GraviGUT
Unification in particle
physics
The Standard Model (SM) unifies the weak and electromagnetic
interactions in the following sense: one starts by postulating
the
existence of a gauge field for the group SU(2)xU(1), that is,
a one
form with values in the Lie algebra of this group. We may call
this the
"electroweak" gauge field because these data are not enough to
tell
which components of the fields have to do with the weak
interactions
and which ones with electromagnetism. One then assumes that
there
exists a "pointer" that identifies a special direction in the
Lie
algebra. The component of the gauge field in this direction is
identified with the electromagnetic potential A, the
components in the
orthogonal directions with the W and Z bosons, the carriers of
the weak
interactions. We do not really know for sure what the pointer
is. It is
usually thought of as the Vacuum Expectation Value (VEV) of a
scalar
field, but more complicated possibilities exist.
While this may be called a unification, in the sense that the
observed
fields A, W and Z are identified as components of a single
geometrical
object, it falls short in another sense, namely the number of
couplings. It would be desirable, in a unified theory, to have
a single
gauge coupling. But in the SM we are allowed to choose the
couplings of
the SU(2) and U(1) parts of the electroweak field
independently. When
the electroweak field is decomposed into its electromagnetic
and weak
components, these couplings get reshuffled and give rise to
the
electromagnetic and weak coupling. Two couplings go into the
theory and
two come out: from this point of view it is sometimes said
that in the
SM we do not have a real unification but rather a mixing.
It did not take long before physicists started to extend the
SM, with a
twofold purpose: one was to include also the strong
interactions, the
other to achieve unification in the sense of having a unique
gauge
coupling. The result of these efforts were the so-called Grand
Unifed
Theories (GUTs). As in the SM, the core idea is to identify
the
electroweak gauge field and the SU(3) strong gauge fields
(known as
gluons) as components of a "unified" gauge field for some
larger group
G.
For this, G must contain U(1), SU(2) and SU(3) as subgroups;
for
coupling unification one would also require the group G to be
simple
(which means, roughly speaking, that it is not a product of
smaller groups). A priori there is an embarrassment of
choices.
Important guidance to identify the correct group G comes from
consideration of the fermionic matter fields. Such fields
(electrons,
neutrinos, quarks etc.) can be classified according to their
charges,
which are integer or fractional numbers that prescribe their
behavior
under gauge transformations. The charges of each field are
known, and
requiring that the corresponding particles be the components
of a
single object transforming properly under the action of the
group G
imposes strong constraints on any attempt at unification.
Using such
constraints, several viable paths towards grand unification
were
identified in the late seventies. The most successful one is
based on
the group G=SO(10). This GUT achieves several goals: 1) it is
based on
one of the smallest groups containing SU(3)xSU(2)xU(1); 2) it
is simple
and therefore achieves coupling unification; 3) all fermions
belonging
to a so-called generation form a single 16-dimensional object
under
SO(10); 4) it predicts the existence of right-handed neutrinos
and 5)
it is not ruled out by experimental bounds on the proton
lifetime.
The Higgs phenomenon
A crucial ingredient in any modern unified theory is the identification of what I called earlier a "pointer". It is crucial that the dynamics of the unified theory be invariant under gauge transformations; the breaking of the symmetry is due entirely to the existence of the pointer, whose physical nature is to some extent immaterial. In the SM it is assumed that the special direction associated with electromagnetism is identified by the vacuum expectation value of a fundamental scalar field in the complex two-component spinor representation of SU(2), the Higgs field. The effect of the VEV is that the W and Z particles get a mass whereas the photon remains massless. While perfectly adequate, this mechanism is by no means necessary: the pointer may be something more sophisticated, like a dynamical condensate of fermion bilinears with the same quantum numbers as the Higgs field. This interpretation, which has been developed extensively in technicolor theories, is supported by the analogy to other instances of symmetry breaking that occur in condensed matter physics and, to some extent, in QCD. The truth is that there is no precedent in particle physics, making the search for the Higgs particle at LHC such an exciting endeavour.
The identification of the pointer with the VEV of a scalar
field is an
adaptation of ideas that had been used previously in condensed
matter
physics. One postulates that the scalar Lagrangian contains a
potential, whose shape can be changed by tuning a mass
parameter.
Typically, when the square of the mass parameter is positive
the
minimum of the potential, and hence the VEV of the scalar
field is
zero. For negative mass squared the potential has its minimum
at a
nonzero value, and the scalar field develops a nonzero VEV. In
the case
of a real scalar field with a potential that depends on the
square of
the field, and hence is invariant under the transformation of
the field
into minus itself, a nonzero VEV breaks this twofold symmetry.
In the
case of
multicomponent scalar fields transforming under some group G,
a
G-invariant potential can be found whose minima break G to any
subgroup
H. So the nice feature of the Higgs phenomenon is that it can
be used
to describe any symmetry breaking phenomenon. Furthermore, by
tuning
the parameters, one can describe several states of the theory
(or
"phases", in the language of condensed matter physics)
characterized by
different symmetries. Once again, note that the symmetry is a
property
of the state and not of the Lagrangian, which is always
invariant under
the full gauge group.
In
addition to breaking the symmetry, each VEV defines a
different energy
or mass scale. We are aware of three different fundamental
scales in
nature, each associated to a fundamental interaction. These
are: the
QCD scale, of the order of 1 GeV, the electroweak scale of the
order of
100 GeV and
the Planck scale, of the order of 10^19 GeV. The origin of
these scales
is still somewhat mysterious. The lowest one lies in an energy
regime
that has been fully explored experimentally, and corresponds
to the
onset of strong coupling phenomena in QCD. In the SM, the
electroweak
scale is associated to the Higgs VEV of 246 GeV. This energy
range is
just being explored by LHC and we will soon know whether the
last bit
of the SM will fall into place as expected. The Planck scale
is way too
far to be accessible directly, so all we can say about it is
based on
low energy gravity phenomena and speculation.
Typical GUTs have a
characteristic scale, given by the VEV of the relevant Higgs
field,
that is only two or three orders of magnitude
below the Planck scale. The hypothesis that there is nothing
between
the electroweak and the GUT scale is known as the "great
desert". If
one believes in coupling unification this hypothesis is
disfavored. In
fact, if one runs the renormalization group for the three SM
gauge
couplings starting from their low energy values, one finds
that they
come close but do
not exactly meet at the GUT scale. One popular fix is
supersymmetry, which would be broken at some scale above the
TeV, and
makes the unification of couplings much more accurate. But the
presence
of other intermediate scales could achieve the same result. In
fact,
typical GUT symmetry breaking chains usually assume the
existence of one or
more such intermediate scales. In this case the SM Higgs VEV
would mark
only the first of a number of scales that would end near the
Planck
scale.
The strength of the gravitational coupling between two
particles is a
dimensionless number given by the product of their center of
mass
energy, or the momentum transfer, times Newton's constant.
This number
is exceedingly small at low energy: even at one TeV its is of
order
10^{-32}. But this number depends quadratically on energy (in
contrast
to the gauge couplings that run only logarithmically) and it
becomes comparable to the other gauge couplings near the
Planck scale. It has been suggested that the nearness
of the GUT and the Planck scales is a sign that gravity also
becomes
unified with the other interactions at that scale.
Einstein
vs. Cartan
One wonders whether the unification mechanism that has been
used in the
SM and in GUTs could be further extended to encompass also the
gravitational interactions. At least up to a point, the answer
is
positive. Since nongravitational interactions are now all
understood to
be gauge interactions, and given that gauge fields can be
interpreted
mathematically as connections, this requires that also gravity
be seen
as the theory of a connection, a point of view that has also
been
emphasized by Ashtekar and in loop quantum gravity.
General relativity, the relativistic theory of gravity, is
typically
described in textbooks as the theory of the metric. A
connection
appears also in general relativity, but it is not an
independent field:
it is given by a formula that relates it to the metric, in
such a way
that given a metric one also automatically has a connection.
This
formula is never given a very convincing justification: it
works, and
it makes things simpler, and that is usually considered to be
enough.
There are extensions of general relativity where the
connection is
treated as an independent field, but the difference between
these
theories and ordinary general relativity could only manifest
itself at
high energy, in a regime that is outside the
experimental domain. Einstein was aware of these extensions,
but given
that all we know about gravity can be successfully modelled
with
general relativity, he never saw much use for them.
Einstein's pragmatic attitude should be contrasted with that
of
mathematicians. A
physicist is interested in the description of reality, but a
mathematician is free to develop the formalism to the full. The
significance
of connections as independent entities has been emphasized
in the mathematical literature beginning with Elie Cartan, who
was one
of the founders of modern differential geometry. He, and his
student
Ehresmann, developed among
others the notion of connection in a fiber bundle , that was
later to
play an important role in the geometry of gauge fields, and
the notion
of moving frame, which is widely used in general relativity.
While
ultimately equivalent to Einstein's in physical terms,
Cartan's
approach is "unification-friendly" in a way that Einstein's is
not.
The point is that viewing the gravitational connection as a
gauge field
for the Lorentz group (or perhaps its extension, the linear
group) we
can try to consider it as a piece of a larger unified gauge
field that
also contains the GUT gauge field. We will call a unified
theory of
this type a "GraviGUT",
to
emphasize
that it is a direct extension to gravity of the unification
philosophy that is used in particle physics. This idea has not
been
developed to the same extent as GUTs have: there are two very
suggestive pieces of evidence in its favor, and several major
obstacles.
The gravitational Higgs
mechanism
Modern unified theories rely heavily on the Higgs phenomenon
as a way
to generate mass for the gauge field in a gauge-invariant way.
It is
therefore very suggestive that a
variant
of Higgs phenomenon is already at work in ordinary
general
relativity, if only we adopt Cartan's point of view.
To make this manifest, one has to think of the connection as a
gauge
field for the linear group. Then, there are two additional
fields in
the theory: a moving linear frame (*not* an orthonormal
frame!) and a
metric. It is unusual to use both metric and frame as
dynamical fields.
This certainly makes for a clumsy and redundant formalism, but
our aim
here is not to do explicit calculations. The redundancy could
be
eliminated by making one of two standard gauge choices: to
choose a
nondynamical coordinate frame, in which case the metric is the
only
degree of freedom, or an orthonormal frame, in which case the
metric is
nondynamical and the frame is the only degree of freedom. But
for now let's
choose to remain in an arbitrary linear gauge.
In Einstein's theory one assumes that the connection is
compatible with
the metric and with the frame, in the sense that the covariant
derivative of the metric and the exterior covariant derivative
of the
frame both vanish. This selects a unique connection, which is
then
given by the formula mentioned earlier. If we do not make
these
assumptions, it is very natural to add to the Lagrangian terms
quadratic in the covariant derivative of the metric and in the
covariant exterior derivative of the frame: such terms are
just the
obvious kinetic terms of the dynamical fields. But then one
sees that
these terms are just a gauge-invariant way of writing masses
for the
linear connection, exactly as the kinetic term of the Higgs
field is a
gauge invariant way of writing a mass term for the W and Z
fields.
This provides an explanation for the fact that an independent
connection does not appear in low energy
gravitational phenomena: it gets mass from a gravitational
Higgs
phenomenon occurring at the Planck scale, and hence cannot be
excited
at low energies. While this Higgs phenomenon
has nothing to do with unification, it is easy to generalize
the
formalism in such a way that it does.
Fermions
The other
piece of evidence is of a rather different nature: it is not
related to
the geometry of the linear connection, but rather to the group
theory
of the fermion representations. It is a rather striking
feature of GUTs
that they do not predict the existence of new fermion fields
in
addition to those that are already known. The known fermions
fall into
three "families" that are completely identical except for the
different
masses. If new fermions existed, one would expect them to come
as
entire additional families, and there are experimental
constraints that
make the existence of additional families unlikely.
Since fermions behave as spinors under Lorentz
transformations, but do
not have a specified behavior under linear transformations, in
order to
meaningfully talk of fermions in general relativity one has to
assume
that the connection is metric, or equivalently that it is a
gauge field
for the Lorentz group. So we leave aside some of the freedom
discussed
above and we assume that the connection is Lorentz and we also
choose
the gauge where the frame is orthonormal.
From here
on we restrict our attention to a single fermion family. It
consists of
16 left-handed Weyl spinors (under Lorentz), representing two
leptons
(in the first family, the electron and the corresponding
neutrino) and
six quarks (in the first family, the up and down quarks, each
in three
colors). It
is one of the most compelling features of SO(10) GUT that
these fields
form a single object under the action of SO(10), namely a Weyl
representation. In GUT, it remains unexplained why the fields
that are
spinors
under the Lorentz transformations are also spinors under the
SO(10)
transformations.
If we consider gravity as a gauge theory for the Lorentz
group, it
seems natural to try and unify it with the already unified
strong and
electroweak interactions by regarding SO(3,1) and SO(10) as
commuting
subgroups in SO(3,11), and the corresponding gauge fields as
components
of a unified SO(3,11) gauge field. A nontrivial test of this
hypothesis
would be to check whether the fermions form representations of
this
GraviGUT group. The 16 Weyl spinors of one family are a set of
32
complex numbers, which in turn can be viewed as 64 real
numbers. It
turns out that
these real numbers can be rearranged into the simplest,
64-dimensional
real spinorial representation of
SO(3,11), called the Majorana-Weyl representation. (For the
cognoscenti, it may at first sound odd that a real
representation of
SO(3,11) corresponds to a complex representation of the
subgroup
SO(3,1)xSO(10). The point is that when one arranges the 64
real numbers
of the Majorana-Weyl representation into 32 complex numbers,
and looks
at the form taken by the representation matrices, the
generators in the
subgroup SO(3,11)xSO(10) correspond to complex linear
transformations
but those not in the subgroup correspond to complex antilinear
transformations.)
Assuming that the bosonic part of the theory generates the
appropriate
nontrivial VEVs to produce the correct symmetry breaking
chain, one can
write a Lagrangian for the fermionic fields that reduces at
low energy
to the correct
one describing fermions coupled to strong, electroweak and
gravitational interactions.
Issues
Given the effort and the ingenuity that has been spent in
attempts to
construct unified theories, it may be surprising that this
idea has not
been put forward before. Actually, one has to distinguish the
old
attempts at unification (meaning the ones in the first half of
the XX
century) from the modern ones (mostly by particle physicists,
in the
second half of the XX century). It turns out that in one of
his many
attempts at a unified theory of gravity and electromagnetism,
Einstein
came close to the idea of GraviGUT, in the sense that the
electromagnetic and gravitational connections were regarded,
in modern
language, as components of a connection in a five dimensional
vectorbundle (Einstein and Maier
1931-32).
Einstein
was not satisfied and abandoned this approach, but it is clear
that at the time one could not even properly formulate the
correct
questions.
Modern approaches to unification have followed, more or less
literally,
the higher dimensional philosophy of Kaluza and Klein. I
believe that
there are two main obstacles, one false and one real, that
have so far
prevented the development of a GraviGUT: the first is a
misinterpretation of the Coleman-Mandula theorem, the second
the issue
of ghosts.
The Coleman-Mandula theorem states that under a number of
hypotheses,
the symmetries of a theory must be the direct product of the
Poincare
group and a group of internal transformations. The pop version
of the
theorem states that "one cannot mix spacetime and internal
transformations", and if one forgets all the hypotheses that
go into
the theorem one would seem to come to the conclusion that a
form of
unification such as GraviGUT is forbidden. But there is a
number of examples
in which spacetime
and internal symmetries get mixed, and they do not violate the
theorem
because in such examples some of the hyptheses of the theorem
do not
hold. Likewise, GraviGUT does not violate the theorem. In
GraviGUT one
can envisage two phases of the theory: the broken phase in
which the
ground state is Minkowski spacetime,
and the
symmetric phase, in which the VEV of the appropriate order
parameter is
zero. In the former, the hypotheses of the Coleman-Mandula
theorem hold
true, and it is indeed the case that the GraviGUT symmetry is
broken to
a product of independent internal and spacetime
transformations. In the
latter, where gravity is undifferentiated from the other
interactions,
the spacetime metric vanishes and the hypotheses of the
theorem do not
hold. Thus GraviGUT is never in contrast with the
Coleman-Mandula
theorem. This has been originally pointed out here.
A much more serious issue is the one of the ghosts. It appears
as soon
as one allows in the Lagrangian terms quadratic in the
curvature
tensor. The simplest such term is the gravitational analogue
of the
Yang-Mills action, the square of the Riemann tensor. Since the
Lorentz
group is noncompact, the Hamiltonian corresponding to this
Lagrangian
is not positive definite. An
analysis
of the generic Lagrangian quadratic in curvature shows that
this is a
generic conclusion in theories with torsion.
The only conceivable way out of this issue is that quantum
effects will
save the day. Perhaps the closest analog would be confinement
in QCD:
there too, the bare Lagrangian is not a good guide to the
physical
spectrum of the theory. The fact that the bare Lagrangian of
QCD does
not suffer from unphysical modes is not the point here; the
point is
that quantum effects prevent quarks and gluons from appearing
as free
particles. Likewise, one may imagine that quantum effects
suppress the
propagation of the ghost states. In fact, a mechanism that
could lead
to this conclusion has been discussed here
and here
and here.
Whether this is really the case is worth studying in greater
detail.
Another issue is that the Lagrangians
that one usually writes require a nondegenerate metric, and in
the
symmetric phase of a GraviGUT the metric vanishes. There is no
problem
writing an
action that describes the theory in the broken phase. It is
harder but
not impossible to write one that makes sense in both phases,
and then
the phases manifest themselves as different solutions of the
equations of motion. To some extent this has been discussed here.
It is
doubtful that a GraviGUT can be constructed where the
transition between phases can be studied by tuning a mass
parameter, as
in the GUT Higgs potentials.
In
SO(3,11) GraviGUT, all fermions in one family form a single
representation. The three families have to be postulated as
three
independent copies of the same representation. This is no
better and no
worse than the SM. Thus the question that was raised by Rabi
in
connection with the muon ("who ordered that?") remains as open
as ever.
In particle physics there have been attempts to answer that
question by
postulating a symmetry that rotates the families into each
other, but
they were not successful. A bold extension of the GraviGUT
idea that
would also answer that question is Lisi's "Exceptionally simple
theory of
everything" based on E8. In Lisi's approach all fields -
bosonic
and fermionic - would be embedded in a single adjoint
representation of
E8. This approach faces considerable challenges in addition to
the ones
that we have already mentioned for any GraviGUT, and remains
for the
time being even more speculative.