2.1.2 p=0, D=4, Relativistic Point-Particle

Following exactly the same steps as in the previous case, we obtain

$$p=0, \quad V_p=0,\quad \sigma^{ \mu_1\dots \mu_{p+1}}\equiv 0 \longrightarrow \hbox{pointlike particle}$$ (13)

$$D=4 \longrightarrow \hbox{Minkowski spacetime}$$ (14)

$$m_{p+1}\, V_p \longrightarrow M_0 = \hbox{particle's mass}$$ (15)

$$\Omega_1=\int_0^T d\tau\, e(\, \tau\, ) \hbox{proper time interval}$$ (16)

$$Z_{\phi , A}\equiv 1 \hbox{Quenching}$$ (17)

$$G\left[\, x- x_0\ ,\sigma\ ; M_0\,\right] = 2 \int_0^\infty d\tau \left( {\pi M_0 \over 2i\tau}\right)^2\exp\left\{ i M_0 \tau +i {M_0 \over 4\tau } \left( x - x_0\right)^2 \right\}$$

$$= 2 M_0\int_0^\infty ds \left(\, {\pi\over 2i s}\, \right)^2\exp\left\{\, i \, M_0^2\, s + {i \over 4s}\, \left(\, x - x_0\, \right)^2 \, \right\} .$$ (18)

Returning now to the general case, one may regard the expression (1) as the propagator of a poly-brane, i.e., a generic $p$-brane combined with its baricentric coordinate, moving in a spacetime with metric:

$$d \Sigma ^{2} = d y ^{\mu}\, d y _{\mu} + \frac{1}{V _{... ...{1} \dots \mu _{p+1}}\, d {Y} _{\mu _{1} \dots \mu _{p+1}}\ .$$ (19)

Equation (19) is a special case of the Clifford line element

$$d \Sigma ^{2} = d y ^{\mu}\, d y _{\mu} + \frac{1}{l ^{2}}\, d {Y... ...} ^{\mu _{1} \mu _{2} \mu _{3}}\, d {Y} _{\mu _{1} \mu _{2} \mu _{3}} + \dots +$$

$$\qquad + \frac{1}{l ^{2p}}\, d {Y} ^{\mu _{1} \dots \mu _{p+1}}\,... ...D-1)}}\, d {Y} ^{\mu _{1} \dots \mu _{D}}\, d {Y} _{\mu _{1} \dots \mu _{D}}\ .$$ (20)

As a mathematical construct, the line element (20) has long been known at least to some practitioners of Clifford algebras. For instance, Pezzaglia has introduced it to discuss the long standing problem of a classical spinning particle [11]. In that classical context, Eq. (20) may be interpreted as an extension of the usual Lorentzian line element.
To our mind, however, the Clifford line element establishes a mathematical and physical link between the theory of relativistic extended objects, as developed over the years by the authors, and the very structure of spacetime geometry at the Planck scale. Indeed, the whole cardinal concept of relativity of motion may be extended to the broader context of relativity of dimensions. By ``relative dimensionalism'', we mean that the new Clifford metric opens the possibility of (generalized Lorentz) transformations between different $p$-branes, so that their effective dimensionality, and the very geometry of spacetime, become resolution dependent.
In order to substantiate this connection between the Clifford line element and the short distance structure of spacetime, note that the generalized Lorentzian metric (20) calls for the introduction of a fundamental length, $l$, or energy scale $l^{-1}$, in the fabric of spacetime, so that, what is described as a scalar, vector, bivector, or $p$-vector, becomes now observer dependent. In the language of $p$-branes, the same fundamental length is necessary in order to include in the line element (20) an ``areal distance'', playing the role of time for each kind of $p$-brane, so that, what is physically perceived as a point, world-line, world-sheet, or $p$-brane really depends on the resolving power of the Heisenberg microscope used to probe the short distance structure of spacetime.
In order to clarify the physical meaning of $l$, let us consider the new tension-shell condition defined by the vanishing of the denominator in Eq. (1):

$$k^2 + {V_p^2\over (p+1)!} k^2_{\mu_1\dots\mu_{p+1}} +(p+1)\, M_0^2=0\ .$$ (21)

Real branes satisfy the constraint (21) which correlates center of mass and volume momentum squared, which are independent quantities for virtual branes. Let us further parametrize the square of the volume momentum in terms of the modulus of the center of mass four momentum:

$$\left[\, k_\mu\,\right]=(\,{\mathrm{length}}\,)^{-1}\ ,\qquad \left[\, k_{\mu_1\dots\mu_{p+1}}\,\right]=(\,{\mathrm{length}}\,)^{-(p+1)}$$ (22)

$${1\over (p+1)!}\, k^2_{\mu_1\dots\mu_{p+1}}\equiv \beta\, k^2 ( k^2 )^p ,$$ (23)

where $\beta$ represents a numerical parameter. Thus, the tension shell condition (21) can be written as a $(p+1)$-degree polynomial in $k^2$. It is convenient, at this point, to restore the explicit presence of Planck's constant in the above equation,

$$\hbar^2\, k^2 + \hbar^2\, V_p^2\, \beta\, (\, k^2\,)^{p+1} +(p+1)\, M_0^2=0\ .$$ (24)

The ``tension shell condition'' (24) can be cast in the more familiar form of mass-shell condition for a pointlike particle by incorporating the extra $k^2$-dependence within an effective, or Running Planck Constant [12]

$$\hbar^{{\mathrm{eff.}}}\equiv \hbar\, \left[\, 1 + V_p^2\, \... ...t(\, \hbar^{{\mathrm{eff.}}} k \,\right)^2 +(p+1)\, M_0^2=0\ .$$ (25)

From here follows the generalized form of the Uncertainty Principle :

$$\Delta x^\mu\, \Delta k_\mu\ge \langle\, \hbar^{{\mathrm{eff.}}}\,\rangle ,$$ (26)

where $\langle\,\dots \,\rangle$ denotes vacuum average.
The physical content of (26) emerges in the non-relativistic limit, by neglecting the timelike components of the four vectors

$$\Delta \vec x\cdot \Delta \vec k\ge \langle\, {\hbar^{{\math... ...\, \left(\,\langle\vec k^2\,\rangle\right)^p + \dots \right) .$$ (27)

Rotational invariance allows a further semplification by rotating the reference frame in such a way that the $x$-axis coincides with the direction of the center of mass position vector $\vec x$:

$$\Delta x\, \Delta k_x\ge \langle\, \hbar^{{\mathrm{eff.}}}\,... ...a\, \left(\,\langle k_x^2\,\rangle\right)^p + \dots \right)\ .$$ (28)

Finally, by taking into account the translational invariance of the vacuum, that is, $\langle \, k_x \,\rangle =0$, we can relate momentum uncertainty and momentum fluctuation as follows,

$$\Delta k_x\equiv \sqrt{\,\langle k_x^2\,\rangle - \langle k_x\,\rangle^2}=\sqrt{\,\langle k_x^2\,\rangle }\ .$$ (29)

Thus, the position uncertainty reads

$$\Delta x \ge {\hbar\over \Delta k_x}\, \left[\, 1 + {1\over ... ..., \beta\, \left(\,\Delta k_x \,\right)^{2p} + \dots \right]\ .$$ (30)

This formula holds for any $p$, and therefore it is applicable to strings as well. Thus, comparing with the ``string uncertainty principle'' [13]

$$\Delta x \ge {\hbar\over \Delta k_x}\, \left[\, 1 + \, \, {\... ...const.}}\times\, G_N\left(\,\Delta k_x \,\right)^2\, \right] ,$$ (31)

we can write the brane invariant volume in terms of the Planck length

$$V_p\propto\left(\, l_{Pl}\, \right)^p =\left(\,\hbar \, G_N\, \right)^{p/2} .$$ (32)

The standard procedure, at this point, is to minimize the position uncertainty

$${d ( \Delta x) \over d ( \Delta k_x)}=0\ \Longrightarrow \le... ...eft[\, {2\over (2p-1)\, l_{Pl}^{2p}\, \beta}\, \right]^{1/p} ,$$ (33)

so that, by inserting (33) in (30), one obtains

$$\left(\,\Delta x\,\right)_{min}= \hbar\,\left(\, 2p+1\,\right) \left(\, 2p-1\,\right)^{-1+ 1/2p}\,\beta^{1/2p}\, l_{Pl} .$$ (34)

By switching once again to natural units, $\hbar=1$, we see that the minimum uncertainty, or quantum of resolution is proportional to the length scale $V_p^{1/p}$, i.e.

$$\left(\,\Delta x\,\right)_{min}\propto l_{Pl}\ .$$ (35)

According to the fundamental condition (35), strings, or any $p$-brane, or test body for that matter, cannot probe distances shorter than $l_{Pl}$. Thus, the length scale $l$, originally introduced in the line element (20) for purely dimensional reasons, can now be given the meaning of a minimum universal length⁵.