Mass Creation From Extra Dimensions

In this work we consider a mechanism for mass creation based on the periodicity condition dictated from the compactification of extradimensions. It is also shown that the existence of Tachyon having negative square mass is closely related to time- like extradimensions.


Introduction
The existence of space-time extradimensions has been a subject of intensive research study during the last decades [1 − 3].
The Topology of extradimension, especially their compactification play a crucial role in many physical aspects, mostly in the construction of various models of Unified theory of interactions, such as Superstring theory, Extended General Ralativity, and so on [4 − 7].
It is worth noting, on the other hand, that is such approaches the particles mass always remains a problem of actual characters.
In this work we propose a mechanism for mass creation through the compactification of spacetime extradimensions. The crucial argument is the proposed periodicity condition dictated from the compactification of extradimensions.
The original field functions depend on all space-time coordinate components including those for extradimensions, the ordinary field functions ordinary 4-dimensional space-time are considered as effective field functions obtained by integration of the original ones over extra space-time.
In section 2, we present some general principles related to the compactification of extradimensions.
In section 3, a mechanism for mass creation is treated.

Periodicity campactification condition
For simplicity let us begin with the case of one extra dimension. Denote the 5-dimensional coordinate vector by x M with M = µ, 5. The Greek indices µ, ν,... will be use as conventional 4-dimensional Lorentz indices (0,1,2, and 3). We do not directly care from the extra dimensions is topologically compactified, but instead a specific periodicity condition is put on the field functions depending on extra dimensions, namely Where f L (L) is some parameter function depending on the compactification lenth L. The condition (1) corresponds to the equation: With the relations: is to be real and therefore θ F = 0, n = 0 . The periodicity condition (1) can be generalized for the case of arbitrary number of extra dimensions in the following manner.

Effective field equation and mass
The general procedure of our treatment is as follows. We start from the (4+d) dimensional Lorentz invariant Kinetri Lagrangian L(x,y) and the action for the field F(x,y) defined as S(y) ≡ d 4 x.L(x, y) Where (dy) ≡ dy 1 .dy 2 ...dy d and the general is performed over the whole extra space time. The principle of minimal action for S(y) then gives the Euler-Lagrange equation Which in turn leads to the equation of Klein-Gordon type: (✷ + m 2 F )F (x) = 0 For the effective field defined as For illustration let us consider in more details the cases of scalar, spinor and vector fields.

Scalar field
The free neutral scalar field Φ(x, y) is described by the Lagrangian Where ∂ a ≡ ∂ ∂y a , ς ab is a MinKonski metric for extra dimensions: By inverting (7) into (12), we obtain: And from here the equation For the effective field Φ(x) = (dy)Φ(x, y) With It is worth nothing that the squared mass m 2 φ is positive if all the extra dimensions are space-live, and can be negative if there exists time-live extra dimensions.
For changed scalar field instead of (12) we take And instead of (13) we have: And from here the same equation as (14) with:

Spinor field
In (4+d)-dimensional space-time, the spinor field is decribed by a 2 4+d 2 component function ψ a (x, y) with the free Lagrangian matrices obeying the ant commutation relations: ψ L a .ψ(x, y) Into (19) we obtain: And from here the equation ψ (L a ).Γ 4+a ψ(x, y) = 0 (23) By acting from the left both sides of this equation by And taking into account the relations (20) we have: And hence We note that

Vector field
We restrict ourselves to the case d=1 and consider the neutral vector field V M (x, y) satisfying the periodicity condition And in correspondence The free vector field V M (x, y) is described by the Lagrangian Where By inverting (27) into (28) we have: Now we define a new physical vector field W µ by putting Expressed in terms of W µ , the Lagrangian (29) has the form: The Lagrangian (31) leads to the equation: Which means that the effective vector field Has squared means It's positive or negative depending upon whether the extra dimension is space-like or time-like.

Conclusion
In this work we have proposed a mechanism for the creation of particle mass. The key idea is that the mass is originated from the compactification of extra dimensions followed by the periodicity condition for the particle fields.
It is worth nothing that according to the mechanism the existence of tachyon having negative squared mass is closely related to the existence of time-like extra dimensions.