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On interaction in extended particle model on (M4 x M4) x Z4

A (M4 x M4) x Z4 model, describing an extended particle composed of two local modes and represented by a field psi(x, xi ;z), is formulated in its most general form (x, xi ; z) belong to (M4 x M4) x Z4. The 'z' argument specifies whether the particle is observable, unobservable, or partially observable (the latter case appears in two forms). In this four-sheeted structure, each sheet posses its own symmetry localized with respect to both space-times M4 inducing thereby connections in the continuous directions. Connections in the discrete direction describe transitions between observable, unobservable, and partially observable states. Curvatures and propagators are determined.


Introduction
The present work is a reconsideration of a previous one, [1] aiming at a description of conversion of external modes into internal ones, and vice-versa, in a geometro-differential conception of extended particles.
Initially, [2] this conception was based on a fiber bundle E D (M, H D , U D (G ′ )). The base space M is ordinary space-time which may have a G symmetry or be curved. The typical fiber H D is a Hilbert space carrying an induced representation [3] U D (G ′ ) of the internal symmetry group G ′ . The particle extension stems from the fact that the latter is no more represented by a point x ∈ M, but by a function Ψ ∈ H D which depends on another spatiotemporal variable ξ with the G ′ symmetry. The function Ψ is not the probabilistic wave function but describes the intrinsic properties of the particle (of which x is a partial representation). [4] The role of probability amplitude, played by the quantum mechanics wave function ψ (x), is guaranteed by a functional [4] X [Ψ, t] in this conception. The function Ψ x (ξ) must be treated according to some realistic model, we assumed that it describes a quantum mode localized at ξ for a particle localized at x (from a partial standpoint). The particle is composed then of two modes. [2] The probabilistic functional X[Ψ](x, ξ) = ψ(x, ξ) has been chosen as a bilocal function representing an external quantum mode localized at x and an internal one localized at ξ. This function was quantized by applying the induced representation method to both the external and internal symmetries. When interaction is absent, the induced representation method leads to a propagator which is a product of the propagators of each local mode. The external mode propagation is determined by a transition from external configuration induced representation to the external momentum one and back to the external configuration representation. Internal mode propagation is realized in the internal spaces. If an interaction is represented by a gauge field (a connection in the fiber), the semigroup induced representations lead to a path integral propagator. [5,6] The assumption that the external mode may transit via internal momentum space and the generalization of this idea to the possibility of transitions between external and internal representations, called mode conversions, led to new physical interpretations and ideas. However, mathematical expressions were deduced by analogy with induced representation results leading thereby to some inconsistencies. [1] The purpose of the present work is to overcome the latter inconsistencies by abandoning the fiber bundle structure in favor of an (M 4 × M 4 ) ⊗ Z 4 , where M 4 is Minkowski space and Z 4 is a discrete space with four elements. The inducing method is applied then between symmetries of the same type only, and connection in the continuous directions is taken into account. Transitions between symmetries of different (external or internal) types is realized by means of connections in the discrete direction.
The idea of discrete structure is drawn from works of Konisi [7] and Kubo [8] who associated connections in the discrete directions with Higgs fields without recourse to noncommutative geometry (NCG). For the sake of definiteness, we mention that our work is neither to be compared with that of Konisi and Kubo nor with those based on NCG. We have just used the discrete structure to provide our idea of conversion with mathematical consistency.
In Sec. 2, the state spaces structure, the connections, and the physical interpretations are presented. In Sec. 3, curvatures are calculated following Kubo's work. [8] In Sec. 4, propagators containing both types of connections are deduced and Sec. 5 is devoted to the conclusion.

The structure and connections
To describe an extended particle composed of two local modes let us consider a (M 4 × M 4 ) ⊗ Z 4 structure, where M 4 is Minkowski space-time and Z 4 is the discrete space with four elements. States Ψ z of the particle belong to Hilbert spaces H z and are considered as physical wave functions in the sense of providing all the physical properties of the particle but not probabilities. The latter are provided by functionals X z [Ψ z ] (x, ξ) = ψ (x, ξ; z). [4] The variables x and ξ belong, respectively, to the first and second space-time and the variable z is an element of Z 4 taking values p for pure, c for crossed, e for external, and i for internal. Each case corresponds to a certain type of localizability of the extended particle composed of a first mode localized at x and a second mode localized at ξ. In the pure case, the first mode is localized in the external space-time and the second mode in the internal space-time. The crossed case is the reverse of the former. The external and internal cases mean that both modes are localized in external or internal space-time, respectively. In other words, the localizability type z attributes a fixed physical meaning to each space in the product (M 4 × M 4 ) as being external or internal space-time. It endows thereby the extended particle with the property of being completely observable as a bilocal object in external space-time (external case), partially observable as a local object in each space-time (pure and crossed cases), or unobservable (internal case).
In the present work, we associate to each type of localizability z a symmetry group G (z) with elements We can assume that where T a and θ a are, respectively, generators and parameters of transformations related with the first space-time. In the same way, T α and θ α are generators and parameters of transformations related with the second spacetime.
These gauge transformations can be either spatiotemporal or unitary and induce a connection corresponding to parallel transport in the continuous directions, i.e. in one or both space-times. In fact, in the covariant derivative [8] the parallel transported field ψ || (x + δx, ξ + δξ; z), from a location (x, ξ) to another location (x + δx, ξ + δξ), can be written The Lie algebra valued connection one-forms corresponding to gauge transformations in each space-time are written in terms of gauge potentials A (x, ξ; z): The covariant derivative takes then the following form In the same manner, a covariant difference can be defined in the discrete direction. Parallel transport is a transition from one type of localizability (say z ′ ) to another (z) Covariant difference is then written as follows but the H (x, ξ; z, z ′ ) field has not a conventional expression in terms of one forms and gauge potentials. It can be interpreted as a transition operator from one type of localizability to another. It corresponds then to a conversion of internal modes to external ones and vice-versa. This conversion can be viewed as a creation of the particle when it passes from an unobservable state to a partially or completely observable one. It is viewed as an annihilation in the inverse transitions. It is then natural to define the conjugate of such a conversion by the following relation

Curvatures
Now, we define and calculate different types of curvatures stemming from the structure considered in this work. The first type of curvature corresponds to parallel transport along closed paths in the continuous direction and is given by the well known strength Subscript brackets [, ] indicates an antisymmetrization over the indices and ordinary ones are commutator of connection forms. If parallel transport takes place in the first space-time, the curvature takes the following form For a path in the second space-time, the curvature is and a closed path lying in the two spaces corresponds to the following curvature If the symmetry groups of each space-time commute, the latter curvature becomes and, if in addition each connection depends only on its corresponding spacetime variable, this curvature vanishes identically. The second type of curvature is concerned with a combination of a parallel transport in the continuous direction with a parallel transport in the discrete direction, Fig.(1).
The curvature is defined as a difference between two paths C 1 and C 2 where We have where It is clear that if parallel transport in the continuous direction concerns one space-time, only the corresponding curvature has to be considered. Parallel transport of the third type curvature links two points in the discrete direction only, Fig. (2). Then For purely discrete curvatures, we adopt the following notation. The initial point z in the diagram is considered as an argument, the intermediate point z ′ as an index, and the end point z as an index between parenthesis. There are also parallel transports linking three and four points in the discrete directions depicted in Figs. (3) and (4), respectively.
They give the curvature of the third type which has the following form and the curvature of the fourth type which has an analogous expression (24) is compatible with (25) since H (x, ξ; z, z) = 1 and that (26) can be derived from (25) The latter relation shows antisymmetry with respect to z ′ and z ′′ . Moreover, it is easy to show that where the dot (•) is to be replaced by the adequate arguments of the purely discrete curvatures. Consequently, curvature (24) is hermitic. All our curvatures are analogous to those of local theories [7,8] except the continuous curvature which contains extra terms due to extension.

Free case
The free case has already been studied [1] by considering a field ψ (x, ξ) with an external space-time variable x, an internal space-time variable ξ, and without the z variable (i.e. on M 4 × M 4 with z = p). The latter field is quantized by applying the method of induced representations to the symmetries of both space-times independently. For that purpose, the momentum representation with functionals ϕ (v, ζ) was considered. According to the induced representation scheme, transitions from a localized state (configuration representation) to another take place via momentum states; the transition from configuration space to momentum space has been called a materialization K, the inverse transition is a localization I. Consideration of all materializations and all localizations between external and internal spaces can be represented by the following diagram for the two modes: In the beginning [2], we considered the case the z = p where the first line of the diagram contained external spaces only (α = β = γ = external), and the second line contained internal spaces only (α ′ = β ′ = γ ′ = internal). The masses m β and µ β ′ take values µ or m depending on whether the momentum space is internal or external, respectively. Afterwards, we have assumed that the first external mode may materialize or localize in the internal space, and vice versa for the second mode [1]. Consequently, the indices α, α ′ , β, β ′ , γ, γ ′ can take any of the values external or internal, independently, so that we have 64 possibilities. A propagation corresponding to a fixed combination of the values of the six indices has been deduced by the method of induced representations [1] ψ and has been written as product of two point-like propagations The point-like propagator has the following form [3] Π where the velocity v β belongs to the positive sheet C 1 of the unit mass hyperboloid with invariant measure dv. The signs (+) and (−) are explicitly shown here and correspond to modes and antimodes, respectively. An identical expression holds for Π µ β ′ (ξ γ ′ , ξ α ′ ) with mass µ β ′ and velocity ζ β ′ . The product form of the extended particle propagator was derived by an analogy with induced representation. This is not consistent since it describes transitions between external and internal spaces of one mode with no interaction. Moreover, these same crossed transitions makes the first fiber bundle structure inadequate [1]. Using the (M 4 × M 4 ) ⊗ Z 4 structure, we can get rid of these problems. We begin by considering the discrete connection first.

Discrete connection case
In the first of our previous works, [2] interaction has been considered in the inducing scheme with no crossed transitions. We have shown that the localization, materialization and propagation processes are locally affected by parallel transport operators containing the interaction. The latter operators are path ordered exponentials, the analogues of the H mappings in continuous directions. And since we have already remarked that the operator H (x, ξ; z, z ′ ) realizing a discrete transition from z ′ to z is a parallel transport operator, let us use that recipe in the context of the present work. Materialization of a type z ′ localized state into a type z real state can then be written as Materialization operators K z are products of materializations of the first and second modes. Consider, for instance, the case where we have a pure localized initial state z ′ = p and a crossed material final state z = c, the materialization is then The rule is the following: mass and spin (0 here) parameters are determined by the real state. In the crossed state ϕ (v, ζ, c), the first mode is internal and the second external. Hence, the first operator K µ0 carries the internal mass µ and intertwines representations of the first group. The second operator K m0 carries the external mass m and concerns the second group. In integral form, we have This relation can be written in the general case where m z and µ z are masses of the first and second modes, respectively. The nature z of the real state specifies these masses (see Table 1 after the conclusion). The mappings H have been defined in the configuration representation, but it is clear that analogous mappings H exist in the momentum representation on (C 1 ⊗ C 1 ) ⊗ Z 4 . Consequently, we can write relation (32) in the form In this case the integral form becomes Compatibility between the two expressions imposes the following relation between materializations on the sheet z : In the same way, localization of a real state z into a localized state z ′ , is written as The integral form is If we begin by a transition from sheet z to sheet z ′ in the momentum representation before realizing the localization, we get and The compatibility relation being then According to the inducing method, the propagation operator is obtained by composing a materialization and a localization. The new processes of materialization and localization (containing transitions from a sheet of M 4 ⊗ M 4 ⊗ Z 4 to another) are combined in the same way. Because of the compatibility relations, the propagator can be deuced in several equivalent ways: and corresponds to a transition from a type z ′ localized state to a type z ′′ localized state via a type z real state. The kernel of the propagation corresponding to the first expression of the propagator Π z (z ′′ , z ′ ) has then the following form The three other expressions are analogous with adequate mappings H.
For particles with non vanishing spin, the propagator is In the D (v L ) I ± jz and K ± jz D v −1 L operators, corresponding to the first momentum space in (C 1 ⊗ C 1 ) ⊗ Z 4 , the D matrix is a representation of the Lorentz subgroup, v L is the velocity boost, and the operators I ± jz and K ± jz are constant matrices verifying K ± jz I ± jz = 1 and whose dimension depends on the spin j z (I ± 0 = K ± 0 = 1). Quantities D (ζ L ) I ± σz and K ± σz D ζ −1 L corresponding to the second momentum space commute with those of the first space. Let us denote the external mass and spin by (m, j) and the internal ones by (µ, σ), we then get masses and spins in the expressions of materialization, localization, and propagation with a momentum state ϕ (v, ζ, z) as shown in Table 1 after the conclusion.

Discrete and continuous connection case
We first consider the case of one sheet with fixed z. The propagator has already been deduced for this case (which is equivalent to (M 4 × M 4 )) by means of trajectory semigroups induced representations in each space-time [6]. Use of trajectory semigroups is imposed as far as the continuous connection is to be taken into account in the inducing method of quantization.
A trajectory in M 4 is a class of parallel curves x(l). It is represented by an element where x ′ = x(s ′ ) and x = x(s) are the initial and final points of the curve x (l), respectively. Right action of trajectories on space-time points is defined in the following way iω i (x, ξ; z) dx i + iω µ (x, ξ; z) dξ µ (55) In our previous works, iω i (x, ξ; z) dx i has been denoted Γ i (x) dx i whereas iω µ (x, ξ; z) dξ µ was not considered since we used a gauging of the internal symmetry with respect to the external space-time only. The situation is quite different now, we have a symmetry of two space-times gauged with respect to both.
The one sheet propagation operator of functions ψ (x, ξ, z) is a path integral expression [6] (θ is the step function)

Conclusion
The (M 4 × M 4 )⊗Z n structure seemed interesting in interpreting the geometrical origin of Higgs fields without recourse to noncommutative geometry. [7,8] The present work reveals another aspect of this structure, which is not concerned with the Higgs phenomenon. It opens the way to the construction of a theory of extended particles interacting with gauge fields and reaches the determination of a path integral form of the propagators. This work explains, in a manner analogous to the Dirac sea for fermions, the creation of particles by admitting their prior existence in unobservable states and the possibility of their transition to observable ones. The (M 4 × M 4 ) ⊗ Z 4 interest is that it provides the mathematical objects representing these transitions, namely the discrete connections H (x, ξ; z, z ′ ). Moreover, the theory allows consideration of gauge fields corresponding to a symmetry localized not only with respect to one space-time (generally, external space-time) but in both space-times.
Symmetries, connections, curvatures, and propagators have been presented in their most general form. Propagators incorporate conversions and effects of gauge fields in the continuous directions. Hence, the particle evolves in space-time and this evolution may be accompanied by its annihilation, its creation, or a transformation of those of its properties which are coupled to the continuous gauge field.
The following step is the adoption of specific physical models and the derivation of equation of motion for ψ (x, ξ; z), A (x, ξ; z), and H (x, ξ; z, z ′ ). The study of this question has already been initiated by the determination of curvatures which may lead to a Lagrangian formulation. However, this is to be carefully analyzed since the validity of a bilocal Lagrangian theory is not well established. z = p (pure) c (crossed) e (external) i (internal) mode 1: (m z , j z ) = (m, j) (µ, σ) (m, j) (µ ′ , σ ′ ) mode 2: (µ z , σ z ) = (µ, σ) (m, j) (m ′ , j ′ ) (µ, σ)