Tag Archives: Canonical quantization

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:






Numerical work with sagemath 23: Wigner Reduced Rotation Matrix Elements as Limits of 6j Symbols

This work is based on the paper “Exact Computation and Asymptotic Approximations of 6j Symbols: Illustration of Their Semiclassical Limits by Mirco Ragni et al which I’ll be reviewing in my next post.

The 6j symbols tend asymptotically to Wigner dlnm functions when some angular momenta are large where θ assumes certain discrete values.







These formulas are illustrated below:




This can be modelled using sagemath.


The routine gives some great results:

For N=320, M=320, n=0, m=0, l=20, L=0,  Lmax=640

Wigner 6j vs cosθL


For N=320, M=320, n=0, m=0, l=10, L=0,  Lmax=640

Wigner 6j vs cosθL


For N=320, M=320, n=0, m=0, l=5, L=0,  Lmax=640

Wigner 6j vs cosθL



Group field theory as the 2nd quantization of Loop Quantum Gravity by Daniele Oriti

This week I have been reviewing Daniele Oriti’s work, reading his Frontiers of Fundamental Physics 14 conference  paper – Group field theory: A quantum field theory for the atoms of space  and making notes on an earlier paper, Group field theory as the 2nd quantization of Loop Quantum Gravity. I’m quite interested in Oriti’s work as can be seen in the posts:


We know that there exist a one-to-one correspondence between spin foam models and group field theories, in the sense that for any assignment of a spin foam amplitude for a given cellular complex,
there exist a group field theory, specified by a choice of field and action, that reproduces the same amplitude for the GFT Feynman diagram dual to the given cellular complex. Conversely, any given group field theory is also a definition of a spin foam model in that it specifies uniquely the Feynman amplitudes associated to the cellular complexes appearing in its perturbative expansion. Thus group field theories encode the same information and thus
define the same dynamics of quantum geometry as spin foam models.

That group field theories are a second quantized version of loop quantum gravity is shown to be  the result of a straightforward second quantization of spin networks kinematics and dynamics, which allows to map any definition of a canonical
dynamics of spin networks, thus of loop quantum gravity, to a specific group field theory encoding the same content in field-theoretic language. This map is very general and exact, on top of being rather simple. It puts in one-to-one correspondence the Hilbert space of the canonical theory and its associated algebra of quantum observables, including any operator defining the quantum dynamics, with a GFT Fock space of states and algebra of operators  and its dynamics, defined in terms of a classical action and quantum equations for its n-point functions.

GFT is often presented as the 2nd quantized version of LQG. This is true in a precise sense: reformulation of LQG as GFT very general correspondence both kinematical and dynamical. Do not need to pass through Spin Foams . The LQG Spinfoam correspondence is  obtained via GFT. This reformulation provides powerful new tools to address open issues in LQG, including GFT renormalization  and Effective quantum cosmology from GFT condensates.

Group field theory from the Loop Quantum Gravity perspective:a QFT of spin networks

Lets look at the second quantization of spin networks states and the correspondence between loop quantum gravity and group field theory. LQG states or spin network states can be understood as many-particle states analogously to those found in particle physics and condensed matter theory.

As an example consider the tetrahedral graph formed by four vertices and six links joining them pairwise


The group elements Gij are assigned to each link of the graph, with Gij=Gij-1. Assume  gauge invariance at each vertex i of the graph. The basic point is that any loop quantum gravity state can be seen as a linear combination of states describing disconnected open spin network vertices, of arbitrary number, with additional conditions enforcing gluing conditions and encoding the connectivity of the graph.

Spin networks in 2nd quantization

A Fock vacuum is the no-space” (“emptiest”) state |0〉 , this is the LQG vacuum –  the natural background independent, diffeo-invariant vacuum state.

The  2nd quantization of LQG kinematics leads to a definition of quantum fields that is very close to the standard non-relativistic one used in condensed matter theory, and that is fully compatible
with the kinematical scalar product of the canonical theory. In turn, this can be seen as coming directly from the definition of the Hilbert space of a single tetrahedron or more generally a quantum polyhedron.

The single field  quantum is the spin network vertex or tetrahedron – the so called building block of space.


A generic quantum state is anarbitrary collection of spin network vertices including glued ones or tetrahedra including glued ones.

gftfig5The natural quanta of space in the 2nd quantized language are open spin network vertices. We know from the canonical theory that they carry area and volume information, and know their pre geometric properties  from results in quantum simplicial geometry.

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