A Variance Inequality for Meromorphic Measurement Functions under Exterior Probability
aa r X i v : . [ m a t h . G M ] J u l A Variance Inequality for Meromorphic Functions underExterior Probability
Swagatam SenJuly 1, 2020
Abstract
The problem of measuring an unbounded system attribute near a singularity has been dis-cussed. Lenses have been introduced as formal objects to study increasingly precise measure-ments around the singularity and a specific family of lenses called Exterior probabilities havebeen investigated. It has been shown that under such probabilities, measurement variance of ameasurable function around a 1st order pole on a complex manifold, consists of two separableparts - one that decreases with diminishing scale of the lenses, and the other that increases.It has been discussed how this framework can lend mathematical support to ideas of non-deterministic uncertainty prevalent at a quantum scale. In fact, the aforementioned variancedecomposition allows for a minimum possible variance for such a system irrespective of howclose the measurements are. This inequality is structurally similar to Heisenberg uncertaintyrelationship if one considers energy/momentum to be a meromorphic function of a complexspacetime.
Emergence of discrete outcome choices upon measurement of a continuously evolving intrinsicsystem has been a longstanding topic of interest. It is particularly of interest in the context ofquantum uncertainty which appears to be fundamental to reality and is most often characterisedby the continuous wave function. However, upon measurement such a continuous system col-lapses to discrete outcome choices.There has been substantial amount of work on developing a mathematical foundation forsuch fundamental uncertainties and evolution/collapse of continuous systems onto discrete ‘purestates’. Almost all of it treats measurements as operators on a Hilbert space. Born rule, put for-ward by Max Born in 1926 [1], proposed an interpretation of the wave function wherein the squareof its amplitude represents the probability of measurement outcome in a quantum experiment.This became a core part of famous Copenhagen Interpretation of Quantum Mechanics. While ithas been deemed the most satisfactory and certainly most popular interpretation for a long time,it poses a serious mathematical challenge as it remains at crossroads with classical framework ofProbability Measures. 1ne of the most significant mathematical groundwork towards addressing this challenge wasconducted by Gleason [2]. Subsequent generalisations led to the formation of Quantum proba-bility via non-commutative *-algebras[3][4][5]. However, the approach merely calls to focus howprobabilities work di ff erently at quantum scale, but doesn’t reconcile the classical and quantumprobabilities under a general framework that applies at all scales. In that sense, it harbours muchof the same concerns around non-emergent nature of quantum mechanics in general. Some of themore recent works in the area[6][7] focuses on measurement as an ‘averaging’ operation, albeit inthe operator space.In contrast to that, our primary concern in this paper is to investigate the possibility of aclassical probability measure reproducing some of the known quantum e ff ects. Also the approachcan be described as a geometric one as opposed to the standard Operator theoretic approach.Motivationally it can be compared with the work of Fuch [8] on Quantum Bayesianism.Particularly within the scope of this paper, we would focus on the nature of the underlying un-certainty as described by Heisenberg Uncertainty principle, and how such a relation can emergefrom a classical probability measure. Heisenberg Uncertainty principle, despite multiple di ff er-ent representations, fundamentally is a probabilistic statement around the limits on accuracy ofmeasurement of certain attributes at a quantum scale[9]. For example, if we allow ∆ x and ∆ µ tobe the standard error in measurement of position and momentum of a point in spacetime, then ∆ x ∆ µ ≥ ~ const A similar relation exists between other conjugate pairs e.g. time and Energy etc.To rigorously define these relationships, we would need to introduce the concept of ‘lenses’.
Definition 1.1.
Let M be an arbitrary manifold, and L = { Ω λ , Σ λ , P λ } λ ≥ be an indexed familyof probability spaces on M such that Ω λ ⊂ Ω λ ′ i ff λ < λ ′ and T λ Ω λ = { p } for some p ∈ M . L would be called a system of lenses around p and each of the underlying probability spaces wouldbe called a lens. p would be called the focus of the system of lenses.Now a conjugate pair of attributes can be represented by f : M C n where M is the manifoldrepresenting the domain for one of the pair of attributes, while f denotes the relationship betweenthe two. Example 1.1.
Let z = x + it be a coordinate on C with x and t representing the space and timecoordinates respectively. Let π : C C be the complex momentum such that π ( z ) = p ( x, t )+ iE ( x, t ).In this case ( z, π ) represents a conjugate pair of attribute with M = C Let’s choose lense L on M with scale index λ such that f is measurable ∀ λ . That wouldmean we can safely talk about expectation E λ and standard deviation σ λ of f measured throughindividual lenses. Definition 1.2.
Let L be a system of lenses on M with scale index λ and f : M C n is measur-able ∀ λ . Then f is called Detectable if E λ ( f ) is independent of scale λ and σ λ ( f ) < ∞ , ∀ λ .2ithout loss generality for any detectable f , we would assume the expectation to be 0 unlessmentioned otherwise.For such a Detectable conjugate pair, Heisenberg-type uncertainty relation can be summarisedas follows. λσ λ ( f ) ≥ const (1.1)If f is continuous in an open neighbourhood U of p then f | U is bounded. In that case σ λ ( f ) ≤ sup U f | U < ∞ would also be universally bounded and would clearly not satisfy desired relationshipsas in (1.1).That implies that for relationships of type (1.1) to hold around a point p , we must have a sin-gularity of f at p . However, it is immediately clear that if we allow P λ to be absolutely continuous(w.r.t Lebesgue measure on C n ), then f can’t be detectable as σ λ = ∞ , ∀ λ . Consequently our searchfor a system of lenses that allow detectable functions to satisfy Heisenberg type relations, has tolimit itself to certain types of probability measures that allows an open null set containing p, ∀ λ .We’d refer these probabilities as Exterior Probability.In the next few sections we would build a system of lenses on a complex manifold and studystandard deviation of a detectable function f under Exterior probabilities. We’ll start with a simple disc of arbitrary length on C , D λ = { w ∈ C | | w | ≤ λ } . Definition 2.1.
For a given open interval I ⊂ [ − π, π ], we can define a Slice as Γ λ ( I ) = { w ∈ C | | w | ≤ λ, arg ( w ) ∈ I } . Let γ λ ( I ) = { w ∈ C | | w | = λ, arg ( w ) ∈ I } be the corresponding arc.It’s easy to check that both Slices and Arcs can be seen as distributive operators on the semi-ring of intervals. Remark. Γ λ ( I ) ∩ Γ λ ( I ) = Γ λ ( I ∩ I ) , γ λ ( I ) ∩ γ λ ( I ) = γ λ ( I ∩ I )2. Γ λ ( ∞ S k =1 I k ) = n S k =1 Γ λ ( I k ) , γ λ ( ∞ S k =1 I k ) = n S k =1 γ λ ( I k )3. Γ λ ( ∞ F k =1 I k ) = n F k =1 Γ λ ( I k ) , γ λ ( ∞ F k =1 I k ) = n F k =1 γ λ ( I k )Space of all Slices of D λ renders a semi-ring structure. Definition 2.2.
Let E λ = { Γ λ ( I ) | ∀ interval I ⊂ [ − π, π ] } be the collection of Slices on D λ . Proposition 2.1. E λ is a semi-ring, ∀ λ Proof.
Proof follows trivially from the fact that the space of all intervals I , forms a semi-ring on[ − π, π ].Let Γ λ ( I ) and Γ λ ( I ) be two Slices. It’s easy to see that, Γ λ ( I ) ∩ Γ λ ( I ) = Γ λ ( I ∩ I ) ∈ E λ . Also, Γ λ ( I ) \ Γ λ ( I ) = Γ λ ( I \ I ) = Γ λ ( S k C k ) = S k Γ λ ( C k ), where { C k } k are a collection of intervals.3f course then we can extend it to the σ -algebra it generates. Definition 2.3.
Let E λ = σ ( E λ ) be the σ -algebra generated by E λ .Now we can start to build the measure, first on the semi-ring. Definition 2.4.
Let Γ λ ( I ) ∈ E λ . Then µ λ : E λ C can be defined as µ λ ( Γ λ ( I )) = πi H γ λ ( I ) 1 w Lemma 2.2. µ λ is σ -additive on E λ Proof.
Let Γ = ∞ F k =1 Γ λ ( I k ) ∈ E λ . That means ∞ F k =1 I k is an interval. That would allow us to write µ λ as, µ λ ( Γ ) = µ λ ( ∞ G k =1 Γ λ ( I k )) = µ λ ( Γ λ ( ∞ G k =1 I k ))= 12 πi I γ λ ( ∞ F k =1 I k ) w = 12 πi I ∞ F k =1 ( γ λ ( I k )) w = ∞ X k =1 πi I γ λ ( I k ) w = ∞ X k =1 µ λ ( Γ λ ( I k )) Corollary. µ λ can be uniquely extended as a complex probability on E λ . Let ( E nλ , µ nλ ) denote the product probability space on D nλ where µ nλ ( (cid:16) Γ k ) = Q k µ λ ( Γ k ) for Γ k ∈ E λ , ∀ k . We’d refer to µ nλ as the Exterior Probability on the poly-disc D nλ . We’d define a Unit Lens asthe filtration of probability spaces L = { ( D nλ , E nλ , µ nλ ) } λ ↓ . Definition 2.5.
A Convergent Lens on C n around 0, is a filtration of probability spaces { D nλ , E nλ , τ λ ) } λ ↓ such that τ λ is absolutely continuous with respect to µ nλ , ∀ λ .For the rest of this paper we’d focus on Unit Lenses. However, similar results can be recoveredfor a more general convergent lenses. Let F be the vector space of all meromorphic function f : C n C k which has a potential pole oforder at most 1 at 0 and is µ nλ -measurable over D nλ . Remark. µ nλ induces an Expectation E ( · ) and an inner product h· , ·i on F as• E ( f ) = R D λ f dµ nλ = πi ) n H D λ f ( w ) Q α w α , 4 h f , g i = R D λ f ∗ gdµ nλ = πi ) n H D λ f ( w ) ∗ g ( w ) Q α w α Remark.
We can write f as a sum of its core, principal and analytical components.• f α = πi ) n H D λ f α ( w ) Q µ =1 w µ is the core of f .• f αP = P nβ =1 η αβ z β is the principal component of f where η αβ = πi ) n H D λ f α ( w ) Q µ , β w µ is the matrix ofresidues.• f A = f − f − f P is the Analytic component of f and can be expressed as a power series withina radius of convergence R i.e. f αA = πi ) n P nβ =1 [ H D λ f α ( w ) w β Q µ w µ ] z β + P α ( z ) = P nβ =1 D αβ z β + P α ( z ), ∀ z such that | z β | < R ∀ β where D αβ = ∂f α ∂w β | w =0 and P α is a power series of degree at least 2 Remark.
For f ∈ F , let η be the residue matrix and f be the core. Then1. E ( f ) = f is independent of λ h z, f i = λ h z , f i = T r ( η )3. h z, f i = λ h z , f i = T r ( D )Additionally we can characterise the higher order terms in f A Proposition 3.1.
Let f ∈ F and let f A be the analytic component of f . Also let f αA ( w ) = D αβ w β + P α ( w ) , ∀ α where P α is a power series of degree at least 2. Then the following statements are true1. H D λ P α ( w ) Q γ w γ = H D λ P α ( w ) Q γ w γ = 0 ∀ α. H D λ P α ( w ) Q γ , δ w γ = H D λ P α ( w ) Q γ , δ w γ = 0 ∀ α, δ. H D λ P α ( w ) w δ Q γ , δ w γ = H D λ P α ( w ) w δ Q γ w γ = 0 ∀ α, δ. H D λ ( P α ( w )) P α ( w ) Q γ w γ = P i,j H D λ ( P αi ( w )) P αj ( w ) Q γ w γ = 0 ∀ α Proof.
Let D ( β ) λ = {| w | = λ } ⊗ ... ⊗ {| w β − | = λ } ⊗ {| w β +1 | = λ } ⊗ ... ⊗ {| w n | = λ } .If f αA ( w ) = D αβ w β + P α ( w ) ∀ α where P α is a power series of lowest degree at least 2, convergentwithin the radius R , that means P α ( w ) = P P αi ( w ) where ∀ α, ∀ i either of these two possibilities aretrue-• Case I - ∃ β i such that P αi ( w ) = w r i β i Q αi ( w ( β i ) ) with r i ≥ Q αi is a convergent power serieson w ( β i ) = ( w , ..., w β i − , w β i +1 , ..., w n ). 5 Case II - ∃ β i | < β i | such that P αi ( w ) = w β i | w β i | Q αi ( w ( β i | ,β i | ) ) where Q αi is a convergent powerseries on w ( β i | ,β i | ) = ( w , ..., w β i − , w β i +1 , ..., w β i | − , w β i | +1 , ..., w n ).1. (a) In that case, I D nλ P α ( w ) Q γ w γ = 0because P α is analytic at 0.(b) Also I D nλ P αi ( w ) Q γ w γ = λ r i I D ( βi ) λ Q αi ( w ( β i ) ) Q γ , β i w γ I | w βi | = λ w r i +1 β i = 0 ∀ α, i. Hence it follows that I D nλ P α ( w ) Q γ w γ = 0 ∀ α.
2. (a) Working with the power series components P αi , we know that, I D nλ P αi ( w ) Q γ , δ w γ = I D ( βi ) λ Q αi ( w ( β i ) ) Q γ , β i ,δ w γ I | w βi | = λ w r i − s i β i = 0where s i = I ( β i , δ ) and r i ≥ ff erent calculation. I D λ P αi ( w ) Q γ , δ w γ = λ r i I D λ Q αi ( w ( β i ) ) w r i + s i β i Q γ , β i ,δ w γ where s i = I ( β i , δ )Unless β i = δ and r i = 1, we could simply write this, I D λ P αi ( w ) Q γ , δ w γ = λ r i I D ( βi ) λ Q αi ( w ( β i ) ) Q γ , β i ,δ w γ I | w βi | = λ w r i + s i β i = 0as r i + s i > β i = δ and r i = 1, we could invoke Case II and assume without loss6f generality that β i | , δ . That would allow us to write, I D λ P αi ( w ) Q γ , δ w γ = λ I D ( βi | λ Q αi ( w ( β i | ,β i | ) ) w β i | Q γ , β i | ,δ w γ I | w βi | | = λ w β i | = 0This of course leads to I D λ P α ( w ) Q γ , δ w γ = X i I D λ P αi ( w ) Q γ , δ w γ = 0 ∀ α, δ (c) For the other identities, I D λ P αi ( w ) w δ Q γ w γ = I D ( βi ) λ Q αi ( w ( β i ) ) w s i δ Q γ , β i w γ I | w βi | = λ w r i + s i − β i = 0 ∀ α, δ. where s i = I ( β i , δ )That implies, I D λ P α ( w ) w δ Q γ w γ = X i I D λ P αi ( w ) w δ Q γ w γ = 0 ∀ α, δ (d) By a similar argument, I D λ P αi ( w ) w δ Q γ w γ = λ r i I D ( βi ) λ Q αi ( w ( β i ) ) w sδ Q γ , β i w γ I | w βi | = λ w r i − s i +1 β i = 0 ∀ α, δ.
3. Also it’s easy to check that, I D λ ( P αi ( w )) P αj ( w ) Q γ w γ = λ r i + r j I D ( βi ,βj ) λ Q αi ( w ( β i ) ) Q αj ( w ( β j ) ) Q γ , β i ,β j w γ I | w βi | = λ w r i +1 β i I | w βj | = λ w r j − β j = 0 , ∀ α, i which means that, I D λ ( P α ( w )) P α ( w ) Q γ w γ = X i,j I D λ ( P αi ( w )) P αj ( w ) Q γ w γ = 0 ∀ α. emma 3.2. Let f ∈ F and let f , f P and f A be the core, principal and analytic components of f . If D λ is equipped with an exterior probability ν λ , then it has following properties1. E ( f P ) = E ( f A ) = E ( f P ) = E ( f A ) = 0 h f A , f P i = h f P , f A i = 0 k f P k = h f p , f p i = T r ( η ) λ k f A k = λ T r ( D ) Proof.
We can look to prove the desired results utilising Proposition 3.1.1. (a) We’ll start by showing the principal component has vanishing expectation. E ( f αP ) = 1(2 πi ) n X β I D λ η αβ w β Q γ w γ = 1(2 πi ) n X β η αβ I D ( β ) λ Q γ , β w γ I | w β | = λ w β = 0 , ∀ α (b) and the same is true for its conjugate. E ( f αP ) = 1(2 πi ) n X β I D λ η αβ w β Q γ w γ = 1(2 πi ) n λ X β η αβ I D ( β ) λ Q γ , β w γ I | w β | = λ , ∀ α (c) Similar calculation can be done for the analytic component as well, with identical out-come. E ( f A ) = 1(2 πi ) n X β I D λ f A ( w ) Q γ w γ = f A (0) = 0as f A is analytic at 0.(d) And the same holds for its conjugate again. E ( f αA ) = 1(2 πi ) n X β I D λ D αβ w β + P α ( w ) Q γ w γ = λ (2 πi ) n X β D αβ I D ( β ) λ Q γ , β w γ I | w β | = λ w β + 1(2 πi ) n I D λ P α ( w ) Q γ w γ = 0 , ∀ α
2. (a) Now we would turn our attention to the inner product between analytic and principal8omponents. h f A , f P i = 1(2 πi ) n X β X δ X α I D λ ( D αβ w β + P α ( w ))( η αδ /w δ ) Q γ w γ = λ (2 πi ) n X β X δ X α D αβ η αδ I D ( β ) λ w sδ Q γ , β w γ I | w β | = λ w − sβ + 1(2 πi ) n X δ X α η αδ I D λ P α ( w ) w δ Q γ w γ = 0where s = I ( β , δ )(b) Change of order, doesn’t make a di ff erence to the outcome, of course. h f P , f A i = 1(2 πi ) n X β X δ I D λ ( D αβ w β + P α ( w ))( η αδ /w δ ) Q γ w γ = 1 λ πi ) n X β X δ X α D αβ η αδ I D ( β ) λ w sδ Q γ , β w γ I | w β | = λ w s +1 β + 1 λ πi ) n X δ X α η αδ I D λ P α ( w ) Q γ , δ w γ = 0where s = I ( β , δ )3. Finally we’ll concentrate on the self interaction terms.(a) First for the principal component. k f P k = h f P , f P i = 1(2 πi ) n X β X δ X α I D λ ( η αβ /w β )( η αδ /w δ ) Q γ w γ = 1 λ πi ) n X β X δ , β X α η αβ η αδ I D | ( β ) λ w δ Q γ w γ I | w β | = λ w β + 1 λ πi ) n X β X α η αβ η αβ I D λ Q γ w γ = 0 + 1 λ X β X α η αβ η αβ = T r ( η ∗ η ) λ (b) And then for the analytic one. 9 f A , f A i = 1(2 πi ) n X α I D λ ( P β D αβ w β + P α ( w ))( P δ D αδ w δ + P α ( w )) Q γ w γ = λ (2 πi ) n X β X δ , β X α D αβ D αδ I D ( β ) λ w δ Q γ w γ I | w β = λ | w β + λ (2 πi ) n X β X α D αβ D αβ I D λ Q γ w γ + λ (2 πi ) n X δ X α D αδ I D λ P α ( w ) w δ Q γ w γ + 1(2 πi ) n X δ X α D αβ I D λ P α ( w ) Q γ , δ w γ + 1(2 πi ) n X α I D λ ( P α ( w )) P α ( w ) Q γ w γ = λ X β X α D αβ D αβ = λ T r ( D ∗ D )With these results, we can now embark on proving the desired relation between the varianceof a meromorphic function over a poly-disc around its singular pole. Theorem 3.3.
For a meromorphic function f ∈ F such that there is a single pole at of order at most1, the variance of f over the Unit Lens L using exterior probability µ nλ is given by, V λ ( f ) = h f , f i − k E ( f ) k = T r ( η ∗ η ) λ + λ T r ( D ∗ D ) Proof.
Self interaction of f can be written as h f , f i = h f , f i + h f P , f P i + h f A , f A i + h f , f P i + h f P , f i + h f , f A i + h f A , f i + h f P , f A i + h f A , f P i = k E ( f ) k + k f P k + k f A k + f ∗ E ( f P ) + E ( f P ) ∗ f + f ∗ E ( f A ) + E ( f A ) ∗ f + h f P , f A i + h f A , f P i = k E ( f ) k + T r ( η ∗ η ) λ + λ T r ( D ∗ D )This would imply V λ ( f ) = T r ( η ∗ η ) λ + λ T r ( D ∗ D ) Corollary. If σ λ is the standard error of f under the exterior probability, then λσ λ ≥ p T r ( η ∗ η ) Let K = (cid:16) M , A (cid:17) be a complex manifold of dimension n where A is a holomorphic structure.Let D λ = { w ∈ C n | | w α | ≤ λ, ∀ α } be a closed poly-disc in C n .10 efinition 4.1. For p ∈ M and λ >
0, Let S = S λ = z − ( D λ ) for some ( z, U ) ∈ A and Z ( S λ ) = { z ′ = g ◦ z | g is holomorphism on z ( U ) } . We’d refer to (cid:16) S λ , Z ( S λ ) (cid:17) as a poly-disc on M around p Remark.
For a poly-disc S λ under a morph z , let S λ = z − ( E nλ ). Since E nλ is a σ -algebra and z isbijective, we can conclude that S λ is a σ -algebra ∀ z ∈ Z ( S λ ). Remark. ( S λ , S λ ) is a measurable space for every morph, z ∈ Z ( S λ ). Proposition 4.1. If ν λ : S λ C is defined as ν λ ( A ) = µ nλ ( z ( A )) , ∀ A ∈ S λ , under a morph z , then ν λ is σ -additive and hence a complex measure on ( S λ , S λ ) Proof.
Let { A i } ∞ i =1 be.a sequence of pairwise disjoint sets in S λ . Since the underlying morph z isbijective, that would mean that { z ( A i ) } ∞ i =1 are also pairwise disjoint. Also, z ( ∞ F i =1 A i ) = ∞ F i =1 z ( A i ). Thatwould naturally yield ν λ ( ∞ G i =1 A i ) = µ λ ( z ( ∞ G i =1 A i )) = µ λ ( ∞ G i =1 z ( A i )) = ∞ X i =1 µ λ ( z ( A i )) = ∞ X i =1 ν λ ( A i ) Definition 4.2.
For a manifold M and p ∈ M , An Unit Lens is the Filtration of probability spacesdefined by L = { ( S λ , S λ , ν λ ) } λ ↓ indexed by a scale parameter λ .Let’s fix a poly-disc S λ and a particular morph z . Let p = z − (0) ∈ S λ . Let P be the vectorspace of all meromorphic function ψ : M C k which has a pole of at most order 1 at p i.e ψ z = ψ ◦ z − : C n C k is analytic everywhere except for a (potential) pole at z ( p ) = 0. Remark. ν λ induces an Expectation E ( · ) and an inner product h· , ·i on P as• E ( ψ ) = R S λ ψdν λ = R D λ ψ z dµ nλ ,• h ψ, φ i = R S λ ψ ∗ φdν λ = R D λ ψ ∗ z φ z dµ nλ Remark.
We can write ψ z = ψ ◦ z − as a sum of its core( ψ ), principal( ψ P ) and analytic( ψ A ) compo-nents. Let η z denote the corresponding matrix of residues and D z be the Jacobian of the analyticcomponent at z = 0. Lemma 4.2.
For any given ψ ∈ P , if η and η ′ denote the residue matrix under two di ff erent morphs z, z ′ ∈ Z ( S λ ) respectively, then ∀ α , η αβ = η ′ αγ ∂z β ∂z ′ γ In other words, η αβ transforms contra-variantly on β .Proof. Since both z and z ′ are morphs for S λ , we can define g = z ′ ◦ z − : D λ D λ , holomorphicwith g (0) = 0. 11et H γβ = ∂z β ∂z ′ γ = ( g − ) ′ (0) be the Jacobian matrix for the coordinate transform at z = 0. So we canwrite, dw β = H γβ dg γ .Now, η αβ = 1(2 πi ) n I D λ ψ z ( w ) Q µ , β w µ Y µ dw µ = 1(2 πi ) n I D λ ψ ′ z ( g ( w )) Q µ , β w µ Y µ dw µ = 1(2 πi ) n I D ( β ) λ Q µ , β w µ Y µ , β dw µ I | w β | = λ ( ψ ′ + D ′ αγ g γ + P α ( g ( w )) + η ′ αγ g γ ) dw β = 1(2 πi ) n I D ( β ) λ Q µ , β w µ Y µ , β dw µ I | w β | = λ η ′ αγ g γ dw β = 1(2 πi ) n I D ( β ) λ Q µ , β w µ Y µ , β dw µ I | w β | = λ η ′ αγ g γ H γβ dg γ = η ′ αγ H γβ = η ′ αγ ∂z β ∂z ′ γ Lemma 4.3.
For any given ψ ∈ P , if D and D ′ denote the Jacobian matrix under two di ff erent morphs z, z ′ ∈ Z ( S λ ) respectively, then ∀ α , D αβ = D ′ αγ ∂z ′ γ ∂z β In other words, D αβ transforms co-variantly on β .Proof. Since both z and z ′ are morphs for S λ , we can define g = z ′ ◦ z − : D λ D λ , holomorphicwith g (0) = 0.Let H γβ = ∂z β ∂z ′ γ = ( g − ) ′ (0) be the Jacobian matrix for the coordinate transform at z = 0. So we canwrite, dw β = H γβ dg γ .Now, D αβ = ∂ψ αA ∂w β = ∂ψ αA ∂g γ ∂g γ ∂w β = D ′ αγ ∂z ′ γ ∂z β This implies we can talk about η and D in a coordinate-free way upto Z . Theorem 4.4.
Let L = { ( S λ , S λ , ν λ ) } λ ↓ be an unit lens on a manifold M and also let ψ ∈ P be ameromorphic ν λ -measurable function on S λ with a potential pole of order 1 at p ∈ S λ . If η and D denote the Residue and Jacobian matrix for ψ on L , then(a) standard error of ψ under the Exterior probability has a lower bound, λσ ψ ≥ q T r ( η ∗ η )12 b) minimum standard error is achieved at a finite scale ∗ λ = h T r ( η ∗ η ) T r ( D ∗ D ) i Proof. (a) Let z ∈ Z λ and ψ z = ψ ◦ z − . Proof follows as consequence of corollary to Theorem 3.3as applied on ψ z and the fact that η and D transforms as tensors under di ff erent choice of z ∈ Z λ (b) By virtue of Theorem 3.3 applied on ψ z , we can write, V ar λ ( ψ ) = T r ( η ∗ η ) λ + λ T r ( D ∗ D )Clearly, this would allow the variance to be minimised for ∗ λ = q T r ( η ∗ η ) T r ( D ∗ D ) . That leads to ∗ λ = h T r ( η ∗ η ) T r ( D ∗ D ) i . In the preceding sections we have been able to formalise the concept of a Lenses around a pointof potential singularity with respect to a particular detectable function, equipped with a specifictype of probability measure called Exterior Probability. Under this structure we’ve seen that thevariance of the function, instead of freely dropping to 0 with increasingly closer measurement, hasa global lower bound. In other words, too close to the point of singularity, the standard deviationtends to increase in proportion with reducing distance.This result was reproduced for a system of lenses over a generic complex manifold, and it wasshown that the variance characterisation can be described through tensors in a coordinate freeway. The set of results produced in the paper is most relatable from the perspective of describingquantum scale e ff ects in the backdrop of a generic space-time curved by gravity. The decompo-sition of variance into parts involving η and D are also significant in that regard, as η representsthe quantum e ff ects near a singularity, while D is tied with the generic smooth curvature of themanifold. They can also be thought of representing matter and force fields respectively.If we consider the special case where η is negligible or zero, then we can recover fully theclassical measurement set up with the lower bound on standard error converging on zero. On theother hand, if D is zero, then we have a pure quantum system on a flat space-time. The resultsshow that the standard error for an unbounded measurement attribute in such a system is inaccordance with Heisenberg uncertainty principle.The other connection that the present work shares with e ff orts towards understanding quan-tum gravity, is obviously in the structural similarity between the Lenses and the Strings. Whileboth are posited as sub-structures of space-time and works as a substitution of the concept ofclassical point mass [10], they di ff er in a fundamental way as well. Strings are inter-dimensionalstructures that spans across a sub-space of a higher dimensional space-time. Lenses constructedas they’ve been in this paper, are intra-dimensional objects plumbing the depth available in each13omplex dimension for additional structure. Hence Lenses, as opposed to Strings do not requireextra large dimensions to produce results consistent with quantum uncertainty.There are several di ff erent directions in which the current results can be progressed or im-proved further. The results so far are based on the simplest, Unit Lens and associated Exteriorprobabilities. However, one might be interested to look into more generic lenses and exteriorprobability structures. Any such probability measures, by definition, would have to be absolutelycontinuous with the unit lens, and hence would allow a density to be incorporated in the resultspresented thus far.Another area of potential interest could be to expand the the set up to attributes with multiplepoles instead of just one, poles with higher orders. In physical terms they would be instrumentalto understand many-body interactions at quantum scale with strong gravitational backdrop. References [1] John Wheeler.
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