (p,p + h2i) for several values of h2i and x (Mathematica 9.0).

1 2 3 4 5 6 7 8 x h2i 10 10 10 10 10 10 10 10 102 11 9 5 5 3 2 2 3 103 51 49 37 34 23 20 17 16 104 270 260 253 224 186 163 142 112 105 1624 1615 1631 1556 1431 1219 1050 918 106 10934 10906 10993 10798 10629 9766 8592 7539 107 78211 78248 78265 77850 77680 76212 71247 63352 108 586811 586908 586516 586587 585883 583976 573938 540323

Table 1: Exact number of prime doubles for the indicated cut-oﬀ x and oﬀ-set h2i.

1 For the Hardy-Littlewood estimate, we have for h2i = 10

1 2 1 2 2 1 − 2 1 − 3 1 − 5 1 − 7 1 − 7 C(2) = · · · · ≈ 1.76 . (A.2) 1 2 1 2 1 2 1 2 1 2 1 − 1 − 1 − 1 − 7

x 1 C dx , (A.3) (2) log(x)2 Z2 the percentage deviation of the Hardy-Littlewood estimate relative to the exact count is tabulated below:

49 1 2 3 4 5 6 7 8 x h2i 10 10 10 10 10 10 10 10 102 64 101 261 261 502 802 802 802 103 19.7 24.6 65 79.6 165 205 259 282 104 5.8 9.8 12.9 27.5 53.5 75.2 101 155 105 2.5 3.1 2.1 7 16.3 36.5 58.5 81.3 106 .6 .8 .2 1.8 3.4 12.6 28 45.8 107 .1 .1 .07 .6 .8 2.8 10 23.6 108 .03 .02 .08 .07 .2 .5 2.3 8.6

Table 2: Percentage deviation between exact and Hardy-Littlewood estimates of prime doubles for the indicated cut-oﬀ x and oﬀ-set h2i.

Since the Hardy-Littlewood estimate is asymptotic, it is not surprising that percentages are fairly high for smaller cut-oﬀs. But notice the general trend of increasing deviation across rows as the ratio h2i/x increases. The estimate of prime doubles from (5.86) requires the Moebius inverse

x1/m ∞ µ(m) (1 − x)(1 − (x + h ) C 2i dx . (A.4) (2) m x(x + h ) log(x) log(x + h ) m=1 2 2i 2i X Z The percentage deviation of the gamma conjecture estimate from the exact count is:

1 2 3 4 5 6 7 8 x h2i 10 10 10 10 10 10 10 10 102 .2 3.7 24.8 5.8 25.7 57.2 34.7 21.4 103 .5 1.9 9.5 6.4 11.2 6.6 7.5 .05 104 .5 3.2 1.2 .6 .4 4.2 5.7 4.6 105 1.1 1.5 .3 .9 .7 .1 .3 .2 106 .2 .4 .5 .7 .5 .2 .5 .5 107 .04 .007 .04 .4 .1 .2 .4 .01 108 .007 .01 .06 .03 .07 .01 .2 .1

Table 3: Percentage deviation between exact and gamma conjecture estimates of prime doubles for the indicated cut-off x and off-set h2i. (The sum over m converges rapidly so only the first 50 terms were used.)

At least in these parameter ranges the gamma estimates are superior, and there is no reason not to expect similar comparisons throughout parameter space and for all prime k-tuples.

B Proof of Proposition 5.1

The proof mimics mostly standard arguments. To begin, integrate (5.99) by parts. The boundary terms won’t contribute because:

50 (k−1)′ 1) A comparison test between the series representations of log (ζ(k)(s)) and log(ζ(s)) (k−1)′ yields a ﬁnite σa. So, for s = c + it with c ∈ R, we have limt→∞ | log (ζ(k)(c + it))| < ∞ provided c> k. To see this, observe that

k−1 m k−1 ′ log (p ) log (p ) | log(k−1) (ζ (c + it))| ≤ (k) = (k) (k) mk pms mpmc pm (k) pm (k) X X k−1 1 log (p(k)) 1 = < m(c−(k−1)) m(k−1) m (c−(k−1)) pm mp p pm m (p ) X (k) (k) X (k) 1 < = | log(ζ((c − (k − 1)) + it))| . (B.1) m (c−(k−1)) pm m (p ) X k−1 2) Next, the inequality log (x)/ log(k)(x) ≤ 1/ log(x) implies Li(k)(x(k)) ≤ li(x(k)). s Hence, limt→∞ |Li(k)(x(k))| = 0 because

s (c+it) lim Li(k)(x(k)) ≤ lim li(x ) t→∞ e t→∞ (k) (c +it) x(k) 1 e = lim e 1+ O t→∞ (c + it) log(x ) (c + it) log(x ) (k) (k) c x(k) 1 1 ≤ lim 1+ O =0 . log(x ) t→∞ (c + it) (c + it) (k)

(B.2)

We will need the truncating integral Lemma B.1 xc c+iT s (k) x 1+ O x(k) > 1 1 (k) T log(x(k)) ds = c . (B.3) x(k) 2πi c−iT s  O 0 < x(k) < 1 Z  T log(x(k)) proof : For x(k) > 1 integrate over a rectangle with left edge (L − iT, L + iT ) such that 0

The top and bottom contribute

c±iT xs 0 −xc−r (k) ds ≤ (k) dr s |(c − r) ± iT | Z−∞±iT Z−∞ 0 −r −x(k) = xc dr (k) |(c − r) ± iT | Z−∞ 0 −x−r < xc (k) dr (k) T Z−∞ xc = (k) . (B.5) T log(x(k))

51 Lastly, the pole at s = 0 contributes Res = 1. For x(k) < 1, integrate over the right edge (R − iT, R + iT ) with c < R. Then

The top and bottom contribute the same order as for x(k) > 1, and the lemma is established. ⊟

Finally, making use of the truncating integral (for c > σa) in the integration by parts yields

(k−1) x c+iT k x x s (−1) log(k)( ) (k−1)′ log ( (k)) (k) lim lim log (ζ(k)(s)) ds ǫ→0 T →∞ 2πi logk(x ) log (x) s (k) Zc−iT (k) 1 c+iTf∞ fλ (n) log(n ) x s = lim lim (k) (k) (k) ds ǫ→0 T →∞ 2πi e ns s Zc−iT n=1 (k) X xg s f(k) ∞ c+iT s Λ(k)(n) 1 n(k) = lim lim log(n(k)) ds ǫ→0 T →∞ log (n) 2πi s n=1 (k) c−iT X Z Λ(k)(n) = lim log(n(k)) ǫ→0 ex log(k)(n) nX≤⌊ ⌋ Λ (n) = (k) log(n ) . (B.7) log (n) (k) n≤x (k) X The third equality follows from the lemma, and the ﬁnal line follows from the deﬁnition x(k) := x(k) + ǫ. Justifying the interchange of the sum and integral is straightforward, and interchange of the T -limit and sum is allowed because the summand contains O(n−c) with c>f 1.

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