{'fraction_non_alphanumeric': 0.059971201697584145, 'fraction_numerical': 0.016453766935843783, 'mean_word_length': 4.076212865997008, 'pattern_counts': {'":': 0, '<': 23, '<?xml version=': 0, '>': 26, 'https://': 0, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 0, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 63, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}}
Strong reactions in quantum super PDE's. I-II

This is a work in three parts, devoted to encode strong reactions of the high energy physics, in the algebraic topologic theory of quantum super PDE's, (previously formulated by A. Pr\'astaro). In particular strong reactions are characterized by means of boundary value problems in quantum super PDE's. In such a way one obtains representations of quantum nonlinear propagators in quantum super PDE's, by means of elementary ones (quantum handle decompositions of quantum nonlinear propagators). These are useful to encode nuclear and subnuclear reactions in quantum physics. Pr\'astaro's geometric theory of quantum PDE's allows us to obtain constructive and dynamically justified answers to some important open problems in high energy physics.


Introduction
The algebraic topologic theory of quantum (super) PDE's formulated by A. Prástaro, allows to directly encode quantum phenomena in a category of noncommutative manifolds (quantum (super)manifolds) and to finally solve the problem of unification, at quantum level, of gravity with the fundamental forces [61,64,65,66,67,68,70,73,75,77,78]. In particular, this theory allowed to recognize the mechanism of mass creation/distruction, as a natural geometric phenomenon related to 1 Nowadays we have also experimental evidences that such fundamental bricks do not exist.
is related to a decomposition theorem for quantum nonlinear propagators that we prove for solutions of boundary value problems in quantum super PDEs, and that will be developed in the second and third parts. This first part is instead devoted to some important preparatory subjects on the geometry of quantum super PDEs that will be utilized in the next parts, to encode strong reaction dynamics. In particular, we will generalize to the category Q S of quantum supermanifolds, the structure of super-bundle of geometric objects, first introduced by A. Prástaro, in the framework of category of commutative manifolds in [57,58]. This structure gives a general framework where characterize physical fields in a fully covariant way. In particular it applies to PDEs defined on some higher order G-structures.
Another subject that will be developed in this first part and that is very important to encode strong reactions, is the characterization of conservation laws for PDE's built in the category Q S . These are functions defined on the integral bordism groups of such equations and belonging to suitable Hopf algebras (full quantum Hopf algebras). In particular, we specialize our calculations on the quantum super Yang-Mills equations, quantum black holes as particular sections there, and the characterization of observed quantum nonlinear propagators by means of conservation laws and observed quantum energy. The last subject considered in this first part, is devoted to PDE's in the category Q hyper,S of quantum hypercomplex supermanifolds, as defined in [83], and will focus our attention on quantum exotic super PDE's, i.e., quantum super PDE's where we can embed quantum exotic super-spheres. For such Cauchy data we will generalize our previous results on quantum exotic PDE's [83]. Exotic boundary value problems are of particular interest in strong reactions encoding quantum processes occurring in high energy physics, as we will show in the second and third parts of this work.
In the following we show as it is organized the paper and list the main results. 2. The concept of super bundle of geometric objects (see [57,58]) is generalized to the category Q S for quantum super manifolds. This is necessary in order to obtain fully covariant structures, as requested in generalized mathematical structures that aim encode physical structures. Theorem 2.9 that characterizes PDEs as superbundles of geometric objects. Theorem 2.10 classifies quantum supermanifolds, with classic limit, with respect to the holonomy groups of these last commutative manifolds. Theorem 2.12 characterizes quantum super manifolds with structured classic limit as super bundles of geometric objects. Theorem 2.16. Here we define quantum Q 7 any quantum (super)manifold M having as classic limit M C = Q 7 . (This is the spinor bundle Q 7 → S 3 , with C 2 fibers.) We say quantum χ-flow on M , any quantum flow that projects on the classic limit in a χ-flow [4]. Then we have that a quantum χ-flow represents diffeomorphically any homotopy 3-sphere, Σ 3 ⊂ Q 7 , onto S 3 ⊂ Q 7 . This also solves a previous conjecture formulated in [4] for commutative manifolds. Definition 2.18 gives useful generalizations of previous, well-known definitions of instantons and solitons in the category Q S . 3. Previous Prástaro's results on the quantum conservation laws of quantum super PDE's are resumed. These are necessary in order to obtain the new results on quantum strong reactions obtained in the second part. Theorem 3.17 characterizes quantum blackhole dynamics by means of quantum integral characteristic supernumbers: these are conserved through a non-weak quantum evaporating black-hole. Theorem 3.20 and Corollary 25 give a precise meaning to the phenomenological concept of conservation of energy during an observed quantum process. 4. Here we explicitly extend to the category Q hyper,S of hypercomplex quantum supermanifolds, previous results, in the category Q hyper of hypercomplex quantum manifolds, given in [83] and in the category of quantum supermanifold [70,75]. In particular, Theorem 4.6 explicitly extends to the category Q hyper,S a previous result about the generalized Poincaré conjecture in the category Q S . (See [75].) Theorem 4.7 classifies diffeomorphic classes of hypercomplex quantum homotopy superspheres. Theorem 4.39 classifies integral bordism groups in quantum hypercomplex exotic super PDEs. Theorem 4.40 characterizes integral h-cobordism in quantum hypercomplex Ricci flow super PDEs.

Super-bundles of geometric objects in the category Q S
The quantum fundamental fields of physics, i.e., electromagnetic, gravitational and nuclear fields, all must obey the full covariance requirement that can be codified generalizing to the category Q S the structure of super-bundle of geometric objects, first introduced by A. Prástaro, in the framework of category of commutative manifolds in [57,58]. 4 Remark 2.1 (The concept of full covariance in Q S ). Similarly to what happens in the classical field theory, the concept of full covariance is fundamental in any quantum field theory. A quantum field is considered a section s of a suitable fiber bundle π : W → M in the category Q S . To say that s is fully covariant it means that for any local diffeomorphism φ of the base M we can calculate the pull-back φ * s of s by means of φ: is a local application on W , canonically associated to φ on M , such that the following diagram is commutative: If this circumstance is verified, we say that π : W → M is a natural fiber bundle (or fiber bundle of geometric objects), in the category Q S . We write: (π : W → M, B). A generalization of this concept is that of super-fiber bundle of geometric objects introduced in [57,58]. A quantum physical field is fully covariant iff it is a quantum superfield, that is a section of such a structure. Definition 2.2. A structure of superbundle of geometric objects in the category Q S , is given by two fiber bundles W πW → M πB ← B over the same base M and a covariant functor B : C(B) → C(W ), where C(B), (resp. C(W )) is the category whose objects are open subbundles of B (respectively, open subbundles of W ) and whose morphisms are the local fiber bundle authomorphisms between those objects such that: 4 See also Prástaro's algebraic topology of PDEs [59,60,61,62,63,67,69,71,74,76,79,80,81,82] in order to better understand its generalization to the category of quantum supermanifolds. Interesting applications can be found in [2,3,39,86,87]. For basic information on the geometry of PDE's see [12,18,23,27,38]. For basic information on algebraic topology see [34,42,43,44,46,49,54,91,92,93,94,97,98,99,100,101,102].
then B(f ) ∈ Hom(C(W )) and satisfies: . W is called the total bundle and B the base bundle. A section of π W is called a quantum superfield of geometric objects. The situation is resumed in the commutative diagram in (1).
3 (G-structures in the category Q S ). G-structures in the category Q S are reductions of principal bundles of (r|s)-frames, on a (m|n)-dimensional quantum supermanifold, natural generalizations of analogous structures on commutative manifolds [59]. In the physical applications it is important to consider the following.
Theorem 2.5. Let (P, M, π; G) be a principal fiber bundle with structure group G in the category Q S . Let W ≡ P × F/G be a fiber bundle associated to P with fibre F . Then, there exists a canonical covariant functor B such that (P, W ; B) is a superbundle of geometric objects if we restrict the category C(P ), where B is defined, to the subcategory C(P ) • , where Hom C(P )• (P |U, P |U ′ ) is the set of fibered quantum diffeomorphisms P |U → P |U ′ , such that the isomorphism on the structure group is the identity.
Proof. The proof can be copied by the classical case [61]. 5 Definition 2.6 (Quantum Lie derivative of section of super bundle of geometric objects). Let (φ M,λ , φ B,λ ) λ∈R , be a (local) 1-parameter group of (local) fiber bundle diffeomorphisms in the category Q S . Let (ξ, ζ) the corresponding couple of quantum vector fields such that the fiber bundle (2) is commutative.
We call quantum Lie derivative of a section s of π W , with respect to (φ M,λ , φ B,λ ) λ∈R , or (ξ, ζ), the infinitesimal variation , ∂ s, of the pull-back s λ ≡ φ * λ s = B(φ −1 λ ) • s • φ M,λ , with s : R × M → W . (Here, for sake of simplicity we have denoted s globally define, but in general it is only locally defined.) Therefore we get Definition 2.7 (Full quantum Lie derivative of section of super bundle of geometric objects). Let (φ M,λ , φ B,λ ) λ∈A , be a (local) 1-parameter group of (local) fiber bundle diffeomorphisms in the category Q S . Let (ξ, ζ) the corresponding couple of vector fields such that the fiber bundle (4) is commutative. We call full quantum Lie derivative of a section s of π W , with respect to (φ M,λ , φ B,λ ) λ∈A , or (ξ, ζ), the infinitesimal variation , ∂ s, of the pull-back s λ ≡ φ * λ s = B(φ −1 λ ) • s • φ M,λ , with s : A × M → W . (Here, for sake of simplicity we have denoted s globally define, but in general it is only locally defined.) Therefore we get Proposition 2.8. Under the same hypotheses of Definition 5 (resp. Definition 2.7) we get that whether π W : W → M is a vector bundle, then we get L ζ s ≡ ∂ s : M → W , (resp. L ζ s ≡ ∂ s : M → W ≡ Hom Z (A; W )).
Theorem 2.9. LetÊ k ⊂ JD k (W ), be a quantum super PDE, with (π B : B → M ← W : π W ; B) a super bundle of geometric objects. Let us assume that the map π k,0 :Ê k → W be surjective, with respect to the natural mapping π k,0 : JD k (W ) → 5 Warn ! Do not consider the concept of super bundle of geometric objects as synonym of fiber bundle in the category C S of supermanifolds. (For a geometric theory of PDEs in C S , see [60].) W , and let assume also thatÊ k is formally integrable and completely integrable. Then for any r-prolongation (Ê k ) +r , r ≥ 0, we get the structure of super bundle of geometric objects reported in (6). where JD k+r (−) • ≡ JD k+r (B • (−))|Ê k+r , with B • (−) the restriction of C(B)(−) to the sub-category C • (B) ⊂ C(B), such that the corresponding morphisms f ∈ Hom(C • (B)) are such that JD k+r (B • (f )) • are symmetries ofÊ k+r . We call (6) a (k + r)-holonomic super bundle of geometric objects.
Proof. This means that for any section u : M → (Ê k ) +r , we can calculate its quantum Lie derivative, with respect to a 1-group of (local) fiber bundle diffeomorphisms of π B : B → M , that should be also symmetries ofÊ k , hence of (Ê k ) +r too. In particular for holonomic sections u = D k+r s, namely s is a solution ofÊ k , we get ∂ D k+r s : M → (D k+r s) * vT JD k+r (W ) ∼ = JD k+r (s * vT W ), such that the diagram (7) is commutative. Theorem 2.10 (Classification of quantum supermanifolds with classic limit). We can classify quantum supermanifolds, having a classic limit, π C : M → M C , on the ground of its classic limit M C . In other words, we can say M has some property whether its classic limit has this property. (For example we can say that M is orientable when M C is so.) In Tab. 2 are reported some quantum supermanifolds classified with respect to the holonomic group of their classic limit.
Proof. This follows from standard properties of classifications of fiber bundles in algebraic topology, (see e.g., [94]), and taking into account that each fiber over p ∈ M C is contractible to a point.
Example 2.11. Quantum (super)manifolds with classic limits having nontrivial structure can be considered of particular interest whether from the mathematical point of view or physical one. In particular, recent works investigate on Calaby-Yau mirror-pairs of submanifolds of G 2 -manifolds and Spin(7)-manifolds. (See e.g., [4].) 7 These are Riemannian manifolds classified on the ground of their holonomy groups. Let us recall in Tab. 1 the classification of possible holonomy 6 Warn ! The set of all solutions of the linearized equationÊ k [s] ≡ (D k s) * vTÊ k is larger than ones obtained as Lie derivative of s, with respect to some 1-parameter symmetry group of some super bundle of geometric objects (π B : B → M ← W : π W ; B). 7 Calabi-Yau manifolds are compact, complex Kähler manifolds that have trivial first Chern classes (over R). Yau proved (1977Yau proved ( -1979 a conjecture by  that there exists on every CY-manifold a Kähler metric with vanishing Ricci curvature [102,13]. One important class of G 2 manifold are the ones from CY -manifolds. Let (X, ω, Ω) be a complex 3-dimensional CY-manifold with Kähler form ω and a nowhere vanishing holomorphic 3-form Ω, then X 6 × S 1 has holonomy groups Hol(M C , g) for simply connected Riemannian manifolds (M C , g) which are not locally a product space and not locally a Riemannian symmetric space (shortly denote them by INS-SCRM here). (This classification was first obtained by M.  and further improved by some other mathematicians (D. Alekseevski, Brown-Gray, Hitchin ...). R.L. Bryant (1987) first proved existence of metrics with holonomy G 2 (resp. Spin (7)) on 7-dimensional manifolds (resp. 8-dimensional manifolds) [6,11,24,33].
The group G 2 can be identified with the group of authomorphisms of octonions O or equivalently the subgroup of GL(7, R) that preserves a suitable 3-form ϕ 0 ∈ Ω 3 (R 7 ), given in (8).
A smooth 7-dimensional manifold M C has a G 2 -structure if its tangent bundle reduces to a G 2 bundle, or equivalently if there is a 3-form ϕ ∈ Ω 3 (M C ) such that at each p ∈ M C the pair (T p M C , ϕ(p)) is isomorphic to (T 0 R 7 , ϕ 0 ). Then the 3-form ϕ identifies an orientation µ ∈ Ω 7 (M C ), and µ determines a metric g ϕ .
is a connected, simply connected group, a connected 7-dimensional manifold with a G 2structure is orientable and admits a spin structure, i.e., its first two Stiefel-Whitney classes vanish.  and R. .) G 2 manifolds can be also characterized as critical points of a suitable functional on the 3-forms (N. .). In [4] it is defined mirror pair a couple of Calabi-Yau manifolds if their complex structures are induced from the same calibration 3-form, ϕ, in a G 2 manifold. It assigns to a G 2 manifold (M C , ϕ, Λ), with the calibration 3-form ϕ and a oriented 2-plane field Λ, a pair of parametrized tangent bundle valued 2-and 3-forms of M C . These forms can be used to define different complex and symplectic structures on certain 6-dimensional subbundles of T M C . More precisely, To any 3-dimensional submanifold Y ⊂ (M C , ϕ), χ assigns a normal vector field, which vanishes when Y is associative. For any associative manifold Y ⊂ (M C , ϕ) with a non-vanishing oriented 2-plane field, χ defines a complex structure on its normal bundle. In particular, any coassociative submanifold X ⊂ M C has an almost complex structure if its normal bundle has a non vanishing section. Two CY-manifolds are mirror pairs of each other if their complex structures are induced from the same calibration 3-form in a G 2 manifold. Furthermore, we call them strong mirror pairs if their normal vector fields ξ and ξ ′ are homotopic to each other through non-vanishing vector fields. One can assign to a G 2 manifold (M C , ϕ, Λ), with the group SU (3) ⊂ G 2 , hence is a G 2 manifold. In this case ϕ = ℜΩ + ω ∧ dt. Similarly, X 6 × R gives a noncompact G 2 manifold. calibration 3-form ϕ and oriented 2-plane field Λ, a pair of parametrized tangent bundle valued 2-and 5-forms of M C . These forms can be used to define differential complex and symplectic structures on certain 6-dimensional subbundles of T M C . When these bundles are integrated they give mirror CY-manifolds. In a similar way one can recognize mirror dual G 2 manifolds inside of a Spin(7) manifold (M 8 C , ψ). In case M C admits an oriented 3-plane field, by iterating this process one can obtain Calabi-Yau submanifolds pairs in M C whose complex and symplectic structures determine each other via the calibration form of the ambient G 2 (or Spin (7)) manifold. Table 2. Classification of quantum supermanifolds on the ground of the Holonomy groups of their classic limit π C : M → M C .
Theorem 2.12 (Quantum super manifolds with structured classic limit as super bundles of geometric objects). Let M be a quantum super manifolds with structured classic limits, π C : M → M C . A super bundle of geometric objects on M , (π C : B → M ← W : π W ) identifies a super bundle of geometric objects on M C iff Hom(C(B)) and Hom(C(W )) are restricted to (local) fiber bundles morphisms and the functor B admits such a restriction, such that the diagrams in (9) are commutative.
Then, when above conditions are satisfied, we say that the super bundle of geometric objects on M , (π C : B → M ← W : π W ), is a classic-regular super bundle of geometric objects on the quantum supermanifold M .
Proof. In fact any super bundle of geometric object on M , i.e., (π C : B → M ← W : π W ; B), one has the natural fiber bundle structures on M C reported in (10).
More explicitly we have the following lemma.
Lemma 2.15. The chi-flow equation admits a solution that diffeomorphically relates any 3-dimensional homotopy sphere Σ 3 , with S 3 .
Proof. Let us consider the following fiber bundle π : Here the fiber is C 2 ∼ = R 4 . One has the canonical embedding of S 3 into W , for any t ∈ R: (t, x k ) 1≤k≤3 → (t, x 1 , x 2 , x 3 , 0, · · · , 0), identified by the zero section of π. In (12) are given the local representations of the geometric objects above introduced.
In (13) is given the χ-flow equation in local coordinates. : Since Σ 3 is diffeomorphic to S 3 , let us consider φ : Σ 3 → S 3 this diffeomorphism. Then a solution of equation (13) is written in (14).
is the determinant of the jacobian of diffeomorphism φ. Therefore, the proof of the lemma is down. Theorem 2.16 (Quantum χ-flow). We define quantumQ 7 any quantum (super)manifold M having as classic limit M C = Q 7 . We say quantum χ-flow on M , any quantum flow that projects on the classic limit in a chi-flow. Then we have that a quantum χ-flow represents diffeomorphically any homotopy 3-sphere, Proof. This follows directly from above definitions and Lemma 2.15.
Theorem 2.17. A gauge theory in the category Q S is fully covariant.
Proof. In fact, any gauge theory in the category Q S , can be identified by the following super bundle of geometric objects on a quantum supermanifold M : where π P : P → M is a G-principal bundle bundle over M , in the category Q S , and g is the Lie superalgebra corresponding to the quantum Lie supergroup G of the P . Therefore one has the commutative diagram reported in (15).
This is a classic-regular super bundle of geometric objects on M . (For details on gauge theory in the category Q S see Refs. [70,75,77,78].) In the following we generalize the definition of instanton and soliton of the classical field theory. (Compare with the definitions usually adopted in commutative differential geometry. 8 Compare also with their non-commutative extension given in [8,50].)

Conservation Laws in Quantum Super PDEs
Conservation laws are considered for PDE's built in the category Q S of quantum supermanifolds. These are functions defined on the integral bordism groups of such equations and belonging to suitable Hopf algebras (full quantum Hopf algebras). In particular, we specialize our calculations on the quantum super Yang-Mills equations and quantum black holes. In this section we shall resume some our fundamental definition and result for PDE's in the category of quantum supermanifolds, Q S , where the objects are just quantum supermanifolds, and the morphisms are maps of class Q k w , k ∈ {0, 1, 2, · · · , ∞, ω} [67,68,70,73,76,77]. A small subcategory is C S ⊂ Q S of supermanifolds as defined in [60]. Let π : W → M be a fiber bundle, in the category Q S , such that dim W = (m|n, r|s), over the quantum superalgebra B ≡ A × E and dim M = (m|n) over A and such that E is a Z-module, with Z ≡ Z(A) ⊂ A, the center of A. The quantum k-jet-derivative space JD k (W ) of π : W → M , is the k-jet-derivative space of sections of π, belonging to the class Q k w . The k-jet-derivative JD k (W ) 8 [53] proving that instantons effects in 3-dimensional QED, coupled to a scalar field (i.e., Higgs field) lead to a massive photon.) In a paper, with the title "Super bundle of geometric objects, Yang-Mills gauge field and Spin G -instantons", announced in [87], and appeared in Section 4.8 of the book [61], were developed some relations between super bundle of geometric objects on classical instantons and gravitational instantons. In particular were there applied the fully covariant theory of classical spinor fields, first previously developed in [57,58].
If (x A , y B ) 1≤A≤m+n,1≤B≤r+s are fibered quantum coordinates on the quantum supermanifold W over M , then (x A , y B , y B A , · · · , y B A1···A k ) are fibered quantum coordinates on JD k (W ) over M , with the following gradiations: |x A | = |A|, |y B | = |B|, |y B A1···As | = |B| + |A 1 | + · · · |A s |. Note, also, that there is not symmetry in the indexes A i . JD k (W ) is an affine bundle over JD k−1 (W ) with associated vector bundle π * k,0 Hom Z (Ṡ k 0 M ; vT W ), whereṠ k 0 M is the k-times symmetric tensor product of T M , considered as a bundle of Z-modules over M , and π k,0 : JD k (W ) → W is the canonical surjection. Another important example isĴ k m|n (W ), that is the k-jet space for quantum supermanifolds of dimension (m|n) (over A) contained in the quantum supermanifold W . This quantum supermanifold locally looks like These are quantum supermanifolds modeled on B ≡ k B k . A quantum super PDE of order k on the fibre bundle π : W → M , defined in the category of quantum supermanifolds, Q S , is a subsetÊ k ⊂ JD k (W ), or E k ⊂Ĵ k m|n (W ). A geometric theory of quantum (super) PDE's can be formulated introducing suitable hypotheses of regularity onÊ k . (See [70].) The characterization of global solutions of a PDEÊ k ⊆Ĵ k m|n (W ), in the category Q S , can be made by means of its integral bordism groups ΩÊ k p|q , p ∈ {0, 1, . . . , m−1}, q ∈ {0, 1, . . . , n − 1}. Let us shortly recall some fundamental definitions and results about. Let f i : , be (p|q)-dimensional admissible compact closed smooth integral quantum supermanifolds ofÊ k . The admissibility requires that N i should be contained into some solution V ⊂Ê k , identified with a (m, n)-chain, with coefficients in A. Then, we say that they areÊ kbordant if there exists a (p + 1|q + 1)-dimensional smooth quantum supermanifolds f : Y →Ê k , such that ∂Y = X 1 ⊔ X 2 , f | Xi = f i , i = 1, 2, and V ≡ f (Y ) ⊂Ê k is an admissible integral quantum supermanifold ofÊ k of dimension (p+ 1|q + 1). We say that N i , i = 1, 2, areÊ k -quantum-bordant if there exists a (p + 1|q + 1)-dimensional smooth quantum supermanifolds f : is an admissible integral manifold ofĴ k m|n (W ) of dimension (p + 1|q + 1). Let us denote the corresponding bordism groups by ΩÊ k p|q and Ω p|q (Ê k ), p ∈ {0, 1, . . . , m − 1}, q ∈ {0, 1, . . . , n − 1}, called respectively (p|q)-dimensional integral bordism group ofÊ k and (p|q)-dimensional quantum bordism group ofÊ k . Therefore these bordism groups work, for (p, q) = (m − 1, n − 1), in the category of quantum supermanifolds that are solutions ofÊ k . Let us emphasize that singular solutions ofÊ k are, in general, (piecewise) smooth quantum supermanifolds into some prolongation (Ê k ) +s ⊂Ĵ k+s m|n (W ), where the set, Σ(V ), of singular points of a solution V is a non-where dense subset of V . Here we consider Thom-Boardman singularities, i.e., q ∈ Σ(V ), if (π k,0 ) * (T q V ) ∼ = T q V . However, in the case whereÊ k is a differential equation of finite type, i.e., the symbolsĝ k+s = 0, s ≥ 0, then it is useful to include also in Σ(V ), discontinuity points, q, q ′ ∈ V , with π k,0 (q) = π k,0 (q ′ ) = a ∈ W , or with π k (q) = π k (q ′ ) = p ∈ M , where π k = π • π ( k, 0) :Ĵ k m|n (W ) → M . We denote such a set by Σ(V ) S , and, in such cases we shall talk more precisely of singular boundary of V , like (∂V ) S = ∂V \ Σ(V ) S . Such singular solutions are also called weak solutions. Let us define some notation to distinguish between some integral bordisms.

Theorem 3.2. [70]
One has the exact commutative diagram (17). Therefore, one has the canonical isomorphisms: LetÊ k ⊂Ĵ k m|n (W ) be a quantum super PDE that is formally quantum superintegrable, and completely superintegrable. We shall assume that the symbolsĝ k+s = 0, s = 0, 1. Proof. In fact, in these cases any weak solution identifies a singular solution, by connecting its branches by means of suitable pieces of fibres. Furthermore, sincê E k+1 is a strong retract ofĴ k+1 m|n (W ), we can deform any quantum bording V ⊂ J k+1 m|n (W ), dim V = (m|n), with ∂V ⊂Ê k+1 , into a (singular) solution ofÊ k+1 , hence into a solution ofÊ k . (For details see Refs. [70].) Corollary 3.4. LetÊ k ⊂Ĵ k m|n (W ) be a quantum super PDE, that is formally superintegrable and completely superintegrable. One has the following isomorphisms: In order to distinguish between quantum integral supermanifolds V representing singular solutions, where Σ(V ) has no discontinuities, and quantum integral supermanifolds where Σ(V ) contains discontinuities, we can also consider "conservation laws" valued on quantum integral supermanifolds N representing the integral bordism classes [N ]Ê k ∈ Ω E k p|q . Definition 3.5. Let us define the space of quantum integral conservation laws of E k ⊂Ĵ k m|n (W ) the Z-module given in (18).
Here C Ω p|q (Ê k ) denotes the space of all quantum (p|q)-forms onÊ k . Then we define quantum integral characteristic supernumbers of N , with Then, one has the following theorems.
Above considerations can be generalized to include more sophisticated quantum solutions of quantum super PDEs. (See Refs. [70].) More preciselyC p (Ê k ; B) is the free two-sided B-module of formal linear combinations with coefficients in B, λ i c i , where c i is a singular p-chain f : △ p →Ê k , that extends on a neighborhood U ⊂ R p+1 , such that f on U is differentiable and T f (△ p ) ⊂Ê k m|n , whereÊ k m|n is the Cartan distribution ofÊ k .
[70] The homologyH •|• (Ê k ; B) of the bigraded bar quantum chain complex ofÊ k is isomorphic to (closed) bar integral singular (p|q)-bordism groups, with coefficients in B, ofÊ k : BΩÊk is formally quantum superintegrable and completely superintegrable, and the symbolsĝ k+s = 0, then one has the following canonical isomorphisms: groups Ω p|q (Ê k ) is an extension of a subgroup of A Ω p|q,s (W ) ∼ = H p|q (W ; A), and the integral (p|q)-bordism group ΩÊ k p|q is an extension of the quantum (p|q)-bordism group.
Corollary 3.10. LetÊ k ⊂Ĵ k m|n (W ) be a quantum super PDE, that is formally superintegrable and completely superintegrable. One has the following isomorphisms: Definition 3.11. The full space of (p|q)-conservation laws, (or full (p|q)-Hopf superalgebra), ofÊ k is the following one: We call full Hopf superalgebra, ofÊ k , the following: Definition 3.12. The space of (differential) conservation laws ofÊ k ⊂Ĵ k m|n (W ), Theorem 3.13. [70] The full (p|q)-Hopf superalgebra of a quantum super PDÊ E k ⊂Ĵ k m|n (W ) has a natural structure of quantum Hopf superalgebra. Quantum Hopf algebras are generalizations of such algebras.
Proposition 3.14. The space of conservation laws ofÊ k has a canonical representation in H m−1|n−1 (Ê ∞ ).
Proof. In fact, one has the following homomorphism j : The proof is obtained directly by duality of the exact commutative diagram (17).
, is the canonical representation of the space of the differential conservation laws in the full Hopf superalgebra ofÊ k , (corresponding to the integral bordism groups for regular smooth solutions), it follows that one has also the follow- Theorem 3.17 (Quantum tunnel effects and quantum black holes). The quantum supergravity equationÊ 2 ⊂ JD 2 (i * C ) admits global solutions having a change of sectional topology (quantum tunnel effects). In general these solutions are not globally representable as second derivative of sections of the fiber bundle i * C → N . E 2 admits solutions that represent evaporations of quantum black holes.
Proof. We shall consider, now, the quantum N = 2 superPoincaré group over a quantum superalgebra A = A 0 ⊕ A 1 , that is a quantum Lie supergroup G having as quantum Lie superalgebra g one identified by the following infinitesimal generators: . The corresponding nonzero Z 2 -graded brackets are the following: Here C αβ is the antisymmetric charge conjugation matrix, σ βµ = 1 4 [γ β , γ µ ], with γ µ the Dirac matrices. Z commutes with all the other ones. One has dim G = (d|N 2 ) = (11|8), and we will consider the following principal bundle in the category of quantum supermanifolds: P is a quantum supermanifold of dimension (15|8); M is a quantum supermanifold of dimension (4|N 1 ) = (4|0), identified, for the sake of simplicity, with a quantum Minkowski space-time. Then a pseudoconnection can be written by means of the following fullquantum differential 1-forms on P : is the usual Levi-Civita connection,θ µ γ is the vierbein,Ā γ is the electromagnetic field andψ aj γ is the usual spin 3 2 field. The blow up structure: π * Ĉ (P ) ֒→ Hom Z (T P ; g) implies that we can identify our fields with sections ⌉ µ of the fiber bundleπ :C ≡ Hom Z (T M ; g) → M . (Ĉ(P ) ∼ = JD(P )/G is the fiber bundle, over M , of principal quantum connections on the G-principal fiber bundle π : P → M .) The corresponding curvatures can be written in the form: , evalued on a macroscopic shell, i.e., an embedding i : N → M , of a globally hyperbolic, p-connected manifold N , 0 ≤ p ≤ 3, is given by the quantum super PDE reported in Tab.3.1, where L : JD(E) → A is a quantum Lagrangian function. Possible Lagrangian densities are polynomial in the curvature, (see example below), hence we can assume that they give formally quantum superintegrable, and completely quantum superintegrable, quantum super PDE's. Then, assuming thatÊ 2 [i] is formally integrable and completely superintegrable, the integral bordism groups ofÊ 2 [i] and its fullquantum p-Hopf superalgebras, can be calculated. More precisely, we use the fact thatĈ(P ) → M is a contractible fiber bundle of dimension (4|0, 44|32) , and that N is topologically trivial. In fact, we can apply Theorem 3.9 and Corollary 3.10, to obtain the quantum and integral bordism groups ofÊ 2 [i]: ΩÊ 2 [i] p,s Therefore, we have that 1-dimensional admissible integral closed quantum submanifolds contained intoÊ 2 [i], (admissible quantum closed strings), can propagate and interact between them by means of 2-dimensional admissible integral quantum manifolds contained intoĴ 2 4 (i * C ), or by means of 2-dimensional admissible integral quantum manifolds contained intô E 2 [i], in such a way to generate (quantum) tunnel effects. Finally, as a consequence of the triviality of the 3-dimensional integral bordism grooups, we get the existence of global quantum solutions of such equations. Let us now see that theorem can be proved by using surgery techniques and taking into account that for the 3-dimensional integral bordism group ofÊ k one has ΩÊ k 3,s = 0 = Ω In fact a boundary value problem forÊ k [i] can be directly implemented in the manifoldÊ in such a way that the boundary ∂B describes a fixed 3-dimensional time-like integral manifold Y ⊂Ê k [i]. (We shall require that the boundary ∂B of B is orientable.) Y is not, in general, a closed (smooth) manifold. However, we can solder Y with two other compact 3-dimensional integral manifolds X i , i = 1, 2, in such a way that the result is a closed 3-dimensional (smooth) integral manifold Z ⊂Ê k [i]. More precisely, we can take We can assume that C is an orientable manifold. Then, from the triviality of the integral bordism group, it follows that Then, again from the triviality of the integral bordism group, it follows also that there exists a 4dimensional integral (smooth) manifold V ⊂Ê k [i] such that ∂V = Z. Hence the integral manifold V is a solution of our boundary value problem between the times t 0 and t 1 , where t 0 and t 1 are the times corresponding to the boundaries where are soldered X i , i = 1, 2 to Y . Now, this process can be extended for any t 2 > t 1 . So we are able to find (smooth) solutions for any t > t 0 , hence (smooth) solutions for any t > t 0 , therefore, global (smooth) solutions. Remark that in order to assure the smoothness of the global solution so built it is enough to develop such construction in the infinity prolongationÊ k [i] +∞ ofÊ k [i]. Finally note that in the set of solutions ofÊ k [i] there are ones that have change of sectional topology. In fact the 3-dimensional integral bordism groups are trivial: Let us, now, consider the dynamics of a quantum black-hole. In order to obtain such solutions we must have a Cauchy integral data with a geometric black hole B embedded in a compact 3-dimensional integral manifold N , B ⊂ N , such that its boundary ∂N propagates with a fixed flow. Then a solution, with quantum tunnel effect of such boundary problem, can describe a evaporation process of such black hole. Above results assure the existence of such solutions and a way to build them. In order to represent such results with respect to a quantum relativistic observer, let us consider the space of observed quantum integral conservation laws.
Here CΩ q (Ê k [i]) denotes the space of all Cartan quantum q-forms onÊ k [i]. (See also [70].) Then we define quantum integral characteristic supernumbers of N , One has the following lemma.
Furthermore, if N is orientable then above condition is sufficient also in order to say that N ′ ∈ [N ]. Therefore a quantum evaporation black-hole process can be described by means of quantum smooth integral manifolds, and therefore for such a process "conservation laws" are not destroyed. By the way, as we can have also weak solutions around a quantum black-hole, we can assume also that interactions with such objects could be described by means of weak-solutions, like shock-waves. Therefore we shall more precisely talk of weak quantum black-holes and non-weak quantum black-holes, according if they are described respectively by means of weak solutions, or nonweak solutions. As a by-product we get that all the quantum integral characteristic supernumbers are conserved through a non-weak quantum evaporating black-hole.
Let us conclude this section by considering the following theorem that gives an important quantum conservation law for what we shall develop in part II. Proof. Let us recall that the observed quantum Hamiltonian of the observed quantum super Yang-Mills equation can be written in the form reported in (19). (See [67,70,75].) where, for sake of simplicity, we have put {x β , y α } the coordinates on the observed configuration bundle for the observed quantum super Yang-Mills equation (Y M ) [i].
Since equation (Y M ) [i] is invariant for time translations, we get that the quantum differential form (20), is a quantum conservation laws of (Y M ) [i], in the sense of Definition 3.18. 9 Here dx β denotes absent with β ∈ {0, 1, 2, 3}. In fact, for any solution s of (Y M )[i], we get dω H = 0. An explicit calculation of ω H in quantum coordinates gives the expression reported in (21). More explicitly one has the following.
Here dx β denotes absent with β ∈ {1, 2, 3}. The restriction of ω H |V on any compact coincides with the restriction of H on σ t . More precisely we get the formula (22).
Then the evaluation of ω H on σ t , gives a time dependent A-valued function H[i|t], defined in (23) This function does not necessitate to be constant. In fact, we get We get In general this last condition is not verified.
Corollary 3.22. For any observed quantum nonlinear propagator V of (Y M ) [i], where t 0 (resp. t 1 ), is the time of N 0 (resp. N 1 ).
Corollary 3.22 gives a precise meaning to the phenomenological statement that with respect to an observer a quantum process must conserve the total mass-energy.
. Steady state solutions of (Y M ) [i] are ones where the partial derivative with respect to the time of the observed fundamental field µ K α is zero: µ K α0 = 0. Then with the symbols used in the proof of Theorem 3.20, we get that the lost quantum energy L[V ] ∈ A of a quantum nonlinear propagator for such a solution must necessarily be zero. In fact, we get since y α 0 = 0. Therefore, in the steady state quantum nonlinear propagators holds the strict conservation of the observed quantum energy: Example 3.24 (Strong reactions with jet quenching in (Y M ) [i]). Quantum nonlinear propagators with lost quantum energy can encode strong reactions between ultrarelativistic heavy-ion collisions, where interactions between the high-momentum parton and the hot, dense medium produced in the collisions, lead to energy loss. This phenomenon is called jet quenching. 10

Quantum hypercomplex exotic super PDE's
In this section we consider PDE's in the category Q hyper,S of quantum hypercomplex supermanifolds, as defined in [83], and focus our attention on "quantum exotic super PDE's", i.e., quantum super PDE's where we can embed "quantum exotic supersheres". For such Cauchy data we will generalize our previous results on quantum exotic PDE's [83]. Such "exotic" boundary value problems are of particular interest in strong reactions encoding quantum processes occurring in high energy physics, as we will prove in part II.
Remark 4.1. Let us remark that in the other sections of this paper we refer to the category Q S instead that Q hyper,S . This is made for three reasons. The first is for convenience, since quantum micro-worlds can be encoded in Q S . The second is that one can directly generalize intrinsic results obtained in the geometry of PDEs in the category Q S to similar ones in the category Q hyper,S , as we have proved in [83], and we will also see in this section. The third reason is that nonassociative algebras, as arise for example in some quantum hypercomplex algebras, can be considered as subsets of their enveloping algebras. These last being necessarily associative, give a general criterion to encode nonassociative algebras in larger associative frameworks.
Definition 4.2 (Quantum homotopy (m|n)-supersphere). We call quantum homotopy (m|n)-supersphere (with respect to a quantum hypercomplex superalgebra A) a smooth, compact, closed (m|n)-dimensional quantum supermanifold M ≡Σ m|n , that is homotopy equivalent to the (m|n)-dimensional quantum supersphereŜ m|n , with classic regular structureπ C : M → M C , where M C is a homotopy m-sphere, and such that the homotopy equivalence between M andŜ m|n is realized by a commutative diagram (26). respect their classic limits: π C :Σ m|n →Σ m|n C . Therefore quantum diffeomorphisms between quantum homotopy (m|n)-superspheres are characterized by a cou- ,Σ m|n 2,C ) of mappings related by the commutative diagram in (27).
There f is a quantum diffeomorphism between quantum supermanifolds and f C is a diffeomorphism between manifolds. Note that such diffeomorphisms of quantum homotopy (m|n)-superspheres allow to recognize thatΣ m|n 1 has alsoΣ m|n 2,C as classic limit, other thanΣ This clarifies that the classic limit of a quantum homotopy (m|n)-supersphere is unique up to diffeomorphisms.
Let us also emphasize that (co)homology properties of quantum homotopy (m|n)superspheres are related to the ones of m-spheres, since we here consider classic regular objects only.
Proof. Let us first calculate the homology groups in integer coefficients Z, of quantum (m|n)-superspheres. In (30) are reported the homology spaces for quantum m = 0.
Furthermore, for m = 0 we get Above formulas can be obtained by the reduced Mayer-Vietoris sequence applied to the triad (Ŝ m|n ,D Therefore we get formulas (30). To conclude the proof we shall consider that H p (Ŝ m|n ; Z) ∼ = Hom Z (H p (Ŝ m|n ; Z); Z). Furthermore, quantum homotopy (m|n)superspheres have same (co)homology of quantum superspheres since are homotopy equivalent to these last ones.
Lemma 4.5. The quantum Euler characteristic numbers for quantum homotopy (m|n)-superspheres are reported in (33). These coincide with the corresponding quantum Euler characteristic numbers of quantum (m|n)-superspheres and with the Euler characteristic numbers of usual m-spheres. Furthermore they are the same of the corresponding total quantum Euler characteristic numbers. See in (33).
So the homological quantum Euler characteristic of the quantum (m|n)-supersphere is the same of the homological Euler characteristic of the usual m-sphere. Furthermore, since quantum homotopy (m|n)-superspheres are homotopy equivalent to quantum (m|n)-superspheres, it follows that the quantum Euler characteristic of a quantum homotopy (m|n)-supersphere is equal to the one ofŜ m|n . Moreover, also the total quantum Euler characteristic numbers for quantum (homotopy) (m|n)superspheres coincide with the ones of S m . In fact, we can consider the quantum total-homological Euler characteristic T otχ (Ŝ m|n ) ofŜ m|n , given by formulas (35).
So also the total homological quantum Euler characteristic of the quantum (m|n)supersphere is the same of the homological Euler characteristic of the usual msphere. Furthermore, since quantum homotopy (m|n)-superspheres are homotopy equivalent to quantum (m|n)-superspheres, it follows that the total quantum Euler characteristic of a quantum homotopy (m|n)-supersphere is equal to the one of S m|n .
Theorem 4.6 (Generalized Poincaré conjecture in the category Q hyper,S ). Let A be a hypercomplex quantum superalgebra with center Z(A) a Noetherian K-algebra, K = R or K = C. Let M be a classic regular, closed compact quantum supermanifold of dimension (m|n), in the category Q hyper,S , homotopy equivalent toŜ m|n , (hence this last is the quantum super CW-substitute of M ). Then we get that M ≈Ŝ m|n , i.e., M is also homeomorphic toŜ m|n , and M C ≈ S m , i.e., the classic limit M C of M is homeomorphic to the classic limit S m ofŜ m|n .
Proof. In [75] we have proved the generalized quantum Poincaré conjecture in the category Q S of quantum supermanifolds, by considering the quantum Ricci flow PDE just in the category Q S . Then, by using similar arguments to prove Theorem 3.10 and Theorem 3.11 in [83], we can state that generalized quantum Poincaré conjecture works also in the category Q hyper,S of quantum hypercomplex supermanifolds. Therefore the quantum (m|n)-supersphereŜ m|n , considered the quantum super CW-substitute of any quantum homotopy (m|n)-supersphereΣ m|n , is just homeomorphic to this last one:Σ m|n ≈Ŝ m|n .
Theorem 4.7. LetΘ m|n be the set of equivalence classes of quantum diffeomorphic quantum homotopy (m|n)-superspheres over a quantum (hypercomplex) superalgebra A (and with Noetherian center Z(A)). 11 InΘ m|n it is defined an additive commutative and associative composition map such that [Ŝ m|n ] is the zero of the composition. Then one has the exact commutative diagram reported in (36).
where Θ m is the set of equivalence classes for diffeomorphic homotopy m-spheres and j C is the canonical mapping j C : One has the canonical isomorphisms: Proof. After above Remark 4.3 we can state that the mapping j C is surjective. In other words we can writeΘ The fiber (Θ m|n ) [Σ m|n C ] is given by all classes [Σ m|n ] such their classic limits are diffeomorphic, hence belong to the same class in Θ m . Furthermore one has ker(j C ) = j −1 C ([S m ]) ≡Υ m|n ⊂Θ m|n . Therefore, we can state thatΘ m|n is an extension of Θ m byΥ m|n . Such extensions are classified by H 2 (Θ m ;Υ m|n ). 12 Table 4. Homology of finite cyclic group Z i of order i.
The composition map inΘ m|n is defined by quantum fibered connected sum, i.e., a connected sum on quantum supermanifolds that respects the connected sum on their corresponding classic limits. More precisely let M → M C and N → N C be connected (m|n)-dimensional classic regular quantum supermanifolds. We define quantum fibered connected sum of M and N the classic regular (m|n)-dimensional . 12 In Tab. 4 are reported useful formulas to explicitly calculate these groups.
Then the additive composition law is + : [Ŝ m|n ] is the zero of this addition. In fact, sinceŜ m|n \D m|n Analogous calculus for M C completes the proof.
In the following remark we will consider some examples and further results to better understand some relations between quantum homotopy superspheres and their classic limits.
These are given by We shall prove that Ext Z (H 1 (Z 28 ; Z);Υ 7|n ) =Υ 7|n /28 ·Υ 7|n . (We have used the fact that H 2 (Z 28 ; Z) = 0.) Let us look in some detail to this Z-module. By using the projective resolution of Z 28 given in (41), we get the exact sequence (42).
In the limit case where the quantum algebra is A = R, for a quantum homotopy (m|n)-supersphereΣ m|n one has justΣ m|n =Σ Theorem 4.10 (Homotopy groups of quantum (m|n)-supersphere). Quantum homotopy (m|n)-superspheres cannot have, in general, the same homotopy groups of m-spheres: 13 Furthermore, S m can be identified with a contractible subspace, yet denoted S m , of S m|n . There exists a mappingŜ m|n → S m , but this is not a retraction, and the inclusion S m ֒→Ŝ m|n , cannot be a homotopy equivalence.
Proof. Since must necessarily be π k (Σ m|n ) ∼ = π k (Ŝ m|n ), k ≥ 0, it is enough prove theorem forŜ m|n . We shall first recall some useful definitions and results of Algebraic Topology, here codified as lemmas.
Definition 4.11. A pair (X, A) has the homotopy extension property if a homotopy is a CW pair, then it has the homotopy extension property.
Lemma 4.13. If the pair (X, A) satisfies the homotopy extension property and A is contractible, then the quotient map q : X → X/A is a homotopy equivalence.
Let us consider that we can represent S m intoŜ m|n by a continuous mapping s : S m →Ŝ m|n , defined by means of the commutative diagram in (44).
Let us yet denote by S m the image of s. So we can consider the canonical couple (Ŝ m|n , S m ) as a CW pair, hence it has the homotopy extension property. S m is not a contractible subcomplex ofŜ m|n , so in general the quotient mapq :Ŝ m|n →Ŝ m|n /S m is not a homotopy equivalence. We have the following lemma.
Since S m is contractible intoŜ m|n , to the point ∞ ∈Ŝ m|n , the quotient map q :Ŝ m|n →Ŝ m|n /S m can be deformed into quotient mappingq t over deformed quotient (45).) 13 In other words, quantum homotopy (m|n)-superspheres are not homotopy equivalent to the m-sphere.
But this does not assure thatq is a homotopy equivalence. 14 Let us, now, consider also some further lemmas.
is a CW pair and we have attaching maps f, g : A → X 0 that are homotopic, then X 0 f X 1 ⋍ X 0 g X 1 rel X 0 (homotopy equivalence).
Lemma 4.16. If (X, A) satisfies the homotopy extension property and the inclusion A ֒→ X is a homotopy equivalence, then A is a deformation retract of X.
Let us emphasize that we have a natural continuous mapping π C :Ŝm|n → S m , i.e., the surjection between the quantum (m|n)-supersphere and its classic limit, identified by the commutative diagram (44). The inclusion i : S m ֒→Ŝ m|n cannot be a deformation retract (and neither a strong deformation retract), otherwise i should be a homotopy equivalence. 15 However, π C :Ŝ m|n → S m , cannot be neither a retraction, otherwise their homotopy groups should be related by the split short exact sequence (46), hence we should have the splitting given in (47). (For details on relations between homotopy groups and retractions see, e.g. [67].) But this cannot work. In fact, in the case A = C, we should have the commutative diagram (48) with exact horizontal lines.
Rally S m is contractible inŜ m|n , but is not a contractible sub-complex ofŜ m|n . This clarifies the meaning of Lemma 4.13. For example, in the case A = C, one has thatŜ 1|n /S 1 is not homotopy equivalent to S 2 ∼ =AŜ 1|n . In fact π 2 (S 2 ) = Z and π 2 ( 15 It is enough to consider the counterexample when A = C andŜ 1|n = C {∞} = R 2 {∞} = S 2 . Then S 1 cannot be homotopy equivalent toŜ 1|n = S 2 , since π 1 (S 1 ) = Z and π 1 (S 2 ) = 0. This should imply that π 1 (S 1 , ∞) = 0, instead that Z, hence the bottom horizontal line in (48) cannot be an exact sequence, hence π C :Ŝ 1 ∼ = S 2 → S 1 cannot be a retraction ! Corollary 4.18. Quantum homotopy superspheres cannot be homotopy equivalent to S m , except in the case that the quantum algebra A reduces to R.
Quantum homotopy groups for quantum supermanifolds are introduced in [75].
Theorem 4.19 (Quantum homotopy groups of quantum (m|n)-supersphere). Quantum homotopy (m|n)-superspheres have quantum homotopy groups isomorphic to homotopy groups of m-spheres: Proof. In fact, we can prove for examples thatπ k (Ŝ m|n ) ∼ = 0, for k < m, and π m (Ŝ m|n ) ∼ = Z. For this it is enough to reproduce the anoalogous proofs for the commutative spheres, by substituting cells with quantum supercells. For example we can have the following quantum versions of analogous propositions for commutative CW complexes.
Lemma 4.20. Let X be a quantum CW-complex admitting a decomposition in two quantum subcomplexes induced by inclusion is an isomorphism for k < m + n, and a surjection for k = m + n.
Remark 4.22. Let us emphasize that Theorem 4.19 does not allow to state that S m is a deformation retract ofŜ m|n , as one could conclude by a wrong application of the Whitehead's theorem, reported in the following lemma. If a map f : X → Y between connected CW complexes induces isomorphisms f * : π m (X) → π * (Y ) for all m, then f is a homotopy equivalence. Furthermore, if f is the inclusion of a subcomplex f : X ֒→ Y , then X is a deformation retract of Y .
In fact, in the case i : S m ֒→Ŝ m|n we are talking about different CW structures. One for S m is the usual one, the other, forŜ m|n is the quantum super-CW structure. In order to easily understand the difference let us refer again to the case A = C. Here one has π 1 (S 1 ) = Z =π 1 (Ŝ 1|n =Ŝ 1|0 =Ŝ 1 ), butπ 1 (Ŝ 1 ) = [Ŝ 1 ,Ŝ 1 ] = [S 2 , S 2 ] = π 2 (S 2 ). Furthermore, π 1 (Ŝ 1 ) = [S 1 , S 2 ] = 0 = π 1 (S 1 ). Therefore,Ŝ 1|n , with respect to the usual CW complex structure, has its first homotopy group zero, hence different from the first homotopy group of its classic limit S 1 . In fact S 1 is not a deformation retract of S 2 =Ŝ 1 . (Therefore there is not contradiction with the Whitehead's theorem.) Moreover, it is useful to formulate the quantum version of the Whitehead's theorem and some related lemmas. These can be proved by reproducing analogous proofs by substituting CW complex structure with quantum CW complex structure in quantum supermanifolds.   Proof. The proof can be done inductively. Let us assume that f has been extended over the quantum (m − 1|n − 1)-superskeleton. Then, an extension over quantum (m|n)-supercells exists iff the composition of the quantum supercell's attaching map As a by-product of above results we get also the following theorems that relate quantum homotopy groups and quantum relative homotopy groups.
for p ≤ m and epimorphisms for p = m + 1. 18 The following propositions are also stated as direct results coming from Theorem 4.19 and analogous propositions for topologic spaces. 2) S i → S n extends to a map D i+1 → S n .
Proof. 1) and 2) follow from the fact that S m is (m − 1)-connected, and 3) and 4) from the fact thatŜ m|n is quantum (m − 1)-connected. Furthermore, let us recall the following related result of Algebraic Topology. 19 Lemma 4.30. The following propositions are equivalent.
Let us, now, consider quantum super PDE's, in the category Q hyper,S , with respect to quantum homotopy (m|n)-superspheres.
Definition 4.31 (Quantum hypercomplex exotic super PDE's). LetÊ k ⊂Ĵ k m|n (W ) be a k-order PDE on the fiber bundle π : W → M in the category Q hyper,S , with dim A M = (m|n) and dim B W = (n|m, r|s), where B = A × E and E is also a Z(A)-module. We say thatÊ k is a quantum exotic PDE if it admits Cauchy integral manifolds N ⊂Ê k , dim N = (m − 1|n − 1), such that one of the following two conditions is verified. 18 Compare with analogous theorem in [75] for quantum supermanifolds. 19 There exists also a relative version of Lemma 4.30, saying that π i (X, A, x 0 ) = 0, for all  20 For complementary information see [83].
Definition 4.32 (Quantum hypercomplex exotic-classic super PDE's). LetÊ k ⊂ J k m|n (W ) be a k-order super PDE as in Definition 4.31. We say thatÊ k is a quantum exotic-classic super PDE if it is a quantum exotic super PDE, and the classic limit of the corresponding Cauchy quantum exotic supermanifolds are also exotic homotopy spheres.
From above results we get also the following one.
Furthermore, the singular integral bordism group for admissible smooth closed compact Cauchy manifolds, N ⊂Ê k , is given in (54).
In the quantum homotopy equivalence full admissibility hypothesis, i.e., by considering admissible only (m − 1|n − 1)-dimensional smooth Cauchy integral supermanifolds identified with quantum homotopy superspheres, and assuming that the space of conservation laws is not trivial, one has ΩÊ k m−1|n−1,s = 0. ThenÊ k becomes a quantum extended 0-crystal super PDE. Therefore, there exists a global singular attractor, in the sense that all Cauchy supermanifolds, identified with quantum homotopy (m − 1|n − 1)-superspheres, bound singular manifolds. Furthermore, if in W we can embed all the quantum homotopy (m − 1|n − 1)superspheres, and all such supermanifolds identify admissible smooth (m − 1|n − 1)dimensional Cauchy supermanifolds ofÊ k ), then two of such Cauchy supermanifolds bound a smooth solution iff they are diffeomorphic and one has the following bijective mapping: ΩÊ k m−1|n−1 ↔Θ m−1|n−1 . Moreover, if in W we cannot embed all quantum homotopy (m−1|n−1)-superspheres, but onlyŜ m−1|n−1 , then in the quantum supersphere full admissible hypothesis, i.e., by considering admissible only quantum (m − 1|n − 1)-dimensional smooth Cauchy integral supermanifolds identified withŜ m−1|n−1 , then ΩÊ k m−1|n−1 = 0. Thereforê E k becomes a quantum 0-crystal super PDE and there exists a global smooth attractor, in the sense that two of such smooth Cauchy supermanifolds, identified witĥ S m−1|n−1 bound quantum smooth supermanifolds. Instead, two Cauchy supermanifolds identified with quantum exotic (m − 1|n − 1)-superspheres bound by means of quantum singular solutions only. All above quantum smooth or quantum singular solutions are unstable. Quantum smooth solutions can be stabilized.
Proof. The relations (53) and (54) can be proved by a direct extension of analogous characterizations of integral bordism groups of PDE's in the category of commutative manifolds and quantum PDEs. (See [63,64,82,83].) Then the rest of the proof follows directly by using above results in this section, and following a road similar to the proof of Theorem 5.38 given in [83].
Similarly one can prove the following theorem that extends in the category Q hyper,S an analogous theorem in the category of commutative manifolds and in the category Q hyper . (See [82,83].) 21 The fiber bundle π : W → M is as in   [i], quantum electromagnetic-Higgs PDE, bounded by a quantum super partial differential relation (Goldstone) [i]w ⊂ (Y M ) [i], quantum electromagnetic Goldstone-boundary. An electric neutral, connected, simply connected observed quantum particle, identified with a Cauchy data of (Y M ) [i], it is proved do not belong to (Y M ) [i]w. Existence of Q-exotic quantum nonlinear propagators of (Y M ) [i], i.e., quantum nonlinear propagators that do not respect the quantum electric-charge conservation is obtained.
By using integral bordism groups of quantum super PDE's, a quantum crossing symmetry theorem is proved. As a by-product existence of massive photons and massive neutrinos are obtained. A dynamical proof that quarks can be broken-down is given too. A quantum time, related to the observation of any quantum nonlinear propagator, is calculated. Then an apparent quantum time estimate for any reaction is recognized. A criterion to identify solutions of the quantum super Yang-Mills PDE encoding (de)confined quantum systems is given. Supersymmetric particles and supersymmetric reactions are classified on the ground of integral bordism groups of the quantum super Yang-Mills PDE (Y M ). Finally, existence of the quantum Majorana neutrino is proved. As a by-product, the existence of a new quasi-particle, that we call quantum Majorana neutralino, is recognized made by means of two quantum Majorana neutrinos, a couple ( νe, νe), supersymmetric partner of (νe,νe), and two Higgsinos.

Introduction
The algebraic topologic theory of quantum (super) PDE's formulated by A. Prástaro, allows to directly encode quantum phenomena in a category of noncommutative manifolds (quantum (super)manifolds) and to finally solve the problem of unification, at quantum level, of gravity with the fundamental forces [65,68,69,70,71,72,74,77,79,81,82]. In particular, this theory allowed to recognize the mechanism of mass creation/distruction, as a natural geometric phenomenon related to the algebraic topologic structure of quantum (super) PDEs encoding the quantum system under study [82]. Aim of this second part is to explicitly prove that nuclear and subnuclear reactions can be encoded as boundary value problems in the Prástaro's algebraic topology of quantum super PDEs, and can be represented in terms of elementary reactions. (For part I and part III see [88,89].) It is important to emphasize that quantum conservation laws do not necessarily produce conservation of quantum charges in quantum reactions. In fact, it is also important consider the topological structure of the corresponding quantum nonlinear propagators encoding these reactions. In [88] (Part I), we have shown this fact for the observed quantum energy. In this second part we characterize observed quantum nonlinear propagators V of the observed quantum super Yang-Mills PDE, (Y M ) [i], with respect to the total quantum electric-charge. Then we define Q-exotic quantum nonlinear propagators ones where there is a non-zero lost quantum electriccharge, Q[V ] ∈ A, in the corresponding encoded reactions. (A is the fundamental quantum superalgebra in (Y M ) [i].) This important phenomenon, that is related to the gauge invariance of (Y M ) [i], was non-well previously understood, since the gauge invariance was wrongly interpreted as a condition that necessarily produce the conservation of electric-charge in reactions. Really just the gauge invariance is the main origin of such phenomenon, but beside the structure of the quantum nonlinear propagator. This fundamental aspect of quantum reactions in (Y M ) [i], gives strong theoretical support to the guess about existence of quantum reactions where the "electric-charge" is not conserved. This was quasi a dogma in particle physics. However, there are in the world many heretical experimental efforts to prove existence of decays like the following e − → γ + ν, i.e. electron decay into a photon and neutrino. In this direction some first weak experimental evidences were recently obtained. 1 With this respect, one cannot remark the singular role played, in the history of the science in these last 120 years, by the electron, a very small and light particle. In fact, at the beginning of the last century was just the electron to cause break-down in the Maxwell and Lorentz physical picture of the micro-world, until to produce a completely new point of view, i.e. the quantum physics. Now, after 120 years the electron appears to continue do not accept the place that physicists have reserved to it in the world-puzzle. In the following we show as it is organized the paper and list the main results. that apparently contradict the conservation of mass-energy, is proved. This is related to the new concept of quantum time (see Definition 4.21) that comes from the interaction between quantum relativistic frame (i.e., an observer) and the quantum system. Theorem 4.27: A criterion to identify solutions of ( Y M ), encoding confined quantum systems, is proved. Lemma 4.31 and Lemma 4.32 state existence of supersymmetric particles and supersymmetric reactions by means of integral bordism groups of (Y M ). Theorem 4.29: Existence of quantum Majorana neutrino and a new quasi-particle, that we call quantum Majorana neutralino, for its similarity with the so-called neutralino, are obtained, by using algebraic topologic properties of (Y M ).

Surgery in Quantum Super PDEs
Let us assume as a prerequisite of this section the knowledge of some previous works by A. Prástaro on quantum (super) PDEs. However, to fix more directly ideas we recall also some fundamental definitions and results that are soon related to the subject here considered. Let A = A 0 × A 1 be a quantum superalgebra in the sense of Prástaro. Theorem 2.2 (Quantum handle decomposition of nonlinear quantum propagator in quantum super PDEs). LetÊ k ⊂Ĵ k m+1|n+1 (W ) be a quantum super PDE in the category Q S and let V ⊂Ê k be a compact solution of a boundary value problem, i.e., ∂V = N 0 ⊔ N 1 . 5 Let us assume thatÊ k is formally integrable and completely integrable. Then there exists a quantum-super-handle-presentation (1) of the nonlinear quantum propagator V .
Proof. Let M be a quantum supermanifold of dimension (m|n) and let us consider an embedding φ :Ŝ p|q ×D m−p|n−q → M. Let us consider the following (m|n)-dimensional quantum supermanifold 5 Let us recall that V such that ∂V = N 0 ⊔N 1 , is called quantum non-linear propagator between the two Cauchy data N 0 and N 1 . This is the non-linear extension of the concepts of quantum propagator, usually used to quantize a classical field theory. For more details see Refs. [79,81,82].
We say that M ′ is obtained from M by a (p|q)-surgery, i.e., cutting out int(Ŝ p|q × D m−p|n−q ) and gluing inD p+1|q+1 ×Ŝ m−p−1|n−q−1 . The process of surgery is related to cobordism and handle attaching ones. Let (X, ∂X) be a (m + 1|n + 1)dimensional quantum supermanifold with boundary ∂X, and let φ :Ŝ p|q ×D m−p|n−q → ∂X ≡ M be an embedding. Set We call X ′ obtained from X by attaching a (m + 1|n + 1)-dimensional quantum superhandle of index p|q. One has We say that M ′ is obtained from M by a quantum (p−1|q −1)-surgery. The surgery allows us to obtain a cobordism We shall use the following lemma. Proof. See Theorem 2.19 in [79]. 7 Let us remark that the proof of Lemma 2.3 is related to the quantum CW substitute structure of quantum nonlinear propagators, hence the relations considered in this lemma are to consider homeomorphisms. On the other hand the handle decomposition of the quantum nonlinear propagator V , is related also to the quantum cell 7 Handle decomposition of bordism for manifolds has been introduced by Stephen Smale [96].

Lemma 2.3 generalizes to quantum supermanifolds an analogous result for commutative manifolds.
However, the proof of Lemma 2.3 is not found on the Morse function, (see, e.g., [48]), but on the quantum CW-substitutes for quantum supermanifolds.
composition that always holds for a (m + 1|n + 1)-dimensional quantum supermanifold, according to its quantum CW-structure. But the relation between a quantum superdiskD r|s and a (r|s)-dimensional quantum superhandleĥ p|q of index p|q, is in general an homeomorphismĥ p|q ≈D p|q ×D r−p|s−q , that cannot be reduced to a diffeomorphism in the category Q S . 8 In Tab Since V is a (m+1|n+1)-dimensional bordism between (m|n)-dimensional quantum supermanifolds N 0 and N 1 , we can determine its handle decomposition (1), that identifies intermediate final Cauchy quantum supermanifolds M j that are obtained from the previous one M j−1 by means of an integral surgery. Let us emphasize that each M j must necessarily be an integral quantum supermanifold having the same dimension of N 0 since it belongs to the same bordism class [N 0 ]. Furthermore, since each quantum supermanifold M j is contained into V , it follows that M j can be identified with a Cauchy (m|n)-dimensional quantum supermanifold ofÊ k , i.e. an admissible (m|n)-dimensional quantum integral supermanifold ofÊ k ⊂ J k m+1|n+1 (W ). Therefore M j belongs to the same bordism class [N 0 ] ∈ ΩÊ k m|n of N 0 inÊ k . The situation is pictured in Fig. 3.  Proof. This follows directly from Proposition 2.6. Definition 2.8 (Fundamental quantum particles). For a quantum system, encoded by a quantum PDEÊ k ⊂Ĵ k m+1|n+1 (W ), we define quantum (m|n)-particles, admissible quantum integral (m|n)-chains N ⊂Ê k ⊂Ĵ k m+1|n+1 . We call fundamental quantum (m|n)-particles for the quantum PDEÊ k , ones that cannot be decomposed into other quantum particles.
Proposition 2.9. The fundamental quantum (m|n)-particles of quantum PDÊ E k ⊂Ĵ k m+1|n+1 (W ), are identified with the bordism classes of the weak integral 9 It is useful to emphasize that quantum exchangions and quantum virtual particles, well interpret the meaning of "exchange particles" introduced in Feynman diagrams (1949) to represent particle interactions, like photons (electro-magnetic interactions), W and Z particles (weak interactions), gluons and gravitons (strong interactions). (See, e.g., Refs. [23,38].) Furthermore, in the framework of Theorem 2.2 we understand also that Regge trajectories and Regge resonances (1959), introduced in the phenomenological theory of strong reactions by the pioneering works of T. Regge [92] (and principally developed also by R. Blanckebecker and M. L. Goldberger [7], G. F. Chew and S. C. Fraustschi [14], V. N. Gribov [28] and G. Veneziano [100]), find their interpretation as quantum exchangions and quantum virtual particles.
Proof. The proof follows directly from Definition 2.8, Theorem 2.2 and Proposition 3.3 in [74].
Remark 2.10. Let us emphasize that the concept of fundamental quantum particles, is strictly related to the quantum (super) PDEÊ k , or in other words, to the quantum theory encoded byÊ k . Whether we aim to forget the framework defined byÊ k , and to discuss about fundamental quantum particles in general, we see that the unique fundamental one is ∅. In fact, according to Lemma 2.3(3), we can build any quantum particles just starting from ∅, and adding quantum handles or quantum-cells. 10 Example 2.11 (Two-body high energy reactions). Let us consider the following typical two-body strong reactions: π − +p → π 0 +n, π + +p → π + +p, π + +p → p+π + . The first reaction can be considered obtained with a quantum nonlinear propagator having an intermediate virtual neutral ρ meson, with charge exchange. In the second reaction the quantum nonlinear propagator has an intermediate virtual double electric charged particle and a neutral quantum pomeron p om , quantum exchangion h 1|1 , representing an elastic scattering. In the third reaction can be considered the quantum nonlinear propagator has an intermediate virtual neutron n, in backward 10 From this geometric structural point of view, we clearly understand that also some particles, that are usually considered fundamental, cannot be considered so. For example electrons and quarks are not quantum fundamental particles whether considered as geometric objects. This agrees with experimental evidences that so-called "quasiparticles" with fractional charges, or with separated quantum numbers (e.g., decay e − → spinon + orbiton), were detected. (See, e.g. [24,94].) Furthermore, it is important to distinguish the concept of quantum fundamental particle, from the one of quantum stable particle. Without any quantum theory, i.e., without any quantum (super) PDE, for the first the unique one is ∅, and for the second, it is a no sense. On the other hand, with respect to a quantum (super) PDEÊ k we can understand that the concept of stability is just related to the integral bordism groups ofÊ k , hence from this point of view the concept of "fundamental particle" and "stable particle" become related each other one.
scattering. In other words, in all these reactions the final Cauchy data can be obtained from the initial one by means of two adjoint elementary bordisms. 11 Example 2.12 (Proton-proton chain reactions). In the following we shall see that solutions, representing proton-proton chain reactions, typical in the Sun for the production of 4 2 He, can be encoded with quantum nonlinear propagators, admitting handle decompositions. In general one can classify such reactions into five basic groups. 12 (I) Production of 4  He + e + + ν e + 18.8 M eV. (V) Production of deuterium from the electron capture by two protons. Fig. 4 we represent some solutions corresponding to reactions in (I). By using the definition introduced in [82], we can say that in both reactions there represented, the quantum nonlinear propagator V , is a quantum matter-solution. Furthermore, in the second reaction, the handle decomposition of V , identifies a Goldstone piece V ∂ = V ( Goldstone). This is the part arriving to γ. 11 The quantum virtual particles there involved can be considered generalizations in the geometric theory of quantum (super)PDEs, of objects called reggeons in analogous reactions considered in the framework of the phenomenological theory of strong reactions developed in the first two decades of the second half of the last century. (See, e.g., [22].) However, let us emphasize that pomeron is better represented as quantum exchangion than a quantum virtual particle. In fact, pomeron, as usually considered in the phenomenological theory of strong reactions, does not carry charge [22]. On the other hand in the elastic scattering π + + p → π + + p it is impossible that the quantum virtual particle should be neutral one. 12 Let us recall the nucleus-notation A Z X, where X denotes the chemical symbol, A = Z + N is the mass number, with Z the atomic number=number of protons and N the number of neutrons. 13 In the Sun this chain reaction is dominant in the temperature range 10−14 M K. In presence of electrons, on has also the secondary reaction e + + e − → 2γ + 1.02M v. 14 In the Sun this chain reaction is dominant in the temperature range 14 − 23 M K. 15 In the Sun this chain reaction is dominant in the temperatures that exceeds 23 M K. Here the quantum nonlinear propagator V has a three step decomposition: Furthermore there is a virtual quantum particle M 3 with a double electric charge. One has V 1 = M 0 ×D 1|1 , V 2 =ĥ 1|1 and V 3 =ĥ 1|1 ĥ1|1 . The final Cauchy data is represented by three disjoint supermanifolds. In the reaction 2 1 D + 1 1 H → 3 2 He + γ, the quantum nonlinear propagator V has a three step decomposition too: There is a double electric charged virtual quantum particle M 3 , and the final Cauchy data is represented by two disjoint quantum supermanifolds.

Surgery in Quantum Super Yang-Mills PDEs
Let us emphasize, now, that in the reactions considered in the above section, we have not really specified the contribution of a specific quantum PDE, i.e., we have not considered some specific integral bordism group. On the other hand, without the introduction of these fundamental structures, what we can do is reproduce some phenomenological theory like dispersion relations and Regge models that cannot be but predictive dynamical theories. Therefore to dynamically encode protonproton reactions we shall use the quantum super Yang-Mills PDE, and to solve suitable boundary value problems. We have formulated this general theory in some our previous works [81,82]. In the following we shall consider some particular applications.  Elementary quantum nonlinear propagators V are (4|4)-dimensional quantum supermanifolds with boundary ∂V = N 0 P N 1 , such that V is homeomorphic tô D 4|4 . One can classify V on the ground of its boundary ∂V . This can be diffeomorphic to a quantum exotic supersphereΣ 3|3 or to a quantum supersphereŜ 3|3 . 16 Proof. In D = 4, the usual N -supersymmetric extension g of the Poincaré algebra p = so(1, 3) ⊕ t, is a Z 2 -graded vector space g = g 0 ⊕ g 1 , with a graded Lie bracket, such that g 0 = p ⊕ b, where b is a reductive Lie algebra, such that its self-adjoint part is the tangent space to a real compact Lie group. 17 Furthermore where ( 1 2 , 0) and (0, 1 2 ) are specific representations of the Poincaré algebra. Both components are conjugate to each other under the * conjugation. s is a N -dimensional complex representation of b and s * its dual representation. 18 Note also that the Lie bracket for the odd part is usually denoted by {, } in theoretical physics. Then with such a notation one has Poincaré algebra in D = 4 too. There a, b and c are constants. This algebra admits the following splitting: so ( the quantum superextension A ⊗ R [so(3, 1) ⊕ osp(1, 4)], where A is a quantum superalgebra, and with respect to the splittingμ = μ + c μ + μ of the fundamental fieldμ, we get: The dynamic equation are resumed in Tab. 3. Table 3. Local expression of (Y M ) ⊂ JD 2 (W ) and Bianchi identity (B) ⊂ JD 2 (W ).
We call quantum graviton a quantum metric g obtained by a solutionμ of (Y M ), via the corresponding quantum vierbein.
Since H 3 (M ; K) = 0, we get So (Y M ) is a quantum extended crystal super PDE. However, in general, (Y M ) is not a quantum 0-crystal super PDE. In fat one has the following short exact sequence such that N = ∂V , where V is some (4|4)-dimensional quantum supermanifolfd identified with a submanifold ofĴ 2 4|4 (W ). However, if we consider admissible only integral boundary manifolds, with orientable classic limit, and with zero characteristic quantum supernumbers, (full admissibility hypothesis), one has: Ω (Y M) 3|3 = 0, and (Y M ) becomes a quantum 0-crystal super PDE. Hence we get existence of global Q ∞ w solutions for any boundary condition of class Q ∞ w . Then we get the exact commutative diagram (11). (For notation see [81,82].) Taking into account the result by Thom on the unoriented cobordism groups [99], we can calculate Ω 6 ∼ = Z 2 Z 2 Z 2 . Then, we can represent Ω 6 as a subgroup of a 3-dimensional crystallographic group type [G(3)]. In fact, we can consider the amalgamated subgroup D 2 × Z 2 ⋆ D2 D 4 , and monomorphism Ω 6 → D 2 × Z 2 ⋆ D2 D 4 , given by (a, b, c) → (a, b, b, c). Alternatively we can consider also Ω 6 → D 4 ⋆ D2 D 4 .
(See Appendix C in [80] for amalgamated subgroups of [G (3)].) In any case the crystallographic dimension of (Y M ) is 3 and the crystallographic space group type are D 2d or D 4h belonging to the tetragonal syngony.
• If the initial and final Cauchy data are non-closed compact bounded 3|3-dimensional quantum supermanifolds, we can consider the exact commutative diagram reported in (12).
It follows the canonical isomorphism: is a free two-sided projective A-module, one has the unnatural isomorphism: Then a ≡ N 0 P and b ≡ N 1 belong to the same bordism group. More precisely one has such that, ∂V = N 0 P N 1 . In general one has the following relation with the (closed) bordism group ΩÊ k 3|3 :B where the cyclism groupCycÊ k 3|3 , is defined by the condition b ∈ [a] ∈CycÊ k 3|3 iff ∂(a−b) = 0. Since, under the full admissibility hypothesis, one has ΩÊ k 3|3 = 0, we get BorÊ k 3|3 ∼ =CycÊ k 3|3 . Therefore, we can say that inequivalent quantum particles inÊ k , are identified with (3|3)-dimensional bounded compact quantum supermanifolds, Cauchy data inÊ k , that are in correspondence one-to-one with bordism classes in BorÊ k 3|3 ∼ =CycÊ k 3|3 . Therefore, we get that the group of such fundamental particles are identified with • Above results mean that all (3|3)-dimensional closed integral quantum supermanifolds X ⊂Ê k are boundary of some 4|4-dimensional integral quantum supermanifold V ⊂Ê k , i.e., X = ∂V . In other words, fixing two Cauchy data N 0 and N 1 in E k , we can consider a quantum non-linear propagator V between them, identified by means of a (3|3)-dimennsional closed compact integral quantum supermanifold X ⊂Ê k , such that X = N 0 P N 1 . Therefore a quantum non-linear propagator between N 0 and N 1 can be identified with a connected (3|3)-dimennsional closed compact integral quantum supermanifold X ⊂Ê k homeomorphic to N 0 P N 1 .
Since Ω 3|3 = 0, it follows that we can represent such quantum non-linear propagators, with a (3|3)-dimensional quantum supersphereŜ 3|3 . Taking into account results contained in [87], we can also distinguish between them (3|3)-dimensional quantum exotic superspheresΣ 3|3 . These last are classified with respect to the equivalence relation induced by quantum diffeomorphisms. In Fig. 5 is represented the boundary X = ∂V , of an elementary quantum nonlinear propagator V , represented as a (3|3)-dimensional quantum supersphereŜ 3|3 . Therefore, such quantum non-linear propagators are homeomorphic to a (4|4)dimensional quantum superdiskD 4|4 . Fixing a (3|3)-dimensional quantum (exotic) supersphere X ≈Σ 3|3 ⊂Ê k , we can embed there (3|3)-dimensional quantum superdisks a i , i = 1, · · · , p and b j , j = 1, · · · , q, and consider the (4|4)-dimennsional quantum integral submanifold V ⊂Ê k , identified by a fixed boundary X. Then we can consider V the quantum non-linear propagator between N 0 ≡ 1≤i≤p a i and • It is important to remark that above quantum non-linear propagators, V , cannot be considered regular solutions in the equation (Y M ) ⊂ JD 2 (W ), but are necessarily singular solutions there. However, by considering the natural embedding JD 2 (W ) ֒→Ĵ 2 4|4 (W ) we can consider V as regular solution of (Y M ) ⊂Ĵ 2 4|4 (W ). But, whether N 0 is not diffeomorphic to N 1 , we cannot say that V is diffeomorphic to N 0 ×D 1|1 . Hence, the integrable full-quantum vector field ζ : V → T V , relating N 0 to N 1 , must necessarily be singular one.
• For any quantum conservation law α of (Y M ) we have < α, X >= 0, if X = ∂V , where V is a quantum nonlinear propagator between N 0 and N 1 . Therefore we get the relation (13) between quantum numbers induced by the quantum conservation laws: 19 Therefore, identifying quantum nonlinear propagators, for N 0 and N 1 , with (4|4)dimensional quantum integral superdisksD 4|4 ⊂Ê k , we get that the relation (13) between the corresponding quantum conservation supernumbers, gives Similar considerations hold for the observed quantum non-linear propagators. (See also [81,82].) This equation is also formally integrable and completely integrable. Furthermore, the 3-dimensional integral bordism group of (Y M ) [i] and its infinity prolongation (Y M )[i] + ∞ are trivial, under the full admissibility hypothesis: 19 In the particular case that P = ∅, i.e., ∂N 0 = ∂N 1 = ∅, then < α, N 0 >=< α, N 1 >.

So equation (Y M )[i]
⊂ JD 2 (i * W ) becomes a quantum 0-crystal super PDE and it admits global (smooth) solutions for any fixed time-like 3-dimensional (smooth) boundary conditions. Whether N 0 ∼ = N 1 , has as a consequence that, considering the analog boundary value problem in the observed PDE (Y M )[i] ⊂Ĵ 2 3 W , cannot exist an observed smooth solution V , representing an observed quantum non-linear propagator between N 0 and N 1 . The corresponding observed quantum nonlinear propagator must necessarily be singular one.

Some open problems in high energy physics solved
In this section we will use the geometric mathematical architecture of the algebraic topology of quantum super PDEs previously developed to answer to some important open problems in High Energy Physics.
• Let us, first, consider the following question: "Does exist a linear dependence between square mass and spin in a solution of the quantum PDE (Y M ) ?". 20 We shall see that in general such simple dependence for quantum solutions is not assured, but there is a very more complex relation between quantum mass and quantum spin. Let us recall our some previous results about quantum torsion. The quantum vierbein curvature R identifies, by means of the quantum vierbein θ a quantum fieldŜ : M → Hom Z (Λ 2 0 M ; T M ), that we call quantum torsion, associated toμ. In quantum coordinates one can write . Furthermore, with respect to a quantum relativistic frame i : N → M , the quantum torsionŜ identifies a A-valued (1, 2)-tensor field on N , S ≡ i * Ŝ : N → A⊗ R Λ 0 2 N ⊗ R T N , that we call quantum torsion of the observed solution. We say that an observed solution has a quantum spin, if the observed solution has an observed torsion (15) with S γ αβ (p) = − S γ βα (p) ∈ A, p ∈ N , that satisfies the following conditions, (quantum-spin-conditions): 20 This question arises from some semiclassical approaches that suggest to consider the following relation J ≃ α ′ M 2 , between angular momentum J and mass M , for a spherical rotating object. This is just the philosophy of so-called Regge trajectories. Such linear trajectories are used to interpret scattering amplitudes strong reactions and are usually seen as gluonic strings attached to quarks at the end points. However there are also more exotic points of view where instead are considered pion excitations of light hadrons. (See, e.g., [16].) whereψ is the velocity field on N of the time-like foliation representing the quantum relativistic frame on N . When conditions (16) are satisfied, we say that the solution considered admits a quantum spin-structure, with respect to the quantum relativistic frame. We call s the quantum 2-form spin of the observed solution. Let {ξ α } 0≤α≤3 be coordinates on N , adapted to the quantum relativistic frame. Then one has the following local representations: We define quantum spin-vector-field of the observed solution where ǫ µνλρ = |g|δ 0123 µνλρ is the completely antisymmetric tensor density on N . One has s ρ (p) ∈ A, p ∈ N . The classification of the observed solution on the ground of the spectrum of | s| 2 ≡ s ρ s ρ , and its (quantum helicity), i.e., component s z , is reported in Tab. 5. Table 5. Local quantum spectral-spin-classification of (Y M ) [i] solutions.

Definition
Name Finally the asymptotic stability study of global solutions of (Y M ), with respect to a quantum relativistic frame, can be performed by means of Theorem 2.46 in [82], since, for any section s : M → W , on the fibers ofÊ[s] → M there exists a non-degenerate scalar product. In fact,Ê[s] ∼ = W , as W is a vector bundle over M . Furthermore, for any section s, we can identify on M a non degenerate metric g, 21 that beside the rigid metric g on g, identifies a non-degenerate metric on each fiberÊ[s] p ∼ = W p = Hom Z (T p M ; g), ∀p ∈ M . In fact we get ξ(p) ·ξ(p) ′ = g KH g AB (p)ξ K A (p) ⊗ ξ ′H B (p) ∈ A. Solutions of (Y M ) that encode nuclear-charged plasmas, or nuclides, dynamics. These are described by solutions that, when observed by means of a quantum relativistic frame have at any t ∈ T , i.e., frame-proper time, compact sectional support B t ⊂ N . The global mass at the time t, i.e. the evaluation of such mass on the space-like section B t , gives the global mass-contents of the nuclear-plasmas or nuclides, in their ground-eigen-states, at the proper time t. Whether such solutions are asymptotically stable, then they interpret the meaning of stable nuclear-plasmas or nuclides.
Lemma 4.1. If a solution admits spin structure, then we get | S| 2 = | s| 2 . Furthermore in a frame-adapted coordinates we can write Then, by taking the pull-back with respect to the embedding i : N → M , identified by the quantum relativistic frame, we get | S| 2 = i * ( S S) = S γ αβ S αβ γ = µ γ K R K αβ R αβ K µK γ . Therefore, taking into account that in the case of solution with spin structure, one has S αβ γ =ψ γ s αβ and S γ αβ =ψ γ s αβ , we get (19). where β is the Laplace transform variable and it does not necessitate to be interpreted as the "inverse temperature", i.e., β = 1 κB θ , where κ B is the Boltzmann's constant. If β = 1 κB θ then the system encoded by the observed solution of (Y M ), is in equilibrium with a heat bath (canonical system). Note that all above objects are local functions on the space-time N . The same holds for β. We can interpret Z(β) as a normalization factor for the local probability density (23) P (E) = 1 Z N (E)e −βE that the system, encoded by the observed solution, should assume the local energy E, with degeneration N (E). In fact we have: As a by-product we get that the local average energy < E >≡ e can be written, by means of the partition function, in the following way: In fact, one has . When we can interpret β = 1 κB θ , then one can write (27) e = κ B θ 2 (∂θ. ln Z).
Then we get also that the local energy fluctuation is expressed by means of the variance of e: Furthermore, we get for the local heat capacity C v the following formula: We can define the local entropy by means of the following formula: In fact one can prove that one has the usual relation by means of the energy. Really we get: Then from the relation s = κ B (ln Z + βe) we get e = θs − κ B ln Z, hence also (∂s.e) = θ. This justifies the definition of entropy given in (30). Furthermore, from (31) we get also 1 β ln Z = e − θs = f , where f is the Helmoltz free energy. It follows the following expression of the local Helmoltz free energy, by means of the local partition function Z: (32) f ≡ e − θs = −κ B θ ln Z.
Conversely, from (32) it follows that the partition function can be expressed by means of the local Helmoltz free energy So we see that the local thermodynamic functions, can be expressed as scalar-valued differential operators on the fiber bundle W   Fig. 4 represent possible smooth integral manifolds, i.e., solutions, of (Y M ), or (Y M ) [i], with respect to a quantum relativistic frame i ?" Taking into account above results on (Y M ) and (Y M ) [i], we can answer, "yes", accepted that the initial and final Cauchy data have the same quantum numbers, identified by the quantum conservation laws of (Y M ) and (Y M ) [i] respectively. Furthermore, we can state that such solutions must necessarily be singular ones, since in order that should be smooth, it is necessary that V should be diffeomorphic to N 0 ×D 1|1 or N 0 × I. Then in such a case we should also have N 0 ∼ = N 1 . Therefore we can conclude that quantum nonlinear propagators representing reactions considered in Example 2.11 cannot be smooth solutions, but singular ones. Lemma 4.3. The quantum mass of a solution of (Y M ) can be written, in coordinates adapted to a quantum relativistic frame, in the form reported in (34).
Proof. The quantum Hamiltonian can be written in the form reported in (35).
This follows directly, by considering the expression for the quantum Hamiltonian, by using the splitting induced by the quantum superalgebra g, and by taking into account the condition that the solution admits a spin structure. The corresponding calculus is reported in (36) and following ones.
In coordinates adapted to the quantum relativistic frame, we get thatψ λ = δ λ 0 , and since µ λ . So, in general, we can write (39) R K αβ R αβ K = | S| 2 −∆ and by using equation (36) we get the expression (40) for the quantum Hamiltonian in terms of quantum torsion.
As a by-product, we get that the quantum mass of an observed solution by means of a quantum relativistic frame i : N → M , of (Y M ) [i], the alternative representation (34) in terms of quantum torsion.
Theorem 4.5. There is the following direct relation between reduced quantum mass and quantum torsion: Definition 4.6. Let us define phenomenological quantum spin of an observed solution by means of a quantum relativistic frame i : N → M , of (Y M ) [i], a A-valued quantum scalar J such that: Corollary 4.7 (Quantum Regge-type trajectories). There is the following direct relation between reduced quantum mass and phenomenological quantum spin: We call quantum Regge-type trajectories the relation (41) between quantum square mass M 2 and the phenomenological quantum spin J.
• Another question, in a sense related to the previous problem, is the dynamical justification of the Gell-Mann-Nishijima formula. This actually has not yet found Fig. 6. Representation of a quantum nonlinear propagator V , for a reaction a + b → c, and its quantum nonlinear anti-propagator V ′ , corresponding to the reaction b → a ′ + c, obtained for quantum crossing symmetry.
a dynamical derivation. 23 Really dynamical characterizations of hypercharge and 3 th isospin component, can be obtained by means of electric charge, by giving the dynamical expression of this last quantity with respect to a quantum relativistic frame. This is obtained in the following theorem.
Theorem 4.8 (Quantum Gell-Mann-Nishijima formula and electric-charge-gap-solutions). We define square-fundamental electric charge of an observed solution V of (Y M ) [i], the littlest value in the spectrum Sp(ŵ) of the observed quantum electromagnetic energyŵ, and denote it by q 2 . We say that V has an electric-charge gap if q 2 > 0. (In general q 2 ≥ 0.) The quantum electric charge,Q(t), of an observed solution V of (Y M ) [i], at the proper time t, is expressed by the formula (42).
This last is the 3-dimensional space-like sub-manifold of V at the time t. Furthermore, we call quantum hypercharge,Ŷ (t), (resp. quantum 3 th -isospin component, I 3 (t)), at the proper time t, the elements of the quantum (super)algebra A, such that holds the formula (43) (quantum Gell-Mann-Nishijima formula).
where △ is the definition time-set of the considered solution of (Y M ) [i]. 23 The more interesting effort in this direction was an heuristic semiclassical justification given in [15].
• The spectral content ofQ(t) is given by Sp(± Q (t) 2 ). • , is a partial differential relation that we call quantum electromagnetic-Goldstone boundary and denote by (Goldstone) [i] w . An electrically neutral connected, simply connected, Let us assume that N r , r = 0, 1, are connected, simply connected particles, hence N 0 has an electriccharge gap. The particle N 1 , instead cannot have electric-charge gap. Thus, V , by crossing (Goldstone) [i] w , loses its electric-charge gap, passing from N 0 to N 1 , and vice versa.
Assuming that the solution V has a quantum spin, one has Ê = 0. (See [82].) Then the quantum electro-magnetic-charge energy of the solution of (Y M ) [i] is given by equation (44).
. Therefore the quantum electromagnetic energy of the space-like set σ t is given by the expression on the right in (42). We define quantum electric charge contained, at the proper time t, into a space-like set σ t ⊂ V ,Q(t) ∈ A, such thatQ 2 (t) = σtŵ ⊗ η. 24 Then equation (43) gives a dynamical definition forŶ andÎ 3 . To prove the last part of the theorem it is enough to consider the continuous mappingŵ : (Y M ) [i] → A, defined by means of equation (44). [i]. Then (Y M ) [i] w is an open quantum PDE of (Y M ) [i], hence retains the same formal properties of this last equation. The rest of the proof follows the same line of the ones of Theorem 3.28 in [82] about solutions with mass-gap. Let us only emphasize that a neutral particles N 0 ⊂ (Y M ) [i], with trivial topology, i.e., connected and simply connected, cannot admit 0 ∈ Sp(Q(t)), [i] w , one has λ > 0, for any λ ∈ Sp(ŵ). Furthermore, if the quantum nonlinear propagator V is such ∂V = N 0 ⊔ P ⊔ N 1 , assuming that N 1 ⊂ (Y M ) [i] w and that it has trivial topology, then N 1 cannot have electric-charge gap, hence V crossing (Goldstone) [i] w must necessarily lose electric-charge gap. and another one c ⊂ (Y M ) [i] w , such that <q, a >= − <q, b > = 0, where < q, a >= af ⊗ η = − bf ⊗ η, withf the function under integral in (42). Furthermore, let <q, c >= 0. Then in the integral bordism classes [a ⊔ b = N 0 ] and [N 1 ], one has <q, [a ⊔ b] >= 0 =<q, c >. Therefore, for a quantum nonlinear propagator V , such that ∂V = N 0 ⊔ P ⊔ N 1 of (Y M ) [i] one has < α, a > + < α, b >= − < α, P >, for α =q, since <q, P >= 0. This is surely down since P is a time-like manifold having 2-dimensional space-like sections σ t , hence <q, P >= From Theorem 4.8 and Remarks 4.9 it follows the following important theorem.
is a term that in general is not zero and that we call lost quantum electric-charge.
We call Q-exotic quantum nonlinear propagators, quantum nonlinear propagators such that Proof. The proof follows the same strategy of one considered to prove Theorem 3.20 in part I [88]. In fact the gauge invariance of equation (Y M ) [i] produces a quantum integral characteristic 3-form, in the sense of Lemma 3.19 in part I, that has a structure similar to ω H . Let us denote such a conservation law by with dξ i , absent, i = 1, 2, 3, and (ω q ) α (p) ∈ A, for p ∈ V . The quantum electric charge, on a space-like section σ t of V , is given by means of ω q , by the following equationQ Therefore we havê . We call Q[V ] ∈ A lost quantum electric-charge of the observed quantum nonlinear propagator V of (Y M ) [i]. In other words, in general one has d dtQ (t) = 0 ∈ A. Remark 4.11. In the Part III we will further characterize Q-exotic quantum nonlinear propagators as nonlinear effects of exotic quantum supergravity. 25 This agrees with the conservation of the observed quantum Hamiltonian. See Theorem 3.20 in [88]. In fact, the observed quantum electromagnetic energy is a form of quantum energy, even if, in general, it does not coincide with the observed quantum Hamiltonian.
• In the following we shall prove that the phenomenological crossing symmetry in particle reactions is well justified in the Prástaro's algebraic topology theory of quantum super PDE.
Theorem 4.12 (Quantum crossing symmetry). If (Y M ) admits a quantum nonlinear propagator V such that ∂V = N 0 P N 1 , with N 0 = a ⊔ b and N 1 = c ⊔ d ⊔ e, then there exists also a quantum nonlinear propagator V ′ such that where the primed symbols denote antiparticles. This property is called quantum crossing symmetry. Similarly there exist quantum nonlinear propagators between a ⊔ b ⊔ c ′ and d ⊔ e or between b and a ′ ⊔ c ⊔ d ⊔ e. 26 Proof. This follows from the general relation between quantum integral bordism groups and quantum integral characteristic conservation laws. (See [74].) In fact if there exists a quantum nonlinear propagator V such that ∂V = N 0 P N 1 , with N 0 = a⊔b and N 1 = c⊔d⊔e, then this means that . Since antiparticles reverse quantum integral characteristic numbers, it follows that for any quantum integral characteristic conservation law α of (Y M ), one has , hence there exists a quantum nonlinear propagator V ′ such that ∂V ′ = N ′ 0 P ′ N ′ 1 . Note also, that as by-product, we get that there exist a (3|3)-dimensional quantum integral supermanifold P ′ , such that ∂P ′ = ∂N ′ 1 ⊔ ∂N ′ 0 , similarly to what happens for P , namely ∂P = ∂N 0 ⊔ N 1 . This means that one has and . Similarly one proves existence of the other reactions obtained from crossing symmetry. For example to the quantum nonlinear propagator V , such that ∂V = (a ⊔ b) P c, corresponds for quantum crossing symmetry, the following quantum nonlinear propagator V ′ , such that ∂V ′ = b P ′ (a ′ ⊔ c), where P ′ is the anti-P . (See Fig. 6 for a representation of P and its anti-P .) There are cases where can exist more than only one anti-P .
• The following theorem answers to the question: "Do exist massive photons ?". Theorem 4.13 (Existence of quantum massive photons). The quantum super Yang-Mills equation (Y M ) admits solutions that starting from a Cauchy data N 0 = a ⊔ b, where a represents a quantum electron and b a quantum positron, bords N 1 , representing a quantum massive particle. Any annihilation e + + e − → γ + γ, must necessarily generate a quantum virtual massive photon, say γ m , before to produce a couple of massless photons γ. Conversely a quantum massive photon decays giving γ m → e + + e − . 26 The crossing symmetry is a well-known property in the reaction particles phenomenology.
These, besides a set of other symmetries and phenomenological conservation laws constitue the Holy Bible for particle-reaction physicists. (See Tab. 7.) In Tab. 7 are reported also two quantum reactions related by means of crossing symmetry. These are the neutron decay and neutrino detection.) Fig. 7. In the figure on the left it is represented a couple of zeromass photons (c ( Higgs)) and massive photons (c ′ ⊂ ( Higgs)). The quantum non-linear propagator V bording a couple of massless photons (yellow particles) with electron-positron (a ⊔ b) has a Goldstone piece (black quantum virtual particle). This is identified with a massive photon. In other words the annihilation of electron-positron generates a virtual massive photon before to produce a couple of massless photons. The quantum non-linear propagator V ′ bording the massive photon (black) with electron-positron (a ′ ⊔ b ′ ) has not a Goldstone piece, since it is completely inside ( Higgs) ⊂ (Y M ). This is a massive photon decay. In the figure on the right, it is represented a quantum nonlinear propagator, V , bording massive couple of antiparticles, and having a part outside (Higgs) (the grey region). In this case we can talk of two Goldstone pieces in V , i.e., there are two massive photons (black particles) related by V .
Proof. Let us first identify a quantum massive photon with a quantum massive particle γ m with decay into an electron-positron couple. We know that from Theorem 3.28 in [82] can exist a quantum propagator V ′ bording a ′ ⊔ b ′ with c ′ , where a ′ ⊔ b ′ is a couple electron-positron and c ′ is a massive particle, hence all contained in ( Higgs) ⊂ (Y M ). On the other hand, from the standard Compton scattering γ + e − → e − + γ we say, for crossing symmetry, that holds the following reaction (annihilation electron-positron): e + + e − → γ + γ. Theorem 3.28 in [82] assures the existence of a quantum nonlinear propagator V such that ∂V = N 0 ⊔ P ⊔ N 1 , where N 0 , is contained in the sub-equation ( Higgs) ⊂ (Y M ), and N 1 ( Higgs).
Therefore V , should cross the Goldstone boundary ( Goldstone) ⊂ (Y M ), hence V should contain a Goldstone piece. This proves that the annihilation electronpositron implies to pass across a quantum massive photon: (Note that whether it is permitted the reaction e + +e − → γ m , for crossing symmetry it is also permitted the decay γ m → e − + e + .) 27 In Fig. 7 are represented productions of quantum massless 27 Actually there are attempts to give experimental evidence to the existence of massive photons. (Visit (Thomas Jefferson National Accelerator Facility, Newport News, Virginia, USA), and [5].) Massive photons can be identified with neutral massive bosonic particles. In the particlezoo these could be identified with so-called vector bosons, like neutral ρ-meson and ϕ-meson and Z 0 -boson, all having spin 1.
photons and quantum massive photons with respect to the constraints ( Higgs) and ( Goldstone) in (Y M ). Whether e + and e − collide at high energy, their annihilation can be reversed into a massive couple of mesons D + ⊔ D − . In other words there exists a quantum nonlinear propagator V , bording e + ⊔e − with D + ⊔D − , such that ∂V = (e + ⊔ e − ) P (D + ⊔ D − ), where P is an integral quantum supermanifold, partially outside (Higgs), but yet contained into (Y M ). (See Fig. 7.) From above considerations, we see that the concept of massive photons can be generalized to more large set of particles.
Definition 4.14 (Quantum anti-particles and quantum massive photons). We say that two quantum, massive, electric charged particles a ⊔ a ′ ⊂ (Y M ) are quantum anti-particles, if they have the same mass and opposite quantum numbers. We call a-quantum massive photon a quantum massive uncharged particle, having a decay into a couple (a, a ′ ) of quantum massive, electric charged anti-particles. We denote such a quantum massive photon with the symbol γ a m , or simply γ m , if no confusion can arise. Therefore, we can write γ a m → a + a ′ . Theorem 4.15 (Existence of a-quantum massive photons). For any couple (a, a ′ ) of quantum massive, electric charged anti-particles, there exists a a-quantum massive photon γ a m . Proof. The proof can be obtained by analogy of the one for Theorem 4.13.
Example 4.16 (Existence of p-quantum massive photon). Since it is permitted the Compton scattering p + nγ → p + nγ, for suitable n ∈ N, it is also permitted for quantum crossing symmetry the following one p +p → 2nγ. Therefore, a virtual p-quantum massive photon γ p m necessarily exists. 28 Definition 4.17 (Quantum massive neutrinos). We say that a quantum, massive, quasi-particle with zero electric charge is a quantum massive neutrino, if it admits a decay into a couple (neutrino,anti-neutrino) of the same type. Proof. In fact, we can find a quantum nonlinear propagator V ⊂ (Y M ), such that ∂V = N 0 P N 1 , where N 0 = a ⊔ b ⊂ (Y M ), with (a, b) = (ν e ,ν e ) ⊂ (Higgs), and N 1 = c ⊔ d ⊂ (Higgs), with (c, d) ia couple of antiparticles in the sense of Definition 4.14. Then, necessarily V must cross (Goldstone), hence identifies a 28 From some experimental results it is well-known that the annihilation p+p produces photons γ, through intermediate reactions and decays coming from massive particles. (For example: p + p → 3π 0 , p +p → 2π 0 + η, p +p → π 0 + 2η, π 0 → 2γ, η → 2γ, Z 0 → e + + e − → 2γ.) Therefore, the virtual p-quantum massive photon γ p m has a very complex structure, made by a collection of massive particles, like mesons π and η, bosons like Z 0 and leptons e ± , all inside (Higgs). Note that in the virtual massive photon γ p m enters also the vector boson Z 0 that is more massive than proton. In 46) the dot-arrows and dash-arrows starting from W − are alternative to the full-arrows and between them. In all these cases there is an intermediate production of a quasi particle Z 0 or W ± . Such quasi-particles can be usually produced when a massive solution cross (Goldstone). Therefore from this point of view we should have a further argument to consider the usual neutrinos massless particles, according to the Gell-Mann's standard model. However, the quasi-particles W ± cannot be considered massive neutrinos, since they do not decay into a massless couple (neutrino,anti-neutrino). Instead Z 0 has all the properties to be considered a massive neutrino ! Example 4.19 (What dark matter is ?). Theorem 4.18 agrees with the experimental fact that the annihilation of the couple (ν e ,ν e ) produces a Z 0 massive boson. This last can again decay in the couple (ν e ,ν e ), or in a couple lepton-antilepton, or quark-antiquark, according to energy level considered. For example: In (47) dot-arrow and dash-arrow starting from Z 0 are alternative to the full-arrow and each other one. Then the Z boson can be considered a massive neutrino when decay into the couple (ν e ,ν e ). The quantum crossed symmetry reaction ν e +ν e → Z 0 , is an example showing how massless particles can cross (Glodstone), producing massive particles inside (Higgs). The reaction, e + + e − → Z 0 → ν e +ν e , obtained, by quantum cross symmetry, from the reaction in the middle line of (47), shows that the annihilation of the couple (e − , e + ), when it is enough energetic, produces a massive quasi-particle Z 0 , before to destroy its mass, crossing (Glodstone), and producing the couple of massless particles (ν e ,ν e ). This clarifies that the energetic level of an electric-charged, massive couple (particle,antiparticle) decides whether in a scattering will produce a massive photons or a massive neutrino. 29 Let us also emphasize that the so-called "dark matter" can be interpreted as massive virtual particles codified in Theorem 4.13, Theorem 4.15 and Theorem 4.18.
In other words "dark matter" can be considered a generic term to identify virtual massive particles produced when a solution of the quantum super Yang-Mills equation (Y M ), crosses the Goldstone boundary, coming inside the Higgs quantum super PDE, (Higgs) ⊂ (Y M ) contained into (Y M ). This interpretation for "dark matter" could theoretically support some recent experimental observations where positron fraction is stably increasing at high energy levels, that are just necessary to produce massive photons, massive neutrinos and a-quantum massive photons, according to the above quoted theorems. See [3] for recent experiments that should suggest existence of "dark matter" in the sense here specified. 30 • The following theorem answers to the question: "Do quarks are fundamental particles ?" Proof. The proof can be obtained easily by considering Theorem 4.13. In fact for any quark q, we can associate an anti-quarkq, so that (q,q) is a couple of massive, electric-charged anti-particles in the sense of Definition 4.14, hence there exist massive q-photons γ q . In other words there exist quantum nonlinear propagators allowing reactions q +q → n γ, n ∈ N. In these reaction quarks are transformed in particles that do not contain quarks, hence quarks are broken-down ! To similar conclusions one arrives by considering Theorem 4.18. In fact, by considering the quantum reactionq + q → Z 0 →ν e + ν e , obtained by means of quantum cross symmetry of the reaction in the bottom in (47), we see that the mass of the couple (q,q) completely desappears and appear particles that are not made by quarks. In other words quarks break-down.
• In the following we give a precise meaning to the concept of observed quantum time and to the so-called time-energy uncertainty principle. This is possible thanks to our geometric theory of quantum (super) PDE's. 31 29 A guess by B. M. Pontecorvo and V. N. Gribov [29,59], in order to explain the so-called mystery of the missing solar neutrinos, assumed that neutrinos have some oscillations that characterize their different types. But such oscillations require massive neutrinos at least for τ -neutrinos and µ-neutrinos. This fact does not agree with the standard model. On the other hand, from Theorem 4.18 we can understand that massive neutrinos can be identified with some energetic levels of the quasi-particle Z 0 , and that non-electron neutrinos can retain their usual property to be massless particles. 30 Production of virtual π + decaying into positrons and neutrinos, can be obtained by means massive photons identified with virtual states into γ-nucleons interactions. This is a new point of view looking to long-studied π + -photoproduction. (For general information on pions photoproduction see, e.g., [18,20,21] and references quoted therein.) 31 The exact interpretation of the time-energy uncertainty principle was remained an open problem, and not even universally accepted, (see, e.g., L. Landau). Really in order to directly apply the Heisenberg uncertainty principle to the (time,energy) couple, it is necessary to define the meaning of the time as a noncommutative variable, with respect to the energy. But the proper time of a quantum relativistic frame is a commutative variable ! (See [79].) (See also some attempts to  (N, g) is a pseudo-Riemannian 4dimensional manifold, endowed with a time-like flow, identifies on N a A-valued metric i * g : N → A S 0 2 N . We define quantum relativistic observed time, between two points p 1 , p 2 ∈ N , belonging to a same flow line of ψ, the A-valued quantum lengtht[p 1 , p 2 ] ∈ A, calculated with respect to i * g. (For more details see [79].) We call spectral content of the quantum relativistic observed time the spectrum Sp(t[p 1 , p 2 ]) oft[p 1 , p 2 ]. Proof. In fact the length of the arc considered in above proposition differs whether measured with respect to g or to i * g, even if one has the case A = R ! This can be seen by a direct calculus, by using coordinates {ξ α } 0≤α≤3 , with ξ 0 = t the proper time of the relativistic quantum frame ψ. Then the arc length between the points p 1 , p 2 belonging to the same flow line φ p1 , passing for p 1 and p 2 , is given by s[p 1 , p 2 ] = [p1,p2] ds = [t1,t2] g αβξ αξβ dt = [t1,t2] √ g 00 dt = t 2 − t 1 , assuming g 00 = 1. On the other hand we havet[p 1 , p 2 ] = [p1,p2] dŝ = [t1,t2] |γ * g (s) dt ∈ A, where γ ≡ φ p1 • i. (For more details see [79].) In the particular case that A = R, we have thatt[p 1 , p 2 ] ∈ R, as well as s[p 1 , p 2 ], but γ * g (s) is different from φ * p1 g. Theorem 4.23 (Quantum virtual anomaly-massive particles). A quantum virtual massive quasi-particle can have an observed mass, in the reaction where it appears, larger than the observed energy of incoming particles entering into the reaction. 32 However this does not really contradict the Einstein's conservation mass-energy equation.
Proof. The justification of this fact, that should appear contradict the Einstein's equation of mass-energy conservation E = mc 2 , is justified by the Heisenberg's uncertainty principle. (This is, really, not more a "principle", but a "theorem". See, [65].) In fact, adopting notation used in [65], the uncertainty relation for the commutator [quantum − time, quantum − energy], can be written as reported in equation (48).
Thus in quantum processes, localized in very short "observed quantum-time", as could happen in the case of virtual particles creation, one could have large masses, solve this problem by Dirac [17] and L. I. Mandelshtam and I. E. Tamm [44].) Actually the timeenergy uncertainty principle is justified by means adopting the Mandelshtam-Tamm's point of view. But that approach is not satisfactory, since it refers to a generic auxiliary non-commutative variable, say B, that has nothing to do with energy and time. Furthermore, in that description time remains a commutative variable, and one identifies the velocity dB dt with a finite ratio, i.e., independently from the initial energy-mass content. However, this argument does not contradict the Einstein's conservation mass-energy equation ! In fact, the concept of "quantum-time" is only an apparent time, different from the proper time of the quantum relativistic frame, and existing with respect to the interaction quantum-system-quantum-relativistic-frame. Instead the proper time of the observer, belongs to R, hence commutes with the hamiltonian. In other words with respect to this proper-time, the Heisenberg's uncertainty principle does not apply.
Definition 4.24 (Apparent quantum observed time in quantum reaction). We call (observed) apparent quantum time in a quantum reaction, σ 2 (t) entering in equation (48). An experimental way to use the concept of quantum time and apparent quantum time, is to estimate the type of interaction present in. In fact, by using the Heisenberg uncertainty relation △t △E ≥ , applied to the quantum time and the strengths of the interactions, it is possible to guess the type of interaction from the observed interaction time. In Tab. 6 such a correlation is reported. There are also reported Planck's constants and quarks masses for any related convenience.
• For example, if one observes a decay into an apparent quantum time σ 2 (t) ≈ 10 −23 s, one can state that it is a strong decay (or quantum-gravity decay) there. Table 6. Quantum time and strengths of the fundamental interactions. • An annihilation quark-antiquark, should be observed into an apparent quantum time σ 2 (t) ≈ 3.61 × 10 −27 , 1.097 × 10 −24 s. (Look the quark masses reported in Tab. 6. The first refers to t +t and the last to u +ū.) • Similarly an annihilation electro-positron, should be observed into an apparent quantum time σ 2 (t) ≈ 6.582 × 10 −22 s. (Look the electron mass reported in Tab. 6.) • In the following we characterize the concept of confinement for quantum systems.   • A solution V of ( Y M ) encodes a confined system if the spectrum Sp(H(q)) of the corresponding quantum Hamiltonian, has non-empty point spectrum: Sp(H(q)) p = ∅, for all q ∈ V . Then we say also that V is a confined quantum solution.
• When a solution V of ( Y M ) has non-empty point spectrum only in some subsets N of V , then we say that V encodes a partially confined system and that N encodes the deconfined subsystem. Then we say also that N is the quantum deconfined part of the solution V .
• Whether a solution V of ( Y M ) is such that the following two conditions are satisfied: (i) Sp(H(q)) p = ∅, for all q ∈ V ; (ii) Sp(H(q)) c = ∅, for all q ∈ V ; then we say that V encodes a confined system that can be deconfined. Then a solution V of ( Y M ) encodes a confined system (or a confined system that can be deconfined), iff ker △ (H(q)) ⊃ {0} ∈ A, ∀q ∈ V . Furthermore, there exists a quantum deconfined part N ⊂ V of V , iff there exist points q ∈ V , such that ker △ (H(q)) = {0} ∈ A.
• Let us conclude this paper by answering to the following question: "Does quantum Majorana neutrino exist ? ". 33 Definition 4.28 (Quantum Majorana neutrino). We define quantum Majorana neutrino a quantum massive, electrically neutral fermionic particle, its own antiparticle, identified with a Cauchy data of the quantum super Yang-Mills PDE (Y M ). (i) A complex quantum quasi-particle exists that is a quasi neutralino, and that we call quantum Majorana neutralino. This contains two Majorana neutrinos other than two Higgsinos and two supersymmetric partner of the couple (ν e ,ν e ). (ii) V is homeomorphic to a quantum (4|4)-superdisk with two attached superhandles. Fig. 8. Rapresentation of the relation between a propagator V and its supersymmetric partner V : 33 This question arises from the paper Symmetrical theory of electron and positron, published by E. Majorana [43], about a solution of the Dirac equation, where he first showed existence of a massive, electric-neutral, spin 1 2 , solution, that coincides with its anti-particle. This solution is now usually called Majorana neutrino. Existence of this particle should allow the so-called neutrinoless double beta decay of some nuclei. (See, e.g., this beautiful link http://thy.phy.bnl.gov/ vogelsan/GGS/Wilkerson.pdf.) Proof. Let V ⊂ (Y M ) be a quantum nonlinear propagator in (Y M ) such that (50) ∂V = N 0 ⊔ P ⊔ N 1 with N 0 = W + ⊔ W − and N 1 = Z 0 ⊔ Z 0 . This is possible since V represents the reaction W + +W − → Z 0 +Z 0 that splits in the intermediate decays W + → ν e +e + , W − →ν e + e − and reactions e + + e − → Z 0 , ν e +ν e → Z 0 . A representation by means of elementary bordisms is given in Fig. 9. Let us, now, consider the following lemmas.
j = 1, · · · ,s, fermionic. Let V be a quantum nonlinear propagator of (Y M ) such that ∂V = N 0 ⊔P ⊔N 1 , and ∂P = ∂N 0 ⊔∂N 1 . Then there exists a quantum nonlinear propagator V of (Y M ) such that ∂ V = N 0 ⊔ P ⊔ N 1 , and ∂ P = ∂ N 0 ⊔ ∂ N 1 , where (N 0 , N 0 ) and (N 1 , N 1 ) are couples of supersymmetric partners. We call V (resp. P ) the quantum supersymmetric partner of V (resp. P ). Therefore to any quantum reaction there corresponds a quantum supersymmetric partner reaction.
Proof. This follows from Lemma 4.31 and from the commutative diagram (51), holding for any quantum conservation law α of (Y M ).
(51) < α, N 0 > + < α, N 1 > − < α, P > < α, N 0 > + < α, N 1 > − < α, P > In Fig. 8 is rapresented the relation between a propagator V and its supersymmetric partner V :  (50), we see that its supersymmetric partner V , represents the following reaction W + + W − → Z 0 + Z 0 . Here Z 0 is just a massive neutral fermionic particle, its own antiparticle. Thus Z 0 can be identified with a Majorana neutrino. 36 Therefore, we can conclude that Majorana neutrinos can be identified with possible products of quantum reactions in (Y M ), and the proof of the first part of the theorem is down. In order to prove (i), let us emphasize that the quantum nonlinear propagator V , identifies also a complex quasi-particle, that we call quantum Majorana-neutralino, ν Maj−neutralino ≡ (g, h, i, l, m, n). Here i and m, are the supersymmetric partners of massive neutrinos, ν m ,ν m , identified in Theorem 4.18, and m and n, are supersymmetric partners of Higgs quasi particles, (say Higgsinos), according to Theorem 4.13 and Theorem 4.15. In other words the quantum super-Majorana-neutrino is a quasi neutralino. This last has two photinos (the supersymmetric partner of photons) instead of the couple ( ν e , ν e ). So it appears that the name is justified ! In order to complete the proof, i.e., to prove proposition (ii), let us remark that ∂ V belongs to the same singular integral bordism class of a quantum exotic homotopy (3|3)-supersphere,Σ 3|3 , since Ω

(Y M)
3|3,s = 0. However, V is not homeomorphic tô D 4|4 , but to a (4|4)-superdisk with two attached super-handles. This means that the corresponding observed solution by means of a quantum relativistic frame i : N → M , is necessarily a singular solution of (Y M ) [i]. 37 36 In the particle-zoo this is usually called zino. 37 Let us emphasize that the quantum nonlinear propagators V and V , considered in this proof are not elementary ones in the sense of Theorem 3.2. In fact V and V are not homeomorphic tô D 4|4 . Fig. 9. Quantum nonlinear propagator representing a reaction generating Majorana neutrinos, identified with supersymmetric partners of the vector boson Z 0 . In the figure one has put: a = W + , b = W − , c = ν e , d = e + , e = e − , f = ν e , h = Z 0 , g = Z 0 . Furthermore, i, l, m, n are the massive quasi-particles identified by means of the intersection of the quantum nonlinear propagator V with the Goldstone quantum partial differential relation (Golstone), (represented by a red-circle in the picture). The quantum nonlinear propagator V , source supersymmetric partner of V , can be similarly pictured.