Hamiltonian dynamics of a quantum of space

In this post I follow up on some of the work reviewed in the post:

The action of the quantum mechanical volume operator plays a fundamental role in discrete quantum gravity models, can be given as a second order difference equation. By a complex phase change this can be  turned into a discrete Schrödinger-like equation.

The introduction of discrete potential–like functions reveals the role of hidden symmetries, first discovered by Regge for the quantum mechanical 6j symbols.

I’ll look at the underlying geometric features. When the spectrum and wavefunctions of the volume operator are discussed from the viewpoint of the Hamiltonian evolution of an elementary quantum of space, and an asymptotic picture emerges of the semiclassical and classical regimes.

The definition of coordinates adapted to Regge symmetry is used to construct a set of discrete orthogonal polynomials, characterizing the oscillatory components of torsion-like modes.


For an elementary spin network as shown below


A quadrilateral and its Regge “conjugate”illustrating the elementary spin network representation of the symmetric coupling scheme: each quadrilateral is dissected into two triangles sharing, as a common side, the diagonal l.


the volume operator K = J1 .J2 × J3,  acts democratically on vectors J1; J2 and J3 plus a fourth one, J4, which closes a not necessarily planar quadrilateral vector diagram J1 + J2 + J3 + J4 = 0.

The matrix elements of K are computed to provide a Hermitian representation, whose features have been studied by many see posts:

By a suitable complex change of phase we can transform the imaginary antisymmetric representation into a real, time-independent Schrödinger equation which governs the Hamiltonian dynamics as a function of a discrete variable denoted l. The Hilbert space spanned by the eigenfunctions of the volume operator can be constructed combinatorially and geometrically, applying polygonal relationships to the two quadrilateral vector diagram which are conjugated by a hidden symmetry discovered by Regge .

Discrete schrodinger equation and Regge symmetry

Eigenvalues k and eigenfunctions Ψl(k) of the volume operator are obtained through the three–terms recursion relationship – see post:

Applying a change of phaseΨl (k)=(-i)lΦl(k) to obtain a real, finite difference Schrodinger–like equation

hamiltonianequ1The Ψl are the eigenfunctions of the volume operator expanded in the J12 = J1 +J2 basis. The matrix elements αl in  are given in terms of geometric quantities, namely

hamiltonianequ2αl is proportional to the product of the areas of the two triangles sharing the side of length l and forming a quadrilateral of sides J1 +½, J2 +½, J3 +½ and J4 +½.

The requirement that the four vectors form a (not necessarily planar) quadrilateral leads to identify the range of l with


which is also the dimension of the Hilbert space where the volume operator acts.

Hamiltonian Dynamics

The Hamiltonian operator for the discrete Schrödinger equation


can be written, in terms of the shift operator


The two-dimensional phase space (l, φ) supports the corresponding classical Hamiltonian function given by

hamiltonianequ7This is  illustrated in below for the two Regge conjugate quadrilaterals of the diagram above.



The quadrilaterals are now allowed to fold along l with φ seen as a torsion angle.

The classical regime occurs when quantum numbers j are large and l can be considered as a continuous variable. This limit for l permits us to draw the closed curves in the (l, k) plane when φ = 0 or φ=π . These curves have the physical meaning of torsional-like potential functions


viewing the quadrilaterals as mechanical systems.

hamiltonianfig4Potential functions U+ and U in  hamiltonianequ8are shown for two cases where the conjugated tetrahedra coincide.

  • left panel:   j1,j2, j3, j4=100,110,130,140 the tangential quadrilateral
  • right panel: j1= j2=j3=j4 =120 the ex-tangential quadrilateral.

During the classical motion, the diagonal l changes its value preserving the energy of the system. The result is a geometric configuration  a tetrahedron changing continuously its shape but preserving its volume as a constant of motion.

Quantum mechanics extends the domain of the canonical variables to regions of phase space classically not allowed. Boundaries of these regions are the so-called potential-energy curves particularly important in applications. They are defined as turning points, namely the points where for each value of energy the classical  changes sign. This happens when the momentum φ is either 0 or π.


The above conditions define closed curved in the l-energy plane. These curves have the physical meaning of torsional like potential functions.

At each value of the possible values E’ of the Hamiltonian are bounded by


and the eigenvalues λk of the quantum system are bounded by



The sagemath code and output for this work is shown below:






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