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space_cadet said:
Thanks! I started a new thread for new questions.
Let me start with one I don't even know makes sense: are there counterparts to the braid group for higher dimensional objects like membranes?
space_cadet said:In GFT the basic building blocks are (n-1)-simplices (a 0-simplex is a point, a 1-simplex is a line-segment, 2-simplex is a triangle, 3-simplex is a tetrahedron ... and so on), which are glued together to form a simplicial complex (a discretized manifold), whose dynamics is given in terms of group elements assigned to each of the n faces of the (n-1)-simplex (see e.g. arXiv:0710.3276v1) The "field" is then taken to be a complex valued functions acting on these (n+1) group elements:
$$ \phi(g_1, g_2, \ldots, g_{n}) : G^n \rightarrow \mathbb{C} $$
Now, given (n+1) copies of a (n-1)-simplex, one can glue these together along their respective faces to form a n-simplex, e.g. for n=3, given four triangles (a triangle is a 2-simplex), one can glue them together along their edges to form a tetrahedron (which is a 3-simplex). One can write down an action for such a theory (see reference above) and explicitly compute various observable quantities. The resulting theory describes the dynamics of an n-dimensional manifold in terms of its constituent (n-1)-simplices.
The connection with the braid proposal arises from the observation that, a priori, there is no restriction on the form of the group G which is used to label faces of the simplices. G could be SU(2), SL(2,C) or even SL(2,Z) (the modular group) or B_3 (the three-stranded braid group). For instance, if one can write down a GFT action for 2-simplices, with edges labeled by representations of B_3, such an action would describe the dynamics of a manifold constructed by gluing the edges of triangles using 3-strand braids. This is the essence of the relationship I see between GFTs and the braid model. It may or may not turn out to technically feasible.
If you have further questions a new thread might be best, since this reply already takes this thread off-topic!
atyy said:Thanks! I started a new thread for new questions.
Let me start with one I don't even know makes sense: are there counterparts to the braid group for higher dimensional objects like membranes?
GFT stands for Group Field Theory. It is a mathematical framework that combines elements of group theory, quantum field theory, and graph theory to study quantum gravity in n-dimensional spacetimes. It is a promising approach to understanding the fundamental nature of the universe at a microscopic level.
The Braid Group is a mathematical group that describes the possible ways in which strands can be intertwined or braided in a given number of dimensions. It has applications in topology, knot theory, and theoretical physics, and is closely related to the GFT approach to quantum gravity.
The GFT approach to quantum gravity relies on the use of Feynman diagrams to represent the interaction of particles in spacetime. These diagrams can be interpreted as braids in higher dimensions, and the Braid Group provides a mathematical framework for studying and manipulating these diagrams. This allows for a deeper understanding of the quantum nature of spacetime.
N-dimensional manifolds are mathematical spaces that have specific properties, such as curvature and dimensionality. By exploring connections in these manifolds, we can gain a better understanding of the underlying structure of the universe and potentially uncover new insights into the nature of reality. This can lead to advances in fields such as physics, mathematics, and computer science.
While the GFT approach and the Braid Group are still in their early stages of development, they have the potential to greatly impact our understanding of the universe and lead to practical applications in fields such as quantum computing and cosmology. By integrating these theories into our understanding of the physical world, we may be able to make advancements in technology and solve complex problems that were previously thought to be unsolvable.