International Journal of Geometric Methods in Modern Physics
Vol. 8, No. 6 (2011) 1239–1258
c World Scientific Publishing Company
DOI: 10.1142/S021988781100566X
GENERALIZED GRAVITY IN CLIFFORD SPACES,
VACUUM ENERGY AND GRAND UNIFICATION
CARLOS CASTRO
Center for Theoretical Studies of Physical Systems
Clark Atlanta University
Atlanta, GA 30314, USA
Received 29 October 2010
Accepted 8 April 2011
Dedicated to the memory of Gustavo Ponce
Polyvector-valued gauge field theories in Clifford spaces are used to construct a novel
Cl(3, 2) gauge theory of gravity that furnishes modified curvature and torsion tensors
leading to important modifications of the standard gravitational action with a cosmolog-
ical constant. Vacuum solutions exist which allow a cancelation of the contributions of a
very large cosmological constant term and the extra terms present in the modified field
equations. Generalized gravitational actions in Clifford-spaces are provided and some
of their physical implications are discussed. It is shown how the 16 fermions and their
masses in each family can be accommodated within a Cl(4) gauge field theory. In partic-
ular, the Higgs fields admit a natural Clifford-space interpretation that differs from the
one in the Chamseddine–Connes spectral action model of non-commutative geometry.
We finalize with a discussion on the relationship with the Pati–Salam color-flavor model
group SU(4)C × SU(4)F and its symmetry breaking patterns. An Appendix is included
with useful Clifford algebraic relations.
Keywords: Clifford algebras; generalized gravity; unification.
1. Introduction
Clifford algebras are deeply related and are essential tools in many aspects in
Physics. The extended relativity theory in Clifford-spaces (C-spaces) is a natural
extension of the ordinary relativity theory [1] whose generalized polyvector-valued
coordinates are Clifford-valued quantities which incorporate lines, areas, volumes,
hyper-volumes, and so on, degrees of freedom associated with the collective parti-
cle, string, membrane, p-brane, and so on, dynamics of p-loops (closed p-branes) in
D-dimensional target spacetime backgrounds.
C-space relativity naturally incorporates the ideas of an invariant length (Planck
scale), maximal acceleration, non-commuting coordinates, supersymmetry, holog-
raphy, higher derivative gravity with torsion; it permits to study the dynamics of
all (closed) p-branes, for different values of p, on a unified footing [1]. It resolves
1239
,1240 C. Castro
the ordering ambiguities in QFT [4]; the problem of time in cosmology and admits
superluminal propagation (tachyons) without violations of causality [1, 5]. The rel-
ativity of signatures of the underlying spacetime results from taking different slices
of C-space [1, 6]. Ideas very close to the extended relativity in Clifford spaces have
been considered by [11, 14].
The conformal group SO(4, 2) in 4D spacetime emerges as a natural subgroup
of the Clifford group; i.e. one can express the conformal algebra generators in terms
of the Clifford Cl(4) algebra ones. Relativity in C-spaces involves natural scale
changes in the sizes of physical objects without the introduction of neither forces
nor Weyl’s gauge field of dilations [1]. A generalization of Maxwell theory of elec-
trodynamics of point charges to a theory in C-spaces involves extended charges
coupled to antisymmetric tensor fields of arbitrary rank and where the analog of
photons are tensionless p-branes. The extended relativity theory in Born–Clifford
phase spaces with a lower and upper length scales and the program behind a Clifford
group geometric unification was advanced by [16].
Furthermore, there is no EPR paradox in Clifford spaces [17] and one does
not need to introduce hidden variables because one can have causality in Clif-
ford spaces despite having non-causality in ordinary Minkowski spacetime. The
electron/positron can exchange information about their spins via the analog of
light signals in C-space; i.e. signals which appear, from the Minkowski spacetime
point of view, to be superluminal. Clifford-space tensorial-gauge fields generaliza-
tions of Yang–Mills theories and the Standard Model allows to predict the existence
of new particles (bosons, fermions) and tensor-gauge fields of higher-spins in the
10 TeV regime [19, 20]. Clifford-spaces can also be extended to Clifford-superspaces
by including both orthogonal and symplectic Clifford algebras and generalizing
the Clifford super-differential exterior calculus in ordinary superspace to the full
fledged Clifford-superspace outlined in [22]. Clifford-superspace is far richer than
ordinary superspace and Clifford supergravity involving polyvector-valued exten-
sions of Poincaré and (anti) de Sitter supergravity (antisymmetric tensorial charges
of higher rank) is a very relevant generalization of ordinary supergravity with appli-
cations in M -theory.
Grand-unification models in 4D based on the exceptional E8 Lie algebra have
been known for sometime [24]. The supersymmetric E8 model has more recently
been studied as a fermion family and grand unification model [29]. Supersymmetric
nonlinear sigma models of exceptional Kähler coset spaces are known to contain
three generations of quarks and leptons as (quasi) Nambu–Goldstone superfields
[30]. A Chern–Simons E8 gauge theory of gravity was proposed [31] as a unified
field theory (at the Planck scale) of a Lanczos–Lovelock gravitational theory with
a E8 generalized Yang–Mills field theory which is defined in the 15D boundary of a
16D bulk space. In particular, it was discussed in [22] how an E8 Yang–Mills in 8D,
after a sequence of symmetry breaking processes based on the non-compact forms
of E8 as follows E8(−24) → E7(−5) × SU(2) → E6(−14) × SU(3) → SO(8, 2) × U (1),
leads to a conformal gravitational theory in 8D based on gauging the non-compact
, Clifford Spaces, Vacuum Energy and Grand Unification 1241
conformal group SO(8, 2) in 8D. Upon performing a Kaluza–Klein–Batakis [32] com-
pactification on CP 2 , involving a non-trivial torsion, leads to a conformal gravity
Yang–Mills unified theory based on the Standard Model group SU(3)×SU(2)×U (1)
in 4D. Furthermore, it was shown [22] how a conformal (super) gravity and (super)
Yang–Mills unified theory in any dimension can be embedded into a (super) Clifford-
algebra-valued gauge field theory by choosing the appropriate Clifford group.
A candidate action for an exceptional E8 gauge theory of gravity in 8D was con-
structed [33]. It was obtained by recasting the E8 group as the semi-direct product
of GL(8, R) with a deformed Weyl–Heisenberg group associated with canonical-
conjugate pairs of vectorial and antisymmetric tensorial generators of rank two
and three. Other actions were proposed, like the quartic E8 group-invariant action
in 8D associated with the Chern–Simons E8 gauge theory defined on the seven-
dimensional boundary of an 8D bulk. To finalize, it was shown how the E8 gauge
theory of gravity can be embedded into a more general extended gravitational the-
ory in Clifford spaces associated with the Cl(16) algebra.
Quantum gravity models in 4D based on gauging the (covering of the) GL(4, R)
group were shown to be renormalizable by [34] however, due to the presence of
fourth-derivatives terms in the metric which appeared in the quantum effective
action, upon including gauge fixing terms and ghost terms, the prospects of unitarity
were spoiled. The key question remains if this novel gravitational model based on
gauging the E8 group may still be renormalizable without spoiling unitarity at the
quantum level.
Most recently it was shown in [36] how a conformal gravity and U (4) × U (4)
Yang–Mills grand unification model in four dimensions can be attained from a
Clifford gauge field theory in C-spaces (Clifford spaces) based on the (complex) Clif-
ford Cl(4, C) algebra underlying a complexified four-dimensional spacetime (eight
real dimensions). Upon taking a real slice, and after symmetry breaking, it leads
to ordinary gravity and a Yang–Mills theory based on the Standard Model group
SU(3)×SU(2)×U (1) in four real dimensions. Other approaches to unification based
on Clifford algebras can be found in [23].
Having presented some of the relevant issues behind the role of Clifford algebras
we outline the contents of this work. In Sec. 2, we construct a novel Cl(3, 2) gauge
theory of gravity that furnishes modified curvature and torsion tensors leading to
important modifications of the standard gravitational action with a cosmological
constant. Vacuum solutions exist which allow a cancelation of the contributions of
very large cosmological constant term with the extra terms present in the modified
field equations. Generalized gravitational actions in C-spaces are provided and some
of their physical implications are discussed. In the final section we describe how the
16 fermions and their masses in each family can be accommodated within a Cl(4)
gauge field theory. In particular, how the Higgs fields admit a natural C-space
interpretation that differs from the one in the Chamseddine–Connes spectral action
model of non-commutative geometry [63]. We finalize with a discussion on the
Vol. 8, No. 6 (2011) 1239–1258
c World Scientific Publishing Company
DOI: 10.1142/S021988781100566X
GENERALIZED GRAVITY IN CLIFFORD SPACES,
VACUUM ENERGY AND GRAND UNIFICATION
CARLOS CASTRO
Center for Theoretical Studies of Physical Systems
Clark Atlanta University
Atlanta, GA 30314, USA
Received 29 October 2010
Accepted 8 April 2011
Dedicated to the memory of Gustavo Ponce
Polyvector-valued gauge field theories in Clifford spaces are used to construct a novel
Cl(3, 2) gauge theory of gravity that furnishes modified curvature and torsion tensors
leading to important modifications of the standard gravitational action with a cosmolog-
ical constant. Vacuum solutions exist which allow a cancelation of the contributions of a
very large cosmological constant term and the extra terms present in the modified field
equations. Generalized gravitational actions in Clifford-spaces are provided and some
of their physical implications are discussed. It is shown how the 16 fermions and their
masses in each family can be accommodated within a Cl(4) gauge field theory. In partic-
ular, the Higgs fields admit a natural Clifford-space interpretation that differs from the
one in the Chamseddine–Connes spectral action model of non-commutative geometry.
We finalize with a discussion on the relationship with the Pati–Salam color-flavor model
group SU(4)C × SU(4)F and its symmetry breaking patterns. An Appendix is included
with useful Clifford algebraic relations.
Keywords: Clifford algebras; generalized gravity; unification.
1. Introduction
Clifford algebras are deeply related and are essential tools in many aspects in
Physics. The extended relativity theory in Clifford-spaces (C-spaces) is a natural
extension of the ordinary relativity theory [1] whose generalized polyvector-valued
coordinates are Clifford-valued quantities which incorporate lines, areas, volumes,
hyper-volumes, and so on, degrees of freedom associated with the collective parti-
cle, string, membrane, p-brane, and so on, dynamics of p-loops (closed p-branes) in
D-dimensional target spacetime backgrounds.
C-space relativity naturally incorporates the ideas of an invariant length (Planck
scale), maximal acceleration, non-commuting coordinates, supersymmetry, holog-
raphy, higher derivative gravity with torsion; it permits to study the dynamics of
all (closed) p-branes, for different values of p, on a unified footing [1]. It resolves
1239
,1240 C. Castro
the ordering ambiguities in QFT [4]; the problem of time in cosmology and admits
superluminal propagation (tachyons) without violations of causality [1, 5]. The rel-
ativity of signatures of the underlying spacetime results from taking different slices
of C-space [1, 6]. Ideas very close to the extended relativity in Clifford spaces have
been considered by [11, 14].
The conformal group SO(4, 2) in 4D spacetime emerges as a natural subgroup
of the Clifford group; i.e. one can express the conformal algebra generators in terms
of the Clifford Cl(4) algebra ones. Relativity in C-spaces involves natural scale
changes in the sizes of physical objects without the introduction of neither forces
nor Weyl’s gauge field of dilations [1]. A generalization of Maxwell theory of elec-
trodynamics of point charges to a theory in C-spaces involves extended charges
coupled to antisymmetric tensor fields of arbitrary rank and where the analog of
photons are tensionless p-branes. The extended relativity theory in Born–Clifford
phase spaces with a lower and upper length scales and the program behind a Clifford
group geometric unification was advanced by [16].
Furthermore, there is no EPR paradox in Clifford spaces [17] and one does
not need to introduce hidden variables because one can have causality in Clif-
ford spaces despite having non-causality in ordinary Minkowski spacetime. The
electron/positron can exchange information about their spins via the analog of
light signals in C-space; i.e. signals which appear, from the Minkowski spacetime
point of view, to be superluminal. Clifford-space tensorial-gauge fields generaliza-
tions of Yang–Mills theories and the Standard Model allows to predict the existence
of new particles (bosons, fermions) and tensor-gauge fields of higher-spins in the
10 TeV regime [19, 20]. Clifford-spaces can also be extended to Clifford-superspaces
by including both orthogonal and symplectic Clifford algebras and generalizing
the Clifford super-differential exterior calculus in ordinary superspace to the full
fledged Clifford-superspace outlined in [22]. Clifford-superspace is far richer than
ordinary superspace and Clifford supergravity involving polyvector-valued exten-
sions of Poincaré and (anti) de Sitter supergravity (antisymmetric tensorial charges
of higher rank) is a very relevant generalization of ordinary supergravity with appli-
cations in M -theory.
Grand-unification models in 4D based on the exceptional E8 Lie algebra have
been known for sometime [24]. The supersymmetric E8 model has more recently
been studied as a fermion family and grand unification model [29]. Supersymmetric
nonlinear sigma models of exceptional Kähler coset spaces are known to contain
three generations of quarks and leptons as (quasi) Nambu–Goldstone superfields
[30]. A Chern–Simons E8 gauge theory of gravity was proposed [31] as a unified
field theory (at the Planck scale) of a Lanczos–Lovelock gravitational theory with
a E8 generalized Yang–Mills field theory which is defined in the 15D boundary of a
16D bulk space. In particular, it was discussed in [22] how an E8 Yang–Mills in 8D,
after a sequence of symmetry breaking processes based on the non-compact forms
of E8 as follows E8(−24) → E7(−5) × SU(2) → E6(−14) × SU(3) → SO(8, 2) × U (1),
leads to a conformal gravitational theory in 8D based on gauging the non-compact
, Clifford Spaces, Vacuum Energy and Grand Unification 1241
conformal group SO(8, 2) in 8D. Upon performing a Kaluza–Klein–Batakis [32] com-
pactification on CP 2 , involving a non-trivial torsion, leads to a conformal gravity
Yang–Mills unified theory based on the Standard Model group SU(3)×SU(2)×U (1)
in 4D. Furthermore, it was shown [22] how a conformal (super) gravity and (super)
Yang–Mills unified theory in any dimension can be embedded into a (super) Clifford-
algebra-valued gauge field theory by choosing the appropriate Clifford group.
A candidate action for an exceptional E8 gauge theory of gravity in 8D was con-
structed [33]. It was obtained by recasting the E8 group as the semi-direct product
of GL(8, R) with a deformed Weyl–Heisenberg group associated with canonical-
conjugate pairs of vectorial and antisymmetric tensorial generators of rank two
and three. Other actions were proposed, like the quartic E8 group-invariant action
in 8D associated with the Chern–Simons E8 gauge theory defined on the seven-
dimensional boundary of an 8D bulk. To finalize, it was shown how the E8 gauge
theory of gravity can be embedded into a more general extended gravitational the-
ory in Clifford spaces associated with the Cl(16) algebra.
Quantum gravity models in 4D based on gauging the (covering of the) GL(4, R)
group were shown to be renormalizable by [34] however, due to the presence of
fourth-derivatives terms in the metric which appeared in the quantum effective
action, upon including gauge fixing terms and ghost terms, the prospects of unitarity
were spoiled. The key question remains if this novel gravitational model based on
gauging the E8 group may still be renormalizable without spoiling unitarity at the
quantum level.
Most recently it was shown in [36] how a conformal gravity and U (4) × U (4)
Yang–Mills grand unification model in four dimensions can be attained from a
Clifford gauge field theory in C-spaces (Clifford spaces) based on the (complex) Clif-
ford Cl(4, C) algebra underlying a complexified four-dimensional spacetime (eight
real dimensions). Upon taking a real slice, and after symmetry breaking, it leads
to ordinary gravity and a Yang–Mills theory based on the Standard Model group
SU(3)×SU(2)×U (1) in four real dimensions. Other approaches to unification based
on Clifford algebras can be found in [23].
Having presented some of the relevant issues behind the role of Clifford algebras
we outline the contents of this work. In Sec. 2, we construct a novel Cl(3, 2) gauge
theory of gravity that furnishes modified curvature and torsion tensors leading to
important modifications of the standard gravitational action with a cosmological
constant. Vacuum solutions exist which allow a cancelation of the contributions of
very large cosmological constant term with the extra terms present in the modified
field equations. Generalized gravitational actions in C-spaces are provided and some
of their physical implications are discussed. In the final section we describe how the
16 fermions and their masses in each family can be accommodated within a Cl(4)
gauge field theory. In particular, how the Higgs fields admit a natural C-space
interpretation that differs from the one in the Chamseddine–Connes spectral action
model of non-commutative geometry [63]. We finalize with a discussion on the
