Abstract
In this chapter we will report on another direction of investigation for SQC, which has received some attention. Instead of using a differential operator representation for the fermionic momenta (see Chap. 5 of either volume), a matrix representation is employed. The ground-breaking importance of the matrix representation approach should not be ignored. Since it was introduced to the SQC community in the late 1980s, this line of research has indeed provided many routes for study. In addition, there are still some unresolved issues to explore, and these deserve due consideration. In this chapter, we thus present this research programme [1–10].
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Notes
- 1.
We are referring to those products of the fermionic components where the indices are all contracted, possibly with the assistance of other variables, e.g., the tetrad (see Sect. 5.2.3 of Vol. I).
- 2.
Recall that the Lorentz constraints act (classically as generators) on the flat (tangent space) indices present in the, e.g., vierbein and the Rarita–Schwinger variables (denoted by the latin letters a, b, … throughout this book).
- 3.
Compare results in Exercise 7.1 with what we retrieve here in Sect. 7.2.1.
- 4.
- 5.
The index 1 in \({\mathcal S}_1\) is the spinor index associated with \(\overline{{\check{\psi}}}{}^{[a]}_0\), [a] = 1.
- 6.
Generically, they will satisfy a Clifford algebra via a matrix representation of the algebras, taking \({\check{\psi}}_{i[a]}\) to contain all Grassmann dependence (and constituting a classical limit of Dirac matrices).
- 7.
Each quantum constraint is written \( \mathcal{S}_{[a]} \varPsi_{[a]} = 0\), where each \( \mathcal{S}_{[a]}\) represents a separate square root for the Hamiltonian constraint \( \mathcal{H}_\bot\), and the \( \mathcal{S}_{[a]}\) are matrices of the smallest possible rank that produces the appropriate algebra for \( \mathcal{S}_{[a]}\) and the \({\check{\psi}}_{i{[a]}}\).
- 8.
Chapter 5 of Vol. I focused on constructing inequivalent Lorentz invariants, i.e., providing scalar wave functions as fermionic and bosonic power series expansions that automatically fulfill the Lorentz condition.
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Moniz, P.V. (2010). SQC Matrix Representation. In: Quantum Cosmology - The Supersymmetric Perspective - Vol. 2. Lecture Notes in Physics, vol 804. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-11570-7_7
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