The question of the origin of mass is one of the more important issues in theoretical physics. The existence (or not) of extra dimensions of space and time will likely affect to the final solution of this unsolved problem.
The topology of extra dimensions, and specially their “compactification” to real space and effective 4D=3d+1 world, plays a crucial role in some physics beyond the Standard Model (SM). The construction of unified models of all interactions from the purely geometric viewpoint is in the target since Einstein seminal work about the nature of the gravitational field. It also arises naturally in superstring theory, M-theory and other BSM models. In those models, particle mass stays as a problem. Nobody can deny that.
Kaluza-Klein theories (KK) are based on a general mechanism for mass creation and unification through the use of a compactification with periodic space-like dimensions (or time-like dimensions if we are general enough). The original field functions depend on all space-time coordinates, but the ordinary field functions in 4D spacetime are considered as effective result by some kind of integration (and Fourier analysis) over the extradimensional spacetime.
Suppose, for simplicity, we treat only one single extra dimension (you can extend the analysis for several extra dimensiones with suitable mathematical background). Let us introduce coordinates with , where the greek indices run over usual 4D spacetime Lorentz indices (0,1,2,3 or 1,2,3,4 if you prefer). A periodicity function condition reads
where is some parameter function depending on the scale L, the compactification length. Let us write
then you can write
If , is real and thus and .
As we have said above, you can generalize the extra dimensional argument in a straightforward fashion. Write and , with . You get
The periodicity condition generalized to the whole extra dimensions (that would be a higher dimensional torus):
The general KK procedure is simple. Start from a general D=4+d dimensional Lorentz invariant lagrangian and the action from the field defined as
is the extradimensional world volume. Then perform over the whole extradimensional space some Fourier transformation or develop the field in terms of suitable eigenfunctions. The minimal action principle for provides the Euler-Lagrange equation
It shows that it leads to a Klein-Gordon field equation
if we define the effective field
Without treating all the possible cases, we will focus our attention on 3 cases: scalar fields, spinor fields and vector fields with extra dimensions. The most “complex” cases (since we do not know yet their quantum particles from experiments) of the Rarita-Schwinger fields (spin 3/2) and the gravitational spin 2 case (beyond higher spin extensions) will be leave for future treatment here.
1. Scalar field in extra dimensions
The free neutral scalar field can be described with a lagrangian
and where , and is a Minkovski metric for the extra dimensional world. It reads
We can write
It yields a Klein-Gordon (KG) equation
for some effective field
Remark: the squared mass
is positive if ALL the extra dimensions are space-like. It could be negative (tachyon-like!) if there exists extra time-like dimensions.
For a charged scalar field, a similar argument provides
and from here you get
2. Spinor field in extra dimensions
This case must be done with care, since there are some conditions for the existence of spinors in D-dimensional spacetime. We will neglect the subtleties related to consider the different kind of spinor fields at this moment(Dirac, Weyl, Majorana, Majorana-Weyl,…)Suppose that in (4+d)-spacetime the spinor field is described by certain component function with a free lagrangian
and where are the (4+d) Dirac (Clifford algebraic) matrices obeying the anticommutation rules (Clifford algebra):
The expanded spinorial lagrangian reads
From here, we obtain
Acting to the left of this last equation with the operator
and using the Clifford algebra relations defined above, we obtain a KG like equation
Remark: if all the extra dimensions are space-like, if all the are real and we get tachyonic modes if there are one or several “big enough” extra time-like dimensions.
3. Vector field in extra dimensions
Finally, the case of a vector field. We only consider here a single extra dimension and a neutral vector field , satisfying the condition
The free vector field is described by the lagrangian
The lagrangian expansion reads
Define a new physical vector field degree of freedom , with
It shows that the previous lagrangian can be rewritten in terms of as follows:
with . Finally, the lagrangian produces the KG field equation
tied to the effective vector field
and the squared vector field mass
Simple induction produces the mass squared vector field formula
Again, it is remarkable that the negative mass of a vector field depends on the number of extra time-like dimensions and its relative “strength”. That is, the existence of tachyonic modes (negative mass terms) is directly related to the existence of extra time-like dimensions. This is a key point that sometimes is not easily found in technical books since they neglect (usually) the (quantum stability) issues of having tachyonic modes, so they are usually ignored. But I like to talk about uncommon ideas like this one in my blog ;).
See you in my next blog post!!!!!!!