4.1 Massless Field, Closed Membrane and Vacuum Energy Density

In order to implement the three properties a), b), c) discussed above, we start from the action functional

$$S=\int d^4x \left({1\over 2\cdot 4!} F_ {\lambda\mu\nu\rh... ...\nu\rho} \right)-\mu^3 \int_M d^3\sigma \sqrt{-\gamma} .$$ (28)

This is a straightforward, but non-trivial, formal extension of the action for the electrodynamics of point charges, or the Kalb-Ramond action of ``string-dynamics.'' More to the point, from our discussion in Section(3), it represents a direct generalization to $(3+1)$ dimensions of the same two-dimensional``electrodynamics'' action [15] in a ``$\sigma$-model'' inspired formulation where a fundamental extended object is coupled to its massless excitations. In order to keep our discussion as transparent as possible, we consider only an elementary, or structureless membrane with vanishing width, interacting with a single massless mode represented by $A_{\mu\nu\rho}$. Thus,

$$J^{\mu\nu\rho}( x )=\int \delta^{4)} \left[ x- Y(\sigma) \right] dY^\mu \wedge dY^\nu\wedge dY^\rho$$ (29)

represents the current density associated with the world history of the membrane. More complex models, in which the membrane is some sort of collective excitation of an underlying field theory, while intriguing, are affected by highly non-trivial technical problems, such as renormalization [17], [8] and bosonization ⁸. That approach, while conceivable in principle, is orthogonal to ours: here, we assume that the membranes under consideration are elementary geometric objects of a fundamental nature, on the same footing as points, strings and other p-branes that constitute the very fabric of quantum spacetime [18]. This principle of geometric democracy, is reflected in the action functional (28) by our choice of the Nambu-Goto-Dirac action for a relativistic bubble in which $\gamma$ stands for the determinant of the induced metric

$$\gamma\equiv det\left( \partial_m Y^\mu \partial_n Y_\mu \right)$$ (30)

and $\mu^3$ represents the bubble surface tension. Notwithstanding the apparent simplicity of our model, we shall see in the next subsection that it is possible to reproduce the correct false vacuum decay rate, without resorting to solitonic computational techniques⁹.
Gauge invariance of the action (28) is guaranteed whenever the bubble embedding equations $x^\mu=Y^\mu( \sigma )$ parametrize a world history without boundary, so that

$$\delta A_ {\mu\nu\rho} =\partial_{[ \mu} \Lambda_{\nu\rho ]}$$ (31)

$$\delta S_\Lambda = 0 \quad\longleftrightarrow \quad \partial_\mu J^{\mu\nu\rho}=0$$ (32)

The divergence free condition for the membrane current is the formal translation of the no-boundary condition. Essentially, it restricts the world history of the membrane to be [20]:
i) spatially closed;
ii) either infinitely extended in the timelike direction (eternal membrane ,) or, compact without boundary (virtual membrane.)
Consequently, the cosmological field $A_{\mu\nu\rho}$ couples in a gauge invariant way only to bubbles whose history extends from the remote past to the infinite future, or to objects that start as a point in the vacuum, expand to a maximum spatial volume and then recollapse to a point in the vacuum. In such a case, variation of the action with respect to $A_{\mu\nu\rho}$ leads to Maxwell's equation

$$\partial_\mu F^ {\mu\nu\rho\sigma}= g J^ {\nu\rho\sigma}$$ (33)

The general solution of (33) is the sum of the free equation solution ($g=0$), and a special solution of the inhomogeneous equation ($g\ne 0$). The complete formal solution is found by inverting the field equation according to the Green function method: taking into account that the Maxwell tensor is proportional to the epsilon-tensor, we find

$$F^{\mu\nu\rho\sigma}= \epsilon ^ {\mu\nu\rho\sigma} \sqrt... ...g \partial^ { [\mu} {1\over\Box} J^{\nu\rho\sigma ]} .$$ (34)

Inserting the above solution back into the action (28), one finds apart from the Nambu-Goto-Dirac term

$$S= - {1\over 2}\int d^4x \left( \Lambda + {g^2\over 3!} ... ...nu\rho\sigma} {1\over \Box} J_ {\nu\rho\sigma} \right) .$$ (35)

Exactly as in the $(1+1)$ dimensional case, we interpret the above expression as follows

$$S=- {1\over 2} \int d^4x \left[ \hbox{\lq\lq Cosmological Cons... ...+ \lq\lq Coulomb Potential''} \right] + \hbox{extra surface terms}$$ (36)

Indeed, as anticipated at the beginning of this section (property (a)), the first term in Eq. (35) is a solution of the free Maxwell equation and represents a constant energy density background. As a free field, that is, in the absence of gravity and any other interaction, that constant term can be ``renormalized away'' since it cannot be distinguished from the vacuum. However, it is equivalent to a cosmological term when gravity is switched on [15], [13].

It may not be immediately evident that, even in the presence of a coordinate dependent metric $g_{\mu\nu}(x)$, the homogeneous solution of Eq. (9) still represents a constant background energy density. Phrased differently, it might appear that ``there is no longer a constant rank-$4$ tensor available to equate $F$ to''.
The loophole is in the covariant form of Eq. (9). Since we are considering the homogeneous solution, we may as well switch-off the coupling to the current, so that:

$$\nabla_\mu F^{\mu\nu\rho\sigma}=0 .$$ (37)

Here, $\nabla_\mu$ represents the covariant derivative compatible with the Riemannian metric $g_{\mu\nu}(x)$, i.e., the connection is chosen to be the Christoffel symbol. In four dimensions there is only one generally covariant and totally anti-symmetric tensor, namely, the covariant Levi-Civita tensor:

$$\varepsilon^{\mu\nu\rho\sigma}(x)\equiv {1\over \sqrt{-g(x)}} \epsilon^{\mu\nu\rho\sigma}$$ (38)

where, $g(x)\equiv det g_{\mu\nu}(x)$ and $\epsilon^{\mu\nu\rho\sigma}$ is the constant Levi-Civita tensor density. Thus, Eq. (37) may be solved by the ansatz

$$F^{\mu\nu\rho\sigma}\equiv {1\over \sqrt{-g(x)}} \epsilon^{\mu\nu\rho\sigma} F(x)$$ (39)

where $F(x)$ is a scalar function to be determined by the field equations. The metric tensor $g_{\mu\nu}(x)$ and its determinant are both covariantly constant with respect to the Christoffel covariant derivative. Thus, the $\varepsilon(x)$ tensor has vanishing covariant derivative. By inserting the trial solution (39) in Eq. (37), one sees that the derivative operator bypasses the $\varepsilon(x)$ tensor and applies directly to the scalar function $F(x)$:

$$\varepsilon^{\mu\nu\rho\sigma}(x)\partial_\mu F(x)=0$$ (40)

Thus, the solution of Eq. (40) is again

$$F(x)= const.\equiv \sqrt\Lambda$$ (41)

To conclude the proof that $\Lambda$ represents a genuine cosmological constant, we need to compute the value of the classical action. This can be done by using the following property of the $\varepsilon(x)$ tensor:

$$\varepsilon_{\mu\nu\rho\sigma}(x)\equiv \sqrt{-g(x)} \epsi... ...n_{\mu\nu\rho\sigma}(x) \varepsilon^{\mu\nu\rho\sigma}(x)=-4!$$ (42)

Thus,

$$-{1\over 2\cdot 4!}\int d^4x \sqrt{-g} F_ {\lambda\mu\nu\... ...u\rho}\longrightarrow \int d^4x \sqrt{-g}{\Lambda\over 2} .$$ (43)

Having clarified the physical meaning of the integration constant $\Lambda$, let us consider the second term in the action (35). Apparently, it describes a long-range, ``Coulomb interaction'' between the bubble surface elements. In reality, it represents the bubble volume energy density written in a manifestly covariant form. In fact, we can re-arrange that ``Coulomb term'' as follows. From the definition (29) and the condition (32), we deduce that

$$J^ {\nu\rho\sigma}(x) = \partial_\mu K^{\mu\nu\rho\sigma}(x)$$ (44)

$$= \partial_\mu \int d^4\xi \delta^{4)}\left[ x -Z(\xi) \right] dZ^\mu\wedge dZ^\nu\wedge dZ^ \rho\wedge dZ^\sigma .$$ (45)

However, in four dimensions

$$K^{\mu\nu\rho\sigma}(x)=\epsilon ^ {\mu\nu\rho\sigma}\int d^... ... -Z(\xi) \right]\equiv \epsilon ^ {\mu\nu\rho\sigma}\Theta(x)$$ (46)

where $\Theta(x)$ is referred to as the characteristic function of the spacetime open sub-manifold bounded by the membrane. Thus, the Coulomb term can be rewritten in terms of $\Theta(x)$

$${g^2\over 3!} J^ {\nu\rho\sigma} {1\over \Box} J_ {\nu\rho\sigma} = {g^2\over 3!}\partial_\mu K^{\mu\nu\rho\sigma} {1\over \Box} \partial^\tau K_{\tau\nu\rho\sigma}$$

$$= {g^2\over 3!} \partial_\mu \epsilon^{\mu\nu\rho\sigma}\Theta(x) {1\over \Box} \partial^\tau \epsilon_{\tau\nu\rho\sigma}\Theta(x)$$

$$= -g^2\partial_\mu \Theta {1\over \Box} \partial^\mu \Theta$$

$$= g^2 \Theta(x)$$ (47)

where we have made use of the formal identity $\Theta^2(x)\equiv \Theta(x)$ and discarded a total divergence. Thus, the classical solution (34) and the action (28) show that the cosmological field $A_{\mu\nu\rho}$ does not describe the propagation of material particles; rather, it represents a constant energy density background with two different values inside and outside the membrane. Indeed, using the previous result, one may calculate the value of the classical action corresponding to the solution (34),

$$S =-{1\over 2}\int d^4x \left[ \Lambda + g^2 \Theta(x) \right] -\mu^3\int_M d^3\sigma \sqrt{-\gamma} .$$ (48)

Once again, we note that in the absence of gravity one is at liberty to choose the ``zero'' of the energy density scale, and thus measure the energy density with respect to the constant background represented by $\Lambda$. With that observation in mind, the classical action turns out to be a pure volume term, as announced:

$$S_\Lambda +\mu^3\int_M d^3\sigma \sqrt{-\gamma} \equiv S( \Lambda; g )- S( \Lambda; g=0 ) = {g^2\over 2} \int d^4x \Theta(x)$$

$$= {g^2\over 2} \int d^4x \int_B d^4\sigma \delta^ {4)}\left[ x - Y(\sigma) \right]$$ (49)