Particle-antiparticle duality from an extra time-like dimension

It is a well known fact that the usual complex structure on the real Clifford Algebra (CA) of Minkowski spacetime can be obtained by adding an extra time-like dimension, instead of the usual complexification of the algebra. In this article we explore the consequences of this approach and reinterpret known results in this new context. We observe that particles and antiparticles at rest can be interpreted as eigenstates of the generator of rotations in the plane formed by the two time-like coordinates. We also find an effective finite scale for the extra dimension when no EM fields are present, whithout postulating compacity. In the case of non-vanishing EM fields, we find a gauge condition to preserve such a scale.


INTRODUCTION
Since the theory of Kaluza-Klein was proposed [9], the idea of a five dimensional spacetime has been widely explored in physics. The Kaluza-Klein theory allows the unification of the Einstein field equations and Maxwell electromagnetism by considering a 5D manifold with a compact fifth coordinate.
In the early nineties Wesson along with collaborators proposed the existence of a non compact extra dimension and presents the theory of space-time-matter (STM) or induced matter theory (IMT) [6], as a way to induce 4D Einstein's equations from a 5D Ricci-flat manifold, making use of the Campbell-Magaard theorem.
Following the spirit of IMT, applications to particle physics and quantum mechanics have also been considered [1]. In these regard, physical quantities (charge, mass) are obtained as geometric parameters from a higher-dimensional spacetime. Since one is adding an extra dimension to the spacetime, the nature of it (space-like or time-like) must be taken into account. Spacetimes with time-like extra dimension (generally refered to as two-times spacetimes, e.g. [10]), and space-like extra dimension (e.g. [12]), has been considered with different applications in cosmology and particle physics.
In this article we are interested in analyzing the Dirac theory and the consequences of an extra time-like dimension in spacetime from a Clifford algebra point of view. We shall also provide an interpretation for what we call the time plane and relate it to the particle/antiparticle character of a Dirac spinor field.
Theories with more than two time-like dimensions has been proposed [7], and also plenty of work has been done on spaces of signature (+ − − − +) in the context of anti de Sitter spaces [8].
Previous studies has been made in 5D spinors [11][12][13][14], but in this article we focus more on the nature of the extra dimension and the structure of the Clifford algebra without studying gravitational cases. A central part of the Dirac theory, which sometimes is not thoroughly studied, is the Clifford algebra (or geometric algebra) of spacetime [2][3][4]. Starting from this algebra, the spinors are defined as representation spaces for it, and then we can talk about the Dirac equation.
Interestingly, an extra dimension naturally emerges on the Dirac theory if we take into account the following fact: the ordinary Minkowski complex Clifford algebra can be obtained as a real algebra with an extra time-like dimension [2,3], as will be detailed in the article. This is a very well known fact in the theory of Clifford algebras. Indeed, we can not distinguish (except for grading) between the complexified 4D Minkowski Clifford Algebra and the real 5D Clifford algebra, they are isomorphic.
In this work we consider the existence of a time-like extra dimension, we analyze its Clifford algebra and Spin group. Then, we obtain the related massive 4D Dirac equation (within and without an electromagnetic potential) from a massless 5D Dirac equation. We see that the particles and antiparticles rest solutions for the Dirac field, can be seen as eigenvectors for the generator of a rotation in the defined time plane. We also see that, for the case with an electromagnetic field in addition to the time plane rotation, we also have a gauge transformation on the spinor. Although we don't ask for the extra dimension to be compact, in the absense of electromagnetic fields, we obtain an effective scale on it. In the presence of the EM field we obtain a gauge condition to keep the scale.

Definitions and general results
Given a real n-dimensional vector space V with a bilinear symmetric form , is the associated quadratic form and we call the pair (V, Φ) a quadratic space. If the form ϕ is non-degenerate we say the quadratic space is regular. Since ϕ(a, b) = , one doesn't lost information when passing from ϕ to Φ.
An important subgroup of the group of linear isomorphisms on this vector space is the group of isommetries that we shall denote by O(Φ), and define by An important subgroup of this group is that composed by positive determinant transformations, SO(Φ): For any quadratic space, one can build an associative unitary real algebra Cl(Φ), called the Clifford algebra (CA) for (V, Φ). It is possible to define that algebra in different equivalent ways. Here, we shall do it as follows: Let {e 1 , .., e n } be a basis for the vector space V , and ϕ ij the matrix elements of the bilinear form ϕ in the given basis. The CA is defined by the generators {E 1 , ..., E n }, with the relations: where 1 is the unit in Cl(Φ). We shall just mention that this algebra can also be constructed as a quotient algebra of the tensor algebra modulo certain ideal [4], but we shall not go further in this subject. Since the tensor algebra is generally considered a real algebra, the CA also happens to be a real algebra. Because there is an injective function from V to the CA, via e i ↦ E i , by abuse of notation we shall refer to the generators of this algebra as e i . In the same way, we shall refer to the subspace span R {E 1 , ..., E n } ⊆ Cl(Φ), as V . It happens that this algebra is finite dimensional with dimension 2 n [3].
We have that a basis for this algebra is the set: Using this fact, we will say that an element of the form ∑ i<j A ij e i e j is a bivector or 2-vector and in general a k-vector is an element of the form: and a 0-vector is an element proportional to 1. We name ⋀ k (V ) or just ⋀ k the vector space of k − vectors, which means ⋀ 1 = V . It can be seen that dim(⋀ k ) = n k . Hence there is just one independent n-vector and also one independent 0-vector for every quadratic space. The unitary n-vector is also called the pseudoscalar of the algebra.
We define the grade involution α on a basis element e i 1 ...e i k as: α(e i 1 ...e i k ) = (−1) k e i 1 ...e i 2 , and extend it to any element as an algebra homomorphism on Cl(Φ).
It can be proven that this morphism induces a Z 2 grading in the algebra, splitting it into We define reversion, t, on a basis element e i 1 ...e 1 k as t(e i 1 ...e 1 k ) = e i k ...e i 1 , and extend it as an algebra anti-morphism (meaning t(a.b) = t(b).t(a), details can be found in [2,4]).
Using the previous functions we define the Clifford conjugation on any CA element x as x = (t ○ α)(x) = (α ○ t)(x). This is an algebra anti-morphism and in a basis element e i 1 ...e i k it can be seen to be e i 1 ...e i k = (−1) k e i k ...e i 1 .
Last we define the norm N (x) of an element x in the CA as N (x) ∶= xx = xx. An important feature of this function is that Although the algebra is constructed as a real algebra, from a real quadratic space, in ocassions we need to work with the complexified version of the algebra. Roughly speaking complexifying the algebra consists in allowing scalars in the linear combinations to be complex, transforming an R-algebra into a C-algebra. Mathematically this is attained by building the algebra Cl C (Φ) ∶= C ⊗ Cl(Φ) with the product trivially defined.
It can be proven that any regular quadratic space admits an orthogonal basis, in the following sense: For every regular quadratic space there's a basis {e 1 , ..., e p , e p+1 , ..., e p+q } such that: with p + q = n. We say that (p, q) or {+, +, ..., We will denote the quadratic form of signature (p, q) as Φ p,q and its CA as Cl p,q .

The Clifford-Lipschitz group and its Lie algebra
In the CA there are certain groups with special properties. All of them are subgroups of the group of units, Cl(Φ) * , of the given Clifford Algebra. These groups are closely related to the isommetries of the quadratic space and to the well known adjoint action of the CA. In this section we will briefly define the Clifford-Lipschitz group and also give its Lie algebra without proof. Detailed calculations can be found in [2]. Let x ∈ Cl(Φ) * , then there exists x −1 such that 1 = xx −1 = x −1 x. The Clifford-Lipschitz group Γ(Φ) is defined as follows: Here we are using the injection V ↪ Cl(Φ) from section 2.1 for xvx −1 to be well defined.
As is explained in [2] the Lie algebra of this Lie group is the set: with Z(Cl(Φ)) the center of the CA: and the Lie algebra bracket being the commutator in the CA, [x, y] = xy − yx.
As in the previous section, we will call Γ p,q the Clifford-Lipschitz group of the bilinear forms considered in (2.3).
Since we are dealing with Lie groups and given the discrepancies between the physics and mathematics literature, it is a good point to state what definition we are going to use of a generator of an element S of a Lie group G.
Recall that if G is a Lie group, then it is a manifold, and its Lie algebra g is defined as the tangent space of that manifold at the point 1 ∈ G. Given an element S ∈ G connected to the identity, there exists a path β ∶ [0, 1] → G, with β(0) = 1 and β(1) = S. We say that dβ(t) dt t=0 is the generator of S. Observe that the generators of elements in G belong to g the Lie algebra of G. It can be seen that this is a good definition.

The Spin group and its Lie algebras
The Spin group of a certain CA, Spin(Φ), is a subgroup of the Clifford-Lipschitz group. This group is defined as follows: (2.7) We will call Spin(p, q) the Spin groups of the bilinear forms defined in (2.3). An important property of this group is that it is a double cover of the isommetry group for the quadratic form Φ. This is stated in the following theorem: This surjective group homomorphism can be extended to the Clifford-Lipschitz group: ..e n (2.8) The function Ad is indeed a Lie group representation. Ad is called the adjoint representation or adjoint action of Spin(Φ) (or the Clifford-Lipschitz group).
It is well known that any real CA is isomoprhic to a finite-dimensional matrix algebra over R, C or H (or direct sum of these algebras), while any complex CA is isomoprphic to a finite-dimensional matrix algebra over C (or direct sum of these algebras) (see [2,3]).
In the framework of the Dirac theory one works with the matrix form of the CA, and with what is called the regular representation of the CA (that induces a representation of the Spin group). In this case the representation space is the space of spinors (actually algebraic spinors), which are the column vectors for which the CA matrices represent a linear transformation. For instance, if Cl(Φ) ≅ M(2, C) then, a spinor is an element in C 2 .
It happens that the space of spinors can be identified with a minimal left ideal in the CA. We will not enter in the details of this construction (see [2,3]) but this is a very well known fact in the CA theory, and indeed would be a more elegant way to introduce spinors; however for sake of simplicity we content ourselves with the definition given above.
Let Ψ be a spinor, then an element in A ∈ Cl(Φ) acts on Ψ just by left multplication using the matrix representation stated above, Ψ ↦ AΨ.
The theorem above is an important result because for every isommetry of the vector space V it provides an element (two actually) in the Spin group. If we further require the element in the Spin group to be connected to the identity, we get uniqueness. In this sense, the morphism ρ tells us how does a spinor change when we perform certain coordinate transformation in SO(Φ).
Reciprocally, if we decide to make a transformation in the space of spinors Ψ ↦ SΦ, with S ∈ Spin then, Ad gives us the corresponding transformation induced by it in V .
As will be clarified in the following sections, the vector space V will model the coordinate spacetime while while the space of spinors correspond to a spin 1 2 physical field. Hence, the relation between the adjoint representation and the regular one reflects the relationship between coordinate space-time and internal space for the configuration of the field. In this sense the elements in the kernel of Ad are somehow an internal symmetry of the theory of spinors.
It can be seen that the Lie algebra spin(Φ) of this Lie group is given by: and that by virtue of the previous theorem the following Lie algebra isomorphism holds: 3 Clifford algebras in physics and applications to the Dirac theory

The Minkowski real CA and its complexification from an extra dimension
As stated in the previous sections, the CA is associated to a particular quadratic space.
In the realm of physics we encounter this kind of spaces in different contexts. For instance in Newtonian mechanics, space is modelled as the quadratic space R 3 with the Euclidean quadratic form (which happens to be a norm), however, the case we will pay more attention in this article is that of special relativity. In this theory space-time is modelled as a Lorentzian 4-dimensional space. This is, the vector space R 4 , with coordinates (x 0 , x 1 , x 2 , x 3 ) (which we will generically refer to as x µ ) and with the Minkowski bilinear form given by the following components in the Cartesian basis: It is said that we adopt the signature {+ − −−} or (+ − −−) or 1, 3. We will call this quadratic regular space the Minkowski space-time, and refer to it as R 1,3 . Thus, the real CA corresponding to this spacetime, Cl 1,3 (R), will be the algebra generated by elements {e µ ∶ µ ∈ {0, ..., 3}} with the relation: According to the well known classification of CA, the real algebra for the Minkowski spacetime with this signature is isomorphic to the algebra of 2 × 2 matrices with entries in the quaternions.
In this article we are going to work with the Dirac theory of spinors. This theory is derived partially from classical quantum mechanics, which postulates the existence of a complex Hilbert space of physical states, hence we need to use the complex CA, Cl 1,3 (C). This algebra is well known to be isomorphic to the complex algebra of 4 × 4 matrices with complex entries, M(C, 4). There are infinite matrix representations for the generators as matrices in M(C, 4), but the more popular are perhaps the Dirac and Weyl representations.
We will pay special attention to the following known fact: the CA of R 1,3 can be "complexified" in an alternative way [3], which allow us to keep working with real Clifford algebras. The complexification is accomplished by adding an extra time-like dimension, x 4 , to the Minkowski spacetime and taking the real CA of the 5D spacetime with signature {+ − − − +}, R 2,3 . This is possible because the following isomorphisms hold: In this case, the imaginary unit in Cl 1,3 (C) is identified with the pseudoscalar e 0 e 1 e 2 e 3 e 4 in the 5D real algebra. This is possible because of two facts: on the one hand the pseudoscalar squares to −1 and on the other hand it lies in the center of the algebra, Z(Cl 2,3 ), this is, it commutes with every element in the CA (a necessary property for scalars). Indeed we have that Z(Cl 2,3 ) ≅ C.
It is important to be careful since, when working in Cl 2,3 (R), we shall use the name i, as referring to the pseudoscalar, but we are by no means complexifying Cl 2,3 (R). Reciprocally, when we complexify the theory in the usual way, this fifth dimension emerges naturally as the matrix element γ 5 , which squares to 1 and is associated with the chirality of the Dirac spinor fields. Since the isomorphism (3.3) holds, the representations of Cl 1,3 (C) and Cl 2,3 (R) are equivalent, and hence the spaces of spinors are isomorphic.
In this article we shall explore the consequences of considering this extra dimension as a real physical one, particularly in the subject related to the particle/antiparticle interpretation of the spinor field. Following the spirit of the theory of Induced Matter [5], we will obtain the mass 4D Dirac equation from a massless 5D Dirac equation.
Definition. We will call a vectorṼ µẽ µ , a tilded 4-vector, in opposition to an ordinary An interesting feature of this treatment is that the elements iẽ µ now belong to the vector space ⋀ 2 (R 2,3 ), which is the Lie algebra of the group Spin(2, 3) and hence, the vectors generating the algebra can also be seen as generators of certain coordinate transformations preserving the 5D metric tensor in the Spin group. Recall that, for instance, e 0 e 1 is the generator of a boost in the direction x 1 and e 1 e 2 is a generator of a rotation in the plane x 1 − x 2 . If we think transformations in a space of functions in R n , then ∂ ∂x k is the generator of a translation in the k direction.
Since iẽ µ = e 4 e µ ∈ ⋀ 2 (R 2,3 ), iẽ 0 is the generator of a rotation in the plane of the two time coordinates and the elements iẽ i ∈ ⋀ 2 (R 2,3 ), are the generators of Lorentz boosts with respect to the extra time and in the spatial direction x i . Furthermore, it can be easily seen thatẽ µẽν = e µ e ν , and hence rather [ẽ µ ,ẽ ν ] or [e µ , e ν ], can be considered as a generator or a Lorentz transformation in the 4D space (x 0 , x 1 , x 2 , x 3 ). Also, if we consider a Lorentz transformation in the 4D spacetime (x 0 , ..., x 3 ) with x ′ µ = Λ ν µ x ν , we know that there exists an element S ∈ Cl, such that Se µ S −1 = Λ ν µ e ν . Now, if we do the same computation on theẽ µ we have the same relation, namely, Sẽ µ S −1 = Λ ν µẽ ν . This is due to the fact that S = exp (ε αβ e α e β ), with ε αβ skew-symmetric. Hence S commutes with e 4 inẽ µ = −ie 4 e µ , and we arrive to the stated conclusion. This is important because it implies that, when restricted to the 4D space-time, a 4-vector V µ e µ transforms in the exact same way as the element V µẽ µ . Note that since iẽ 0 = e 4 e 0 commutes with e 1 e 2 , we can have a spinor that is simultaneously a spin eigenvector and an eigenvector for the 2-time plane rotation generator e 4 e 0 .

Dirac equation in Minkowski spacetime R 1,3 from R 2,3 for neutral matter
If we consider the four dimensional Minkowski spacetime R 1,3 . The Dirac equation in the complexified Clifford algebra Cl 1,3 (C) is: with Ψ a Dirac spinor, m the mass of the spinor field and i the complex imaginary unit. In this section we shall consider the Minkowski 5D spacetime with an extra time-like dimension, R 2,3 described in the previous section. An important thing to state about the coordinates is that, although x 0 and x 4 are time-like, all the coordinates have dimensions of length; hence x 0 = ct 0 and x 4 = ct 4 . Since x 0 and x 4 are both time-like, we will call the plane x 0 − x 4 the time-plane. Following the ideas of Induced Matter Theory (IMT) [1,5,6], a way to treat mass in the 4D spacetime is to induce it from a 5D spacetime, as a property of the particle motion in the fifth direction. In our case, we shall apply this to the Dirac equation. We shall propose a massless 5D Dirac equation in order to arrive to a massive 4D Dirac equation: The equation can be written in the following form: Using the elementsẽ µ defined in the previous section and assuming (3.8) and due to the fact the elementsẽ µ obey the rules for the Clifford algebra Cl 1,3 , we recover the familiar 4D Dirac equation. Let's solve the Dirac equation (3.8), subject to the condition ∂ i Ψ = 0 ∀i ∈ {1, 2, 3}. This is, the spinor field is homogeneous in space, but may change in time. Then the equation is written: Multiplying by e 0 e 4 on both sides we get:  With some care we will write where the left hand side must be interpreted as: taking the spinor Ψ as a function of x 0 , applying exp(α∂ 0 ) (with α an external parameter) and then substituting α by the value x 0 and x 0 = 0, in the initial state. This is important because the operator exp(x 0 ∂ 0 ), where x 0 is a variable, is NOT the translation of magnitude x 0 in the direction x 0 . Since we are to work with rest solutions (∂ k Ψ = 0), p k = 0 holds for every k in equation (3.11), and since the integral in the same equation is taken over p A p A = 0 we get that (p 0 ) 2 = m 2 c 2 , which implies p 0 = ±mc and we obtain for the full spinor: and for x 0 = 0: , and all of these numbers are complex constants. By virtue of equation (3.10), and due to the fact that [∂ 0 , e 0 e 4 ] = 0, we have: The quantity exp mcα ̵ h e 0 e 4 ∈ Cl 2,3 can be computed via the series expansion in equation (3.17). We know that (ie 0 e 4 ) 2 = 1, which implies Using the identities cosh(ix) = cos(x) and sinh(ix) = i sin(x), together with the fact that cosh is even and sinh is odd, we have In this fashion we obtain the "coordinate-time evolution" for the field from the initial value, using an element on the Spin group of the 5D Minkowski space.

Time evolution in coordinate space-time
The advantage of the treatment here is that the "time-evolution" operator represents also a coordinate transformation and not only an internal transformation of the spinor. Since this operator is the exponential of a real multiple of e 4 e 0 , it belongs to the real Lie group Spin(2, 3) and hence the adjoint action will transform a vector V A e A into another vector . Using this, a vector V A e A transforms according to This transformation is a rotation of angle 2mcα ̵ h on the plane of two times x 0 − x 4 . Note that if we take the projection of this 5-vector on the hyperplane orthogonal to the direction x 4 (the 4D space {x 4 } ⊥ ), V µ e µ and compute the adjoint action on it, and then project again on the same hyperplane we obtain . This is clearly not an isometry, since V ′µ V ′ µ − V µ V µ = sin 2 (2mcα ̵ h)V 0 V 0 , which in general is not zero. This is not a problem since the quantity that should be preserved by the Spin(2, 3) group is the 5D norm.

The rest solution and the particle-antiparticle character
Let us recall that in the usual CA Cl 1,3 , the generators e µ admit the Dirac matrix representation given by the gamma matrices. In particular the matrix γ 0 representing e 0 is written in the form which clearly means that the basis of spinors {u 1 , ..., u 4 } is a basis of eigenvectors for the matrix γ 0 . The interpretation of the Dirac spinor as a particle or antiparticle for the rest solution (p i = 0) is given precisely by the eigenvalue of γ 0 , associated to the eigenstate in consideration. If it's +1, it is a pure particle state and if it is −1, it is a pure antiparticle state. Now, the treatment given in this article is different. Here we have decided to add an extra time-like coordinate but to interprete the ordinary 4D Clifford algebra as a subalgebra of it. Hence we have the elementsẽ 0 = −ie 4 e 0 appearing in the Dirac equation, instead of e µ , and because of this fact, the particle/antiparticle interpretation of the rest solution should be given by the eigenvalues ofẽ 0 instead of e 0 alone. As was stated before, seeing the 4D CA as this particular subalgebra generated byẽ µ , has the advantage that ie µ lies in the Lie algebra of the Spin(2, 3) group and hence it is the generator of certain coordinate isometry. Due to the facts stated above, and in order to work in a spinor basis of pure particle/antiparticle rest states, we need to change our matrix representation: instead of taking e 0 to be diagonal and equal to γ 0 , we demandẽ 0 = −ie 4 e 0 = γ 0 . By consistency of the CA commutation relations, this implies also thatẽ µ = γ µ , for every µ ∈ {0, ..., 3}. Recall that e 4 = γ 5 , hence we haveẽ µ = γ µ = −iγ 5 e µ , and multiplying by iγ 5 on both sides, we have e µ = iγ 5 γ µ . With this, the new representation for the generators e A is given by where the matrices σ k are the well known Pauli matrices (with σ 2 i = 1). Sinceẽ 0 is diagonal, we have that the basis is a basis of eigenstates ofẽ 0 = −ie 4 e 0 with u 1 and u 2 having eigenvalue +1, and u 3 and u 4 having −1 as eigenvalue.
Let us use this fact for the rest solution (3.21), with the integral expansion (3.16) In the following, when writing u i we will be assuming i ∈ {1, 2} and when writing u j j ∈ {3, 4}. Applying the operator exp( −imcx 0 ̵ hẽ 0 ) on the u i and u j we have: (3.31) Putting this back in equation (3.21), and with (3.30), we have Hence, a full particle rest solution will be characterized by A j 0 = 0, and a full antiparticle rest solution by A i 0 = 0, as usual. We can see that in both cases the solution oscillates in ordinary time, while it's damped in the extra-time. Let us note, that we have solved only the rest case because it was sufficient for us and it simplified the calculus. However, one can build any solution in the series expansion (3.11) with p i ≠ 0 by just taking a representant S in the Spin group of the Lorentz transformation Λ µ ν that takes (mc, 0, 0, 0) to certain (p 0 , p 1 , p 2 , p 3 ), and then transforming the spinors u i into the states u ′ i = Su i .

The effective scale of the spinor field in the extra dimension
Since ∂ 4 Ψ = −mc ̵ hΨ holds for any spinor solution, all massive spinors decay exponentially in the extra dimension. We see that if we assume that the extra coordinate doesn't extend to −∞, then after some extra-time the field values get very small, the exponential damping constant being mc ̵ h, the inverse of the reduced Compton wavelength for the particle. Then for a given particle after a time of order ̵ h mc 2 =λ c the field is 90% smaller than initially. This would mean that even though we didn't propose the extra dimension to be bounded, the field would only extend over a finite region of it, of the order of the Compton wavelength (orλ c in units of time).

Charged Dirac fields with an electromagnetic field
The 5D massless Dirac equation has the global U (1) symmetry given by Ψ ↦ e iα Ψ, where α is a real number. QED introduces the electromagnetic field as the gauge field required to make this symmetry a local one. This is, the theory is invariant for Ψ ↦ e iα(x) Ψ, with α(x) a function on the coordinates of the space-time. Usually α is a function of the four variables of space-time, but since now our spacetime has five coordinates, let's consider α = α(x B ). This induces the gauge covariant derivative: with the transformation for the spinor and gauge field A B : where q is the charge of the field Ψ in consideration. In this way we get an electromagnetic 5-vector potential. Hence, we have the massless Dirac equation Proceeding as in the previous section we multiply on the left by e 4 and get iẽ µ D µ Ψ + D 4 Ψ = 0, (3.35) withẽ µ = −ie 4 e µ , as in the previous sections. In order for the 4D Dirac equation to be fulfilled, we require D 4 Ψ = − mc ̵ h Ψ and now the 4-electromagnetic potential vector in the 4D induced CA is the vector A µẽ µ , since is the quantity appearing in the Dirac equation. If we further require the solution to comply with D i Ψ = 0, then what is left is the equation Multiplying by −iẽ 0 = −e 4 e 0 = e 0 e 4 on both sides, and using D 0 Let's note that the condition we are asking for (D i Ψ = 0) doesn't make the field homogeneous. Hence now we have Ψ = Ψ(x 0 , x 1 , x 2 , x 3 , x 4 ).
In the previous section we went from equation ( In this case, we are in the conditions of neutral matter and by the same procedure we write: In the previous section we interpreted the right hand side of equation (3.21) as an equation in the algebra ⋀ 2 (R 2,3 ) which is the Lie algebra so(2, 3), however, we see now in equation (3.37) that the right hand member doesn't lie in ⋀ 2 (R 2,3 ) but rather in ⋀ 2 (R 2,3 ) ⊕ Z(Cl(2, 3)), which is the Lie algebra of the Clifford-Lipschitz group Γ 2,3 (cf. section 2.2). This group contains all the elements that under the adjoint action on the CA transform vectors into vectors. Indeed the adjoint action of exp(−i q ̵ h A 0 x 0 ) on every element of the CA is the identity, since this element sits on the center of the CA. This can be put as follows: Additionally to the coordinate transformation that we have in the absence of electromagnetic field, there is also an internal transformation, which has no coordinate consequences and that corresponds to the internal local gauge freedom of the theory. Note that this case contains every electrostatic potential A 0 (x 1 , x 2 , x 3 , x 4 ). If we further ask for the cyclic condition to hold [5] (potentials not depending on the 5th coordinate), we have A 0 = A 0 (x i ). Now, if we decide to use a basis of eigenvectors of e 0 e 4 for the spinor, as in section (3.4), we have that a +i eigenvector solution is and a −i eigenvector solution This is quite different from what we obtained in section 3.4, where the eigenvalue of e 0 e 4 alone determined the particle/antiparticle state. Now this interpretations has to change, since an antiparticle state has an opposite charge and, as we see in the equations above, solutions associated to different eigenvalues of e 0 e 4 have the same charge. It happens that the interpretation of a particle/antiparticle in the charged case has not only to do with the coordinate rotation in the time-plane, but also with the local gauge transformation of the spinor in (3.38). In the neutral matter case it was sufficient with changing the sense of a rotation in the time-plane, but now we also have to change the sense of "rotation" in the local gauge transformation, passing from exp . Of course, the antiparticle covariant derivative is now different, with the opposite sign charge, and in consequence the Dirac equation fulfilled is different. Note that when putting q = 0, we recover everything in section (3.4).
In this case we have: which makes a little harder to see the interpretation of the time evolution as a coordinate transformation. However, it is simple to solve directly the equation (3.37), and then interprete the time evolution accordingly. Combining (3.37) with D i Ψ = 0 and D 4 Ψ = −mc ̵ hΨ, we have the solution with ∂ B F = A B . If we assume the solution to be separable, then F is written as F (x A ) = F 0 (x 0 ) + F i (x i ) + F 4 (x 4 ) and consequently we can separate the dependence on x 0 , from the rest of the coordinates Therefore, we recover the interpretation given in the previous case: the time evolution is a rotation in the time plane, together with an internal local gauge transformation, which now is not linear in time. And also, the antiparticle solution is obtained by changing the sense of rotation in the time-plane and in the internal transformation. The same observations about particle and antiparticles for the case ∂ 0 A 0 = 0 can be made here.

Propagation in the extra dimension and 5D gauge fixing
Since D 4 Ψ = (∂ 4 + iq ̵ hA 4 )Ψ = −mc ̵ hΨ holds for the charged spinors, then and in addition to the exponential decay in the case of neutral matter, there is also a propagation in the extra dimension, if and only if, the 5th component of the vector potential is a non zero real number. In order to have the same confinement as in the non-charged case we could ask for the gauge condition A 4 = 0, and in this case the extra dimension has the same effective extension for both the charged and the neutral case. Let's note that this condition, would imply that the theory is invariant only under the local gauge transformation exp(iα), with ∂ 4 α = 0. This is, the field have the same phase along any line (x 0 , x i , x 4 = λ), with λ ranging over R. Note also that if in addition we ask for ∂ 4 A µ = 0 ∀µ then F AB F AB = F µν F µν and we recover the classical QED Lagrangian.

On ghosts and tachyons in this two-times physical theory
The typical problems in theories with more than one time-like dimension are the existence of tachyons and the existence of ghost fields (negative norm states), which difficult the process of quantization of the theory. [7] As exposed in [10], the momentum in the extra dimension is related to the rest mass of the particle, hence tachyons are not a problem since there's no superluminical propagation in the 4D spacetime.
Regarding ghost fields, if we assume the following Lagrangian density: then the corresponding 5-current associated to the phase shift symmetry is given by: and if we look at the 0th component, we get J 0 = iΨ † γ 0 e 0 Ψ = Ψ † γ 5 Ψ, which is not possitive definite. Indeed, none of the componets J A equal the probability density Ψ † Ψ. However, recall that we have defined the 4D induced CA, as the one having generators {ẽ µ }. In this context, the same equations of motion can be obtained from the following Lagrangian density: which, after imposing ∂ 4 Ψ = − mc ̵ h Ψ is written as: This is the usual Dirac Lagrangian, sinceẽ µ = γ µ , and hence we have the usual conserved 4-current:J µ = Ψ † γ 0 γ µ Ψ = Ψ † γ 0ẽµ Ψ. (4.5) In this way, the problem of ghosts is solved by consideringJ as the physical quantity, and not J. Of course, J A is a 5-vector, whileJ µ is a 4-vector. This means that when performing a Lorentz transformation involving the extra coordinate, in generalJ µ won't obbey a vector transformation rule, while J A will. Indeed, using this tilded current, the Dirac theory is completely unchanged.
Note that althoughJ µ is not obtained as the vector bilinear covariant of the field, it is certain projection of the 3-vector bilinear covariant of the 5D CA (since eachẽ µ is a 3-vector), hence, the information is contained within a bilinear covariant of the theory. This change from the, let's say, "vector current" to this "3-vector current" is related to the equivalence of Lagrangians (4.1) and (4.3) once we impose the constraint ∂ 4 Ψ = − mc ̵ h Ψ. This also implies that the 5 current J A is always conserved, whileJ µ is only conserved once we introduce the mentioned constraint.

Conclusions and prospects
In this article we have obtained the 4D Dirac equation for spinors from a 5D massless equation. In order to do so we have exploited the fact that the Dirac theory naturally comes equiped with an extra time like dimension. Hence, following the principles of induced matter theory [5,6] we have studied the Dirac equation in a 5D space-time of signature 2, 3. Analyzing the Dirac equation for neutral matter (or absence of EM fields) and the Clifford algebra structure of the 5D space-time, we have provided an interpretation for a particle/antiparticle states, as eigenstates of the generators of rotations in the time-plane. In the charged case (with the presence of an EM field), by performing a similar analysis we have seen that in addition to the rotation in the time-plane, there is an internal gauge transformation in two opposite directions that separates particles from antiparticles. In the absence of EM fields, the interpretation given in this article could be analyzed by studying high order representations of the group SO (2,3). In just the same way spin up and down are the two states associated to the s = 1 2 representation of the SO(3) group, the particle and antiparticle states could be the two states of a particular representation of the group (or some subgroup of) SO (2,3). If the extra dimension is a physical one and representation theory predicts more states of matter, it should be possible to observe them in experiments.
Additionally to this we have seen that, although the extra dimension was not proposed to be bounded, the effective extension of the field on it is of the order of the Compton wavelength of the particle. The main success of the Kaluza-Klein theory lies on its unification of Einstein field equations and Maxwell electromagnetism. It also introduces the Lorentz force on a test particle in the geodesic equations of motion. Due to these facts an interesting approach to induce an EM field in the Dirac theory (alternative to the used above) would be given by working with the original Kaluza-Klein metric [5]. In this way, we would work with a non trivial 5D manifold and by working on the Clifford and spinor bundles we could try to obtain QED in absence of gravity. This is something we would like to do in the future since it would also allows us to work with gravitational fields.