LOG#168. D-dimensional laws(III).

Posted on by April 6, 2015

flatland

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 X^M with M=\mu,5, 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

    \[\boxed{F(X^\mu,X^5+L)=f_L(L)F(X^\mu,X^5)}\]

where f_L(L) is some parameter function depending on the scale L, the compactification length. Let us write

    \[\dfrac{\partial}{\partial x^5}F(X^M)=\partial_5F(X^M)=g_L(L)F(X^M)\]

Define

    \[f_L(L)=e^{Lg_M(L)}\]

    \[g_L(L)=\dfrac{1}{L}\left[\ln f_L(L)+2\pi i n\right](n\in Z)\]

then you can write

    \[f_L(L)=\rho_F(L)e^{i\theta_F(L)}\]

and

    \[g_L(L)=\dfrac{1}{L}\left[\ln\rho_F(L)+i(\theta_F(L)+2\pi n)\right]\]

If F=F^+, f_L(L) is real and thus \theta_F=0 and n=0.

As we have said above, you can generalize the extra dimensional argument in a straightforward fashion. Write X^5,X^6,\ldots,X^{4+d} and Y^a\equiv X^{4+a}, with a=1,2,\ldots,d. You get

    \[F(X^M)=F(X^\mu,Y^a)\equiv F(X,Y)\]

The periodicity condition generalized to the whole extra dimensions (that would be a higher dimensional torus):

    \[F(X,Y^a+L^a)=f_F^{(a)}(L_a)F(X,Y)\]

and so

    \[\dfrac{\partial}{\partial Y^{a}}F(X,Y)=\partial_{Y^a}F(X,Y)\equiv \partial_aF(X,Y)=g_F^{(a)}(L^a)F(X,Y)\]

with

    \[f_F^{(a)}(L^a)=f_F^{(a)}(L^a)e^{i\theta_f^{a}(L^a)}\]

    \[g_F^{(a)}(L^a)=\dfrac{1}{L^a}\left[\ln f_F^{(a)}(L^a)+i(\theta_F^{(a)}(L^a)+2\pi n)\right]\]

 The general KK procedure is simple. Start from a general D=4+d dimensional Lorentz invariant lagrangian L(X,Y) and the action from the field F(X,Y) defined as

    \[S=\int S(Y)(dY)\]

    \[S(Y)=\int d^4xL(X,Y)\]

and where

    \[(dY)=dY^1\cdots dY^d\]

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 S(Y) provides the Euler-Lagrange equation

    \[\dfrac{\partial L(X,Y)}{\partial F(X,Y)}-\partial_\mu\dfrac{\partial L(X,Y)}{\partial(\partial_\mu F(X,Y))}=0\]

It shows that it leads to a Klein-Gordon field equation

    \[(\square^2+m_F^2)F(X)=0\]

if we define the effective field

    \[F(X)=\int (dY)F(X,Y)\]

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 \Phi(X,Y) can be described with a lagrangian

(1)   \begin{equation*} L(X,Y)=\dfrac{1}{2}\partial^M\Phi(X,Y)\partial_M\Phi(X,Y) \end{equation*}

or equivalently

(2)   \begin{equation*} L(X,Y)\equiv \dfrac{1}{2}\{\partial^\mu\Phi(X,Y)\partial_\mu\Phi(X,Y)+\sum_{a=1}^d \delta_{aa}\partial_a\Phi(X,Y)\partial_a\Phi(X,Y)\} \end{equation*}

and where \partial_a=\dfrac{\partial}{Y^a}, and \delta_{ab} is a Minkovski metric for the extra dimensional world. It reads

    \[\delta_{ab}=\begin{cases}0,\mbox{if}\;\;\; a\neq b\\ +1,\mbox{if},\;\;\; a=b(timelike)\\ -1,\mbox{if}\;\;\; a=b(spacelike)\end{cases}\]

We can write

(3)   \begin{equation*} L(X,Y)=\dfrac{1}{2}\left[\partial^\mu\Phi(X,Y)\partial_\mu\Phi(X,Y)+\sum_{a=1}^d\delta_{aa}(g^{(a)}(L^a))^2\Phi^2(X,Y)\right] \end{equation*}

It yields a Klein-Gordon (KG) equation

(4)   \begin{equation*} (\square^2+m_\Phi^2)\Phi(X)=0 \end{equation*}

for some effective field

    \[\Phi(X)=\int (dY)\Phi(X,Y)\]

with

(5)   \begin{equation*} m_\Phi^2=-\sum_{a}\delta_{aa}(g^{(a)}(L^a))^2 \end{equation*}

Remark: the squared mass

    \[\boxed{m_\Phi^2=-\sum_{a}\delta_{aa}(g^{(a)}(L^a))^2}\]

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

(6)   \begin{equation*} L(X,Y)=\dfrac{1}{2}\partial^M\Phi^+(X,Y)\partial_M\Phi(X,Y) \end{equation*}

(7)   \begin{equation*} L(X,Y)\equiv \dfrac{1}{2}\{\partial^\mu\Phi(X,Y)\partial_\mu\Phi(X,Y)+\sum_{a=1}^d \delta_{aa}\partial_a\Phi^+(X,Y)\partial_a\Phi(X,Y)\} \end{equation*}

(8)   \begin{equation*} L(X,Y)=\dfrac{1}{2}\left[\partial^\mu\Phi^+(X,Y)\partial_\mu\Phi(X,Y)+\sum_{a=1}^d\delta_{aa}\vert g^{(a)}(L^a)\vert^2\Phi^+(X,Y)\Phi (X,Y)\right] \end{equation*}

and from here you get

    \[\boxed{m_\Phi^2=-\sum_{a}\delta_{aa}\vert g^{(a)}(L^a)\vert^2}\]

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 2^{(4+d)/2} component function \Psi_\alpha(X,Y) with a free lagrangian

(9)   \begin{equation*} L(X,Y)=\dfrac{i}{2}\overline{\Psi(X,Y)}\Gamma^M\overleftrightarrow{\partial_M}\Psi(X,Y)=\dfrac{i}{2}\left(\overline \Psi\Gamma^\mu\overleftrightarrow{\partial_\mu}\Psi+\sum_{a=1}^d\overline\Psi\Gamma^{a+4}\overleftrightarrow{\partial_a}\Psi\right) \end{equation*}

and where \Gamma^M are the (4+d) Dirac (Clifford algebraic) 2^{(4+d)/2}\times 2^{(4+d)/2} matrices obeying the anticommutation rules (Clifford algebra):

    \[\{\Gamma^\mu,\Gamma^\nu\}=2\delta^{\mu\nu}\]

    \[\{\Gamma^M,\Gamma^{4+d}\}=0\]

    \[\{\Gamma^{4+a},\Gamma^{4+b}\}=2\delta^{ab}\]

    \[\overline{\Psi}=\Psi+\Gamma^0\]

We write

    \[\partial_a\Psi(X,Y)=g_\Psi^{(a)}L_a\Psi(X,Y)\]

    \[\partial_a\overline{\Psi(X,Y)}=g_\Psi^{(a)}L_a\overline{\Psi(X,Y)}\]

The expanded spinorial lagrangian reads

(10)   \begin{equation*} L(X,Y)=\dfrac{i}{2}\overline{\Phi(X,Y)}\Gamma^\mu\overleftrightarrow{\partial_\mu}\Psi(X,Y)-\mathcal{I}(g_\Psi^{(a)}(L^a))\overline{\Psi}\Gamma^{4+a}\Psi \end{equation*}

From here, we obtain

(11)   \begin{equation*} \left(i\Gamma^\mu\partial_\mu-\sum_{a=1}^d\mathcal{I}(g_\Psi^{(a)}(L^a))\Gamma^{4+a}\right)\Psi(X,Y)=0 \end{equation*}

Acting to the left of this last equation with the operator

    \[i\Gamma^\nu\partial_\nu-\sum_{b=1}^d\mathcal{I}(g_\Psi^{(b)}(L^b))\Gamma^{4+b}\]

and using the Clifford algebra relations defined above, we obtain a KG like equation

    \[\left[\square^2-\sum_{a=1}^d\delta_{aa}(\mathcal{I}(g_\Psi^{(a)}(L^a)))^2\right]\Psi(X,Y)=0\]

and hence

    \[\boxed{m_\Psi^2=-\sum_{a}\delta_{aa}(\mathcal{I}(g_\Psi^{(a)}(L^a))^2}\]

Remark: m_\Psi^2>0 if all the extra dimensions are space-like, m^2_\Psi=0 if all the g_\Psi^{(a)} are real and we get tachyonic modes m_\Psi^2<0 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 V_M(X,Y), satisfying the condition

    \[V_M(X,Y+L)=f_V(L)V_M(X,Y)\]

and

    \[\dfrac{\partial}{\partial Y}V_M(X,Y)=g_V(L)V_M(X,Y)\]

    \[f_V(L)=e^Lg_V(L)\]

The free vector field V_M(X,Y) is described by the lagrangian

(12)   \begin{equation*} L(X,Y)=-\dfrac{1}{4}F_{MN}F^{MN}=-\dfrac{1}{4}\left(F^{\mu\nu}F_{\mu\nu}+2F_{\mu 5}F^{\mu 5}\right) \end{equation*}

and thus

(13)   \begin{equation*} L(X,Y)=-\dfrac{1}{4}F_{\mu\nu}F^{\mu\nu}-\dfrac{1}{2}\delta_{55}\left(\partial_\mu V_5\partial^\mu V_5+\partial_5V_\mu\partial_5V^\mu-2\partial_\mu V_5\partial_5 V^\mu\right) \end{equation*}

and where

    \[F_{\mu\nu}=\partial_\mu V_\mu-\partial_\nu V_\mu\]

    \[F_{\mu 5}=\partial_\mu V_5-\partial_5V_\mu\]

The lagrangian expansion reads

(14)   \begin{equation*} L(X,Y)=-\dfrac{1}{4}F_{\mu\nu}F^{\mu\nu}-\dfrac{1}{2}\delta_{55}\left(\partial_\mu V_5\partial^\mu V_5+g_V^2(L)V_\mu V^\mu-2g_V(L)\partial_\mu V_5V^\mu\right) \end{equation*}

Define a new physical vector field degree of freedom W^\mu, with

    \[W^\mu=V_\mu-\dfrac{1}{g_V(L)}\partial_\mu V_5\]

It shows that the previous lagrangian can be rewritten in terms of W^\mu as follows:

(15)   \begin{equation*} L(X,Y)=-\dfrac{1}{4}G_{\mu \nu}G^{\mu\nu}-\dfrac{1}{2}\delta_{55}g_V^2(L)W^\mu W_\mu \end{equation*}

with G_{\mu\nu})=\partial_\mu W_\nu-\partial_\nu W_\mu. Finally, the lagrangian produces the KG field equation

    \[(\square-\delta_{55}g_V(L)^2)W^\mu=0\]

tied to the effective vector field

    \[W_\mu (X)=\int_0^L dYW_\mu (X,Y)\]

and the squared vector field mass

    \[\boxed{m_W^2=-\delta_{55}g_V^2(L)}\]

Simple induction produces the mass squared vector field formula

    \[\boxed{m_W^2=-\sum_{a=1}^d\delta_{aa}g_V^2(L^a)}\]

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!!!!!!!

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