This week I have been reading a couple of PhD thesis
- Complete Loop Quantum Gravity Graviton Propagator by Emanuele Alesci
- Semiclassical analysis of Loop Quantum Gravity by Claudio Perini
They are both quite nice works and contain content relevant to my work on the quantum tetrahedron. In this post I be looking at some of Alesci’s work whilst I will write about Perini’s work in a later post.
Quantum tetrahedron in 3d
Here the author describes the quantum geometry of a tetrahedron in 3 dimensions as given by a spin network independently from the LQG approach. This description is in the context of spinfoam models and is used in the calculation of the graviton propagator.
Consider a compact, oriented, triangulated 3-manifold , and the complex Δ∗ dual to it, so having one node for each tetrahedron in and one link for each face – a triangle.
Considering a single tetrahedron in a 3d reference system R³; its geometry is uniquely determined by the assignment of its 4 vertexes. The same geometry can be determined by a set of 4 bivectors Ei – elements of ∧²R³ obtained taking the wedge product of the
displacement vectors of the vertexes) normal to each of the 4 triangles satisfying the closure constraint
where the last constraint simply says that the triangles close to form a tetrahedron. The quantum picture proceed as follows. Each bivector corresponds uniquely to an angular momentum operator in 3 dimensions , so an element of SU(2) (using the isomorphism between
∧²R³ and so(3)), and we can consider the Hilbert space of states of a quantum bivector as given by H = ⊕j, where j indicates the spin-j representation space of SU(2). Considering the tensor product of 4 copies of this Hilbert space, we have the following operators acting
with I = 1, 2, 3, and the closure constraint is given by
But now the closure constraint indicates nothing but the invariance under SU(2) of the state ψ so that the Hilbert space of a quantum tetrahedron is given by:
or in other words the sum, for all the possible irreducible representations assigned to the triangles in the tetrahedron, of all the possible invariant tensors of them, i.e. all the possible intertwiners.
It isalo possible to define 4 area operators;
and a volume operator;
and find that all are diagonal on;
the area having eigenvalue;
We can think of a spin network living in the dual complex Δ∗ of the triangulation, and so with one 4-valent node, labelled with an intertwiner, inside each tetrahedron, and one link, labelled with an irreducible representation of SU(2), intersecting exactly one
triangle of the tetrahedron. We then immediately recognize that this spin network completely characterizes a state of the quantum tetrahedron, so a state in T , and gives volume to it and areas to its faces – matching the results from LQG.
Geometrical interpretation and quantum tetrahedron
Now we look at an approach leading to the Barrett-Crane model; based on the geometrical interpretation of the discretized B fields on a fixed triangulation.
The first step is to turn the bivectors associated to the triangles into operators. To do this use is made of the isomorphism between the space of bivectors ∧²R4 and so(4) identifying the bivectors associated with the triangles with the generators of the algebra:
Then we proceed turning these variables into operators by associating to the different triangles t an irreducible representation ρt of the group and the corresponding representation space, so that the the generators of the algebra act on it as derivative operators. Unitary representations are chosen.
in the Riemannian case. However, this is not yet the Hilbert space of a quantum geometric triangle, simply because a set of bivectors does not describe a simplicial geometry unless it satisfies constraints equal to the Plebanski one for the discretized B fields. The task is then to translate these constraints in the language of group representation theory into the quantum domain.
Of the constraints given for the bivectors,only
The most important constraint is however the simplicity constraint, that forces the bivectors to be simple, i..e formed as wedge product of two edge vectors. Classically this was expressed by: Bt ·∗Bt = 0. Using the quantization map and substituting the bivectors with
the Lie algebra elements in the representation ρt for the triangle t, we obtain the condition:
The condition is then of having vanishing pseudoscalar Casimir i.e the condition is to have simple representations.
The simplicity constraint forces the selfdual and anti-selfdual parts to be equal, and we are restricted to considering only representations of the type (j, j).
The Hilbert space of a quantum triangle is given by:
In this way the geometrical meaning of the parameters labelling the representations becomes clear: Classically the area of a triangle is given in terms of the associated bivector by:
that in the quantum case translates into
i.e. the area operator is the scalar Casimir and it is diagonal on each representation space associated to the triangle. Its eigenvalues are then in the Riemannian case, for simple representations
The representation label characterizes the quantum area of the triangle to which that representation is associated.
States of the quantum theory are assigned to 3-dimensional hypersurfaces embedded in the 4-dimensional spacetime, and are thus formed by tetrahedra glued along common triangles. We then have to define a state associated to each tetrahedron in the triangulation, and then define a tensor product of these states to obtain any given state of the theory. In turn, the Hilbert space of quantum states for the tetrahedron has to be obtained from the Hilbert space of its triangles, since these are the basic building blocks at our disposal corresponding to the basic variables of the theory.
Each tetrahedron is formed by gluing 4 triangles along common edges, and this gluing is naturally represented by the tensor product of the corresponding representation spaces; Spin(4) decomposes into irreducible representations which do not necessarily satisfy the simplicity constraint.
The third of the constraints composition on bivectors translates at the quantum level as the requirement that only simple representations appear in this decomposition. Therefore we are considering for each tetrahedron with given representations assigned to its triangles a tensor in the tensor product:
of the four representation spaces for its faces, with the condition that the tensored spaces decompose pairwise into vector spaces for simple representations only. We have still to impose the closure constraint; this is an expression of the invariance under the gauge group of the tensor we assign to the tetrahedron, thus in order to fulfill the closure constraint we have to associate to the tetrahedron an invariant tensor, i.e. an intertwiner between the four simple representations associated to its faces:
Therefore the Hilbert space of a quantum tetrahedron is given by:
with Hi being the Hilbert space for the i-th triangle defined above. Each state in this Hilbert space turns out to be the Barratt-Crane intertwiner.
This construction was rigorously perfomed in the Riemannian case, using geometric quantization methods. In any case, this invariant tensor represents is the state of a tetrahedron whose faces are labelled by the given representations; it can be represented graphically as:
A vertex corresponding to a quantum tetrahedron in 4d, with links labelled by the representations and state labels associated to its four boundary triangles.
The result is that a quantum tetrahedron is characterized uniquely by 4 parameters, i.e. the 4 irreducible simple representations of so(4) assigned to the 4 triangles in it, which in turn are interpretable as the oriented areas of the triangles. This result is geometrically rather puzzling, since the geometry of a tetrahedron is classically determined by its 6 edge lengths, so imposing only the values of
the 4 triangle areas should leave 2 degrees of freedom, i.e. a 2-dimensional moduli space of tetrahedra with given triangle areas. For example, this is what would happen in 3 dimensions, where we have to specify 6 parameters also at the quantum level. So why does a tetrahedron have fewer degrees of freedom in 4 dimensions than in 3 dimensions, at the quantum level, so that its quantum geometry is characterized by only 4 parameters? The answer is that the essential difference between the 3-dimensional and 4-dimensional cases is represented by the simplicity constraints that have to be imposed on the bivectors in 4 dimensions. At the quantum level these additional constraints reduce significantly the number of degrees of freedom for the tetrahedron, as can be shown using geometric quantization see the post The Quantum Tetrahedron in 3 and 4 Dimensions , leaving us at the end with a 1-dimensional state space for each assignment of simple irreps to the faces of the tetrahedron, i.e. with a unique quantum state up to normalization.
Then the question is: what is the classical geometry corresponding to this state? In addition to the four triangle areas operators, there are two other operators that can be characterized just in terms of the representations assigned to the triangles: one can consider the parallelograms with vertices at the midpoints of the edges of a tetrahedron and their areas, and these are given by the representations entering in the decomposition of the tensor product of representations labelling the neighbouring triangles. Analyzing the commutation relations of the quantum operators corresponding to the triangle and parallelogram area operators, it turns out that while the 4 triangle area operators commute with each other and with the parallelogram areas operators among which only two are independent, the last ones have non-vanishing commutators among themselves. This implies that we are free to specify 4 labels for the 4 faces of the tetrahedron, giving 4 triangle areas, and then only one additional parameter, corresponding to one of the parallelogram areas, so that only 5 parameters determine the state of the tetrahedron itself, the other one being completely randomized, because of the uncertainty principle.
Consequently, we can say that a quantum tetrahedron does not have a unique metric geometry, since there are geometrical quantities whose value cannot be determined even if the system is in a well-defined quantum state. In the context of the Barrett-Crane spin
foam model, this means that a complete characterization of two glued 4-simplices at the quantum level does not imply that we can have all the informations about the geometry of the tetrahedron they share. This is an example of the kind of quantum uncertainty relations
that we can expect to find in a quantum gravity theory.
A generic quantum gravity state is to be associated to a 3-dimensional hypersurface in spacetime, and this will be triangulated by several tetrahedra glued along common triangles; therefore a generic state will live in the tensor product of the Hilbert spaces of the tetrahedra of the hypersurface, and in terms of the Barrett-Crane intertwiners it will be given by a product of one intertwiner for each tetrahedron with a sum over the parameters -angular
momentum projections – labelling the particular triangle state for the common triangles -the triangles along which the tetrahedra are glued. The resulting object will be a function of the simple representations labelling the triangles, intertwined by the Barrett-Crane intertwiner to ensure gauge invariance, and it will be given by a graph which has representations of Spin(4) labelling its links and the Barrett-crane intertwiner at its nodes; in other words, it will be
given by a simple spin network.
We are left with a last ingredient of a quantum geometry still to be determined: the quantum amplitude for a 4-simplex χ, interpreted as an elementary change in the quantum geometry and thus encoding the dynamics of the theory, and representing the fundamental building block for the partition function and the transition amplitudes of the theory.
This amplitude can be constructed out of the tensors associated to the tetrahedra in the 4-simplex, so that it immediately fulfills the conditions necessary to describe the geometry of the simplicial manifold, at both the classical and quantum level, and has to be invariant under the gauge group of spacetime Spin(4). The natural choice is to obtain a C-number for each 4-simplex, a function of the 10 representations labelling its triangles, by fully contracting the tensors associated to its five tetrahedra pairwise summing over the parameters associated to the common triangles and respecting the symmetries of the 4-simplex, so:
The amplitude for a quantum 4-simplex is thus, in the Riemannian case:
So we obtain the amplitude as the 10j-symbol.
This amplitude is for fixed representations associated to the triangles, i.e. for fixed triangle areas; the full amplitude involves a sum over these representations with the above amplitude as a weight for each configuration. In general, the partition function of the quantum theory describing the quantum geometry of a simplicial complex made of a certain number of 4- simplices, will be given by a product of these 4-simplex amplitudes, one for each 4-simplex in the triangulation, and possibly additional weights for the other elements of it, triangles and tetrahedra, with a sum over all the representations assigned to triangles. We can then envisage a model of the form:
or, if one sees all the data as assigned to the 2–complex Δ∗ dual to the triangulation with faces f dual to triangles, edges e dual to tetrahedra, and vertices v dual to 4-simplices:
clearly with the general structure of a spin foam model.
The last aspect that needs to be implemented is sum over triangulations; the restriction to a fixed triangulation of spacetime has to be lifted, since in this non-topological case it represents a restriction of the dynamical degrees of freedom of the quantum spacetime, and a suitably defined sum over triangulations has to be implemented.
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