tag:blogger.com,1999:blog-2381900080106522441.post3459522189695876338..comments2018-04-16T20:13:13.025+02:00Comments on relatively quantum: Updates on some criticisms of loop quantum gravityNorberthttp://www.blogger.com/profile/04965554394207041308noreply@blogger.comBlogger5125tag:blogger.com,1999:blog-2381900080106522441.post-29256122970216975282016-04-28T08:51:40.126+02:002016-04-28T08:51:40.126+02:00Matter propagates in a spin foam according to the ...Matter propagates in a spin foam according to the discretized form of the corresponding matter-field action. In the case of fermions there is a problem to formulate such an action because the fermions couple to the tetrads, and the tetrads are not well-defined in a spin-foam geometry. That was the reason to propose a spin-cube formulation, which is a generalization of the spin-foam formulation, based on the substitution of the spin-foam group by a 2-group (in the QG case the Lorentz group is replaced by the Poincare 2-group). The consequence is that the tetrads enter the formalism explicitely, through the edge lengths, so that a spin foam (a colored 2-complex) is replaced by a spin cube, which is a colored 3-complex. Hence in a spin cube the edges carry the 2-group irreps, the triangles carry the corresponding 1-intertwiners and the tetrahedrons carry the corresponding 2-intertwiners as oposed to a spin foam where the triangles carry the group irreps and the tetrahedrons carry the corresponding intertwiners.A. Mikovichttps://www.blogger.com/profile/03175906801121515444noreply@blogger.comtag:blogger.com,1999:blog-2381900080106522441.post-22811942066855870582016-04-27T11:45:40.255+02:002016-04-27T11:45:40.255+02:00Thank you for the clarification. It still seems to...Thank you for the clarification. It still seems to me that the notion of local Lorentz invariance that you refer to is not what one means when one asks whether LQG is locally Lorentz invariant from a phenomenological point of view. There, the question of how matter propagates, morally from 4-simplex to 4-simplex, is the relevant one. Norberthttps://www.blogger.com/profile/04965554394207041308noreply@blogger.comtag:blogger.com,1999:blog-2381900080106522441.post-7067440309221112032016-04-26T14:52:53.431+02:002016-04-26T14:52:53.431+02:00In canonical LQG the local Lorentz invariance can ...In canonical LQG the local Lorentz invariance can be established by constructing the generators of local Lorentz rotations from holonomy and flux operators (if there is fermionic matter one has to add the matter contributions to the generators) and then verify that the algebra closes. This is a difficult task and anomalies may appear. In the spin-foam case, the analog of the local Lorentz transformations would be the Lorentz transformations that preserve the flat metric in each 4-simplex (provided that such a metric exists in every 4-simplex).A. Mikovichttps://www.blogger.com/profile/03175906801121515444noreply@blogger.comtag:blogger.com,1999:blog-2381900080106522441.post-35822346223921822432016-04-25T11:38:02.938+02:002016-04-25T11:38:02.938+02:00Thank you for pointing out these differences betwe...Thank you for pointing out these differences between canonical LQG and the spinfoam formulation. I didn’t include them in my post, because I thought that they would distract too much from the core issues. <br /><br />About Lorentz invariance: I am not sure if we speak about the same thing. As emphasised in the post, the notion of “Lorentz invariance” appears in LQG both when choosing the internal gauge group (which can be the Lorentz group), and in the context of the question how the quantum geometry described by your physical state behaves (e.g. how matter fields propagate on it). While there are spinfoam models which are “Lorentz invariant” w.r.t. the the former notion, this is not clear for the second one, and thus for what people usually mean by Lorentz invariance. Norberthttps://www.blogger.com/profile/04965554394207041308noreply@blogger.comtag:blogger.com,1999:blog-2381900080106522441.post-19602610209561224622016-04-24T12:28:04.800+02:002016-04-24T12:28:04.800+02:00In order to better understand the theory called LQ...In order to better understand the theory called LQG, it is necessary to make the distinction between the spin-foam (covariant) and the canonical formulation. Although many people beleive that these two are the same theory, they are really two different theories. In the canonical LQG<br />the spacetime is topologically Sx[a,b] and there are colored graphs (i.e. spin networks) embedded in a smooth 3-manifold S. A physical state is a linear combination of the spin-network states such that it satisfies the hamiltonian constraint. The classical limit in this case is<br />defined by veryfying whether (hbar)Im(log Psi(A)) satisfies the Hamilton-Jacobi equation for the canonical GR in the limit hbar --> 0 and A is the Ashtekar-Barbero connection on S. This is a very difficult task, so that one uses the spin-foam formulation. In the spin-foam formulation, the theory is formulated on a triangulation of Sx[a,b], which is not the same as the smooth manifold Sx[a,b]. Hence obtaining the classical limit requires performing 2 limits, as you point out, i.e. the limit hbar --> 0, or equivalently j(triangle) --> infinity, and the smooth limit, i.e. N --> infinity, where N is the number of 4-simplices in T(Sx[a,b]). The easiest way to study these two limits is to use the effective action formalism. One then obtains that the first limit gives the area-Regge action. This is a problem, because area-Regge theory does not define always a metric geometry, so it is not equivalent to the usual Regge discretization of GR. Hence one needs to modify the usual spin-foam formulation in order to implement the constraints which reduce the area-Regge theory to the length-Regge theory. <br /><br />As far as the Lorentz invariance is concerned, it is not problematic for the spin-foam formulation, because the unitary irreps of the Lorentz group are (j,p) where j is an SU(2) spin and p is a real number. Also one can assume that the area of a triangle in a spacetime triangulation is determined by the flat metric in a 4-simplex which contains that triangle. The Lorentz invariance is more problematic in the canonical case, because it is not manifest.<br />A. Mikovichttps://www.blogger.com/profile/03175906801121515444noreply@blogger.com