2 Symmetry Breaking Revisited

``Topological symmetry breaking'' and the concomitant mechanism of mass generation have never been discussed in the physics of point-particles for the simple reason that the world history of a material particle is usually assumed to have no boundary, that is, it is usually assumed to be infinitely extended in time⁶. Typically, in a world of classical point particles described by a local field theory, reparametrization invariance of the world-trajectory is tacitly assumed while gauge invariance is explicitly broken only by introducing a mass term in an otherwise invariant action. The asymmetry of the vacuum with respect to some global transformation (spontaneous symmetry breaking) provides a second possibility which, in turn, leads to the celebrated Nambu-Goldstone-Higgs mechanism of mass generation. The existence of a boundary in the world history of an object provides an additional possibility of symmetry breaking.
For the sake of illustration, consider the familiar case of a charged particle: the gauge invariance of the free Maxwell action may be broken either by introducing a mass term into the action,

$$S\equiv S_m + S_A$$ (1)

$$S_m= -\mu_0 \int_0^\infty d\tau \sqrt{-{dx^\mu\over d\tau} {dx_\mu\over d\tau}}$$ (2)

$$S_A=\int d^4x \left[-{1\over 4} F_{\mu\nu} F^{\mu\nu} + {m^2\over 2} A_\mu A^\mu -e J^\mu A_\mu \right]$$ (3)

or, more generally, by coupling the gauge potential $A_\mu(x)$ to a non-conserved current $J^\mu(x)$, i.e., $\partial_\mu J^\mu(x)\ne 0$. Thus, in either case: i) $m\ne 0$, or, ii)$m=0$, $\partial_\mu J^\mu\ne 0$, gauge invariance is violated. The massless, non-conserved current case corresponds to a classical point-like particle whose world-line $\Gamma_0$ has a free end-point. By definition, this represents a boundary condition that is not explicitly encoded into the action. Therefore, to the extent that there are no apparent symmetry violating terms in the action, we refer to this case as ``topological symmetry breaking.'' For instance,

$$\Gamma_0: x^\mu= x^\mu(\tau) ,\qquad 0\le \tau\le \infty$$ (4)

with

$$J^\mu(x)=\int_0^\infty {dx^\mu\over d\tau} \delta^{4)}... ...t_{\Gamma_0} dx^\mu \delta^{4)}\left( x-x(\tau) \right)$$ (5)

represents a semi-infinite spacetime trajectory $\Gamma_0$ that originates at $x_0$ and then extends forever. An extremal free end-point physically represents a ``singular'' event in which a particle is either created or destroyed, so that the covariant conservation of the associated current is violated⁷,

$$J(x)\equiv \partial_\mu J^\mu(x)= \partial_\mu \int_{\... ...au) \right)\right] =\delta^{4)}\left( x-x_0 \right)\ne 0$$ (6)

Furthermore, under a gauge transformation of the action integral, the interaction term transforms as follows

$$\delta_\Lambda S^{int} =e \int_{x_0}^\infty dx^\mu \partial_\mu \Lambda= -\Lambda(x_0)$$ (7)

assuming, as usual, that the gauge function vanishes at infinity.
In Section(4) we shall extend the above considerations to the case of a relativistic bubble in $(3+1)$-dimensions. That is the natural setting for discussing the new inflationary scenario. There we shall argue that the corresponding classical action represents an effective action for the quantum bubble nucleation process that takes place within the background vacuum energy represented by the cosmological constant. The novelty here is that the cosmological constant is disguised as a ``Maxwell field strength''. Due to the presence of a boundary of the bubble trajectory in spacetime, the process of nucleating an inflationary bubble [10], [11], [12], [13] must be accompanied by the excitation of massive spinless particles. A possible quantum formulation of the same boundary mechanism using the path integral approach to the dynamics of a generic p-brane in an arbitrary number of spacetime dimensions is given in Ref. [7].