Abstract
This paper aims at reviewing various topics and different issues related essentially to topological quantum field theories and string theory. We present recent work and discuss new ideas and results from these topics. We focus on the subject of the geometric and topological structures and invariants, which enriched in a remarkable way relativistic and quantum field theory in the last century, starting from Einstein’s general relativity until string theory. In the last three decades, new and deep developments in this direction have emerged from theoretical physics. We stress the crucial fact that many physical phenomena, at the quantum and at the cosmological level as well, appear to be related to deep geometrical and topological invariants, and furthermore that they are effects which emerge, in a sense, from the geometric structure of space–time. Topological quantum field theory (TQFT) arisen in the eighties as a new relation between mathematics and physics, and appears as a very rich and promising research program in theoretical physics. Two conceptual points appear to be very significant in TQFT, and likely promising for physics. The first is the assumption of an effective correlation between knots and link invariants and the physical observables and states of quantum field theories and gauge theories. The second is, on the one hand, the idea of the fuzziness of space–time and of its emergence from the dynamical fluctuations of physical phenomena, on the other, the idea of the geometric and topological nature of physical process at different scales.
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Notes
- 1.
Finite groups are spacial cases of compact Lie groups. For example, the rotation group SO(3) of the three-dimensional Euclidean space or the gauge group U(1) × SU(2) × SU(3) of the Standard Model in elementary particle physics are compact Lie groups.
- 2.
Recall that if A is an involutive algebra over the complex numbers C, then a Fredholm module over A consists of an involutive representation of A on a Hilbert space H, together with a self-adjoint operator F, of square 1 and such that the commutator [F, a] is a compact operator for all a ∈ A.
- 3.
A good example is quantum gravity in 3-dimensional space–time. Classicaly, Einstein’s equations predict qualitatively very different phenomena depending on the dimension of space–time. If space–time has 4 or more dimensions, Einstein’s equations imply that the metric has local degrees of freedom. In other words, the curvature of space–time at a given point is not completely determined by the flow of energy and momentum through that point: it is an independent variable in its own right. For example, even in the vacuum, where the energy–momentum tensor vanishes, localized ripples of curvature can propagate in the form of gravitational radiation. In 3-dimensional space–time, hovewer, Einstein’s equations suffice to completely determine the curvature at a given point of space-tume in terms of the flow of energy and momentum through that point. We thus say that metric has no local degrees of freedom. In particular, in the vacuum the metric is flat, so every small patch of empty space–time looks exactley like every other.
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Boi, L. (2022). Topological Quantum Field Theory and the Emergence of Physical Space–Time from Geometry. New Insights into the Interactions Between Geometry and Physics. In: Wuppuluri, S., Stewart, I. (eds) From Electrons to Elephants and Elections. The Frontiers Collection. Springer, Cham. https://doi.org/10.1007/978-3-030-92192-7_23
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