Supersymmetric-field theory remains the best new physics beyond the SM and also “builds a bridge between the low energy phenomenology and high-energy fundamental physics“. Moreover, I will show here that SuSy can incorporate aspects of loop quantum gravity to shed deep light on fundamental issues in quantum cosmology. Before getting deep, recall that the central action for supersymmetric field theory is:
with the Grassmannian variable and total action is given by:
and the super-field/anti-super-field theory expansion in the Landau type gauge is given by:
with the super-covariant derivatives of
are defined by:
and the super-derivative is:
where the Ashtekar-Barbero connection , for a, b = 1, 2, 3, is given in terms of co-triads :
and the extrinsic curvature being:
For definitions of terms, see my ‘SuperSymmetric Field Theory and the Quantum Master Equation‘ post. Some philosophical points are in order first.
In this, part 1, I will take the analysis up to but not inclusive of deriving a supersymmetric generalization of group field cosmology and derive the group field cosmology action and the corresponding equation of motion. Aside: I highly recommend this book.
First, although loop quantum gravity is provably not an adequate quantum gravity theory nor a successful unification paradigm, some of its mathematical-physics insights can, via super-group-field theory, shed light, in conjunction with supersymmetric-field-cosmology, on a GUT quantum cosmology theory. LQG theory is background independent quantization of gravity (though not successful) where the Hamiltonian constraint is given via the Ashtekar-Barbero connection and densitized triad mentioned above. Hence, the curvature of the connection is determined by the holonomy around a loop whose corresponding Hilbert space is a space of spin networks and the spin foam arises analytically from the time evolution of these spin networks, implying that it is a second quantized formalism. One, as usual, gets via a third quantization of LQG the corresponding group field theory that is homeomorphic to a quotiented orfibold in supersymmetric field theory. Read this for a great exposition and background for this technical post.
Crucial is to realize that in loop quantum gravity, the holonomies of the connection are given by operators. Given the Ashtekar-Barbero connection, the metric is:
with the scalar factor and the lapse function discussed in my post here. With the crucial Barbero-Immirzi parameter and the Levi-Civita connection , we have:
and is completely determined in loop quantum cosmology by:
with the time derivative of . The operators:
being the conjugate to , a function of . Thus,
is a basis of the eigenstates of the volume operator and we have
with the dimensions of length being
Now, the curvature of is given via the holonomy around a loop and the area of such a loop cannot be smaller than a fixed minimum area since the smallest eigenvalue of the area operator in loop quantum gravity is nonzero! Hence, in Planck units, the Hamiltonian constraint for a homogeneous isotropic universe with a massless scalar field is given as:
where is defined as:
with and for the gauge-choice in loop quantum cosmology, we have as usual:
Setting , the canonical quantization scheme yields:
and with , solvably, we have: