The Impact of the Higgs on Einstein’s Gravity and the Geometry of Spacetime

The Impact of the Higgs on Einstein’s Gravity and the Geometry of Spacetime The experimental observation of the Higgs particle at
the LHC has confirmed that the Higgs mechanism is
a natural phenomenon, through which the particles of
the standard model of interactions (smi) acquire their
masses from the spectrum of eigenvalues of the Casimir
mass operator of the Poincaré group. The fact that the
masses and orbital spins defined by the Poincaré group
appear in particles of that model, consistent with the
internal (gauge) symmetries, naturally suggests the existence
of some kind of combination between all symmetries
of the total Lagrangian. However, such “symmetry
mixing” sits at the core of an acute mathematical
problem which emerged in the 1960’s, after some “no-go”
theorems showed the impossibility of an arbitrary combinations
between the Poincaré group with the internal
symmetries groups. More specifically, it was shown that
the particles belonging to the same internal spin multiplet
would necessarily have the same mass, in complete
disagreement with the observations [1, 2].
It took a considerable time to understand that the
problem was located in the somewhat destructive “nilpotent
action” of the translational subgroup of the Poincaré
group over the spin operators of the electroweak symmetry
U(1) × SU(2) [3, 4]. Among the proposed solutions,
one line of thought suggested a simple replacement of the
Poincaré group by some other Lie symmetry, like for example
the 10-parameter homogeneous de Sitter groups.
Another, more radical proposal suggested the replacement
of the whole Lie algebra structure by a graded Lie
algebra, in the framework of the super-string program.
Such propositions have impacted on the subsequent development
of high energy physics and cosmology during
the next four or five decades, lasting up to today.
Here, following a comment by A. Salam [5], we present a
new view of the symmetry mixing problem, based on the
Higgs vacuum symmetry. In order to assign masses to
all particles of the smi, in accordance with the eigenvalues
of the Casimir mass operator of the Poincaré group,
the vacuum symmetry must remain an exact symmetry
mixed with the Poincaré group. Admittedly, this is not
too obvious because the Higgs mechanism requires the
breaking of the vacuum symmetry and consequently also
of the mixing. We start with the analysis of the Higgs vacuum symmetry, and its relevance to the solution of the
symmetry mixing problem. In the sequence, we explore
the fact that the mixing with the Poincaré group also
implies in the emergence of particles with higher spins,
including the relevant case of the Fierz-Pauli theory of
spin-2 fields in the smi. We end with the proposition of a
new, massive spin-2 particle of geometric nature, acting
as a short range carrier of the gravitational field, complementing
the long range Einstein’s gravitational interaction.
We begin by tracing an analogy between the “Mexican
hat” shape of the Higgs potential with a cassino roulette.
The roulette works by the combined action of gravitation
with the spin produced by the action of the croupier over
the playing ball. The energy of the ball eventually ends as
it “naturally falls” into one of the numbered slots at the
bottom of the roulette, producing a winning number. In
our analogy, the playing ball represents a particle of the
standard model and the numbered slots at the bottom of
the roulette corresponds to Higgs vacuum represented by
a circumference at the bottom of the hat, whose symmetry
group is SO(2). A difference is that while the slots in
the roulette are labeled by the integers, the bottom circle
of the Mexican hat is a continuous manifold parametrized
by an angle, assuming specific real values in the interval
[0, ∞). When a particle falls into the vacuum, it “wins a
mass” so to speak, not any mass, but only a discrete, positive,
isolated real mass values which correspond to one
of the eigenvalues of the Casimir mass operator of the
Poincaré group [27]. In other words, the measurement of
one particle mass in its vacuum state is an “observational
condition” of the Higgs theory, which in our analogy corresponds
to stopping the roulette, so that every player
can read and confirm who is the winner, does not end
the game. The roulette will spin again, so that all other
particles also may have the chance of winning a mass.
The spontaneous breaking of the vacuum symmetry will
does not eliminate that symmetry. Consequently, the
Higgs mechanism requires that the vacuum symmetry is
exact, braking only at the moment of assigning the mass
to any given particle.