Abstract
In a class of quantum gravity approaches it is indicated that our observable world emerges out of a fundamental structure that appears highly resistant to any clear spatial or temporal interpretation. In this work we are examining an analogue quantum system that appears to simulate such an unintuitive structure: the emergence of the so called topological phase of matter depicted by the Chern–Simons gauge theory. By investigating the proposed analogy from the lens of category theory, we offer a clear interpretation of the way in which space and time act at the fundamental level.
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Notes
Here by existence we mean physical existence linked with concrete entities.
Here objects refer to the elements of a category, and they are not necessarily linked with physical entities.
In this context, we employ the concept of analogue simulation in a less stringent manner by focusing solely on establishing the plausibility of a hypothesis, rather than using it as a means of confirming specific phenomena that may exist within the inaccessible target system as proposed by Dardashti et al. (2017).
Low energy and short distance means that deformation invariance is only violated by terms that vanish as \(\exp (-\Delta /T)\), where \(\Delta \) denotes some energy gap in the system. Therefore, generally systems at topological phase admit energy gap that separates (degenerate) ground states from the lowest excited states.
An important clarification needs to be made here in regards to the distinction between topological quantum field theories, which serve as effective descriptions of topological phases, and diffeomorphic invariant (general covariant) classical field theories such as the general theory of relativity (GR). The difference lies in the fact that topological quantum field theories do not admit any local degrees of freedom, while GR is fundamentally a local theory. The reason for this difference arises from the quantization or path integral transformation of the Chern–Simons action. In fact, if we were to perform a path integral of the Einstein–Hilbert action over all possible metric fields, it would similarly give rise to a topological field theory.
The ambient space refers to the environment in which the systems is embedded.
Here by a degree of freedom we refer to any dynamical variable or observable the perturbation of which induces a measurable change in the state of the system.
Here the gauge field \(\mathcal {A}\) is not a local degree of freedom, but rather emerges from the collective behavior of the underlying many-body system. Ideally \(\mathcal {A}\) can be thought of as analogous to the way quasiparticles such as phonon degrees of freedom emerge from the collective motion of particles.
In the abelian case \(\mathcal {A}\) is a vector of numbers for which Tr is trivial and \(\mathcal {A}\wedge \mathcal {A}\wedge \mathcal {A}\) vanishes.
Such quasi-particles are referred to as (non-abelian) anyons which exhibit exotic statistical properties. Unlike bosons and fermions the exchange of two identical anyons can induce any unitary operation on the original state.
Two categories C and D are equivalent if for a functor \(\mathcal {F}: C \rightarrow D\) when there exists an other functor \(\mathcal {G}: D \rightarrow C\) such that \(\mathcal {G}.\mathcal {F}\) is isomorphic to the identity functor from C to itself, and \(\mathcal {F}.\mathcal {G}\) is isomorphic to identity functor from D to itself.
Here by distinguishable states we refer to states of which the relative entropy is non-zero (Farshi & DeBianchi, 2022).
In a world where the underlying quantum field theory is believed to be geometric, the possibility of reducing the background Lorentzian metrical structure in which quantum systems operate to pure relations in a category is not clear. In attempting to formalise the structure of QFTs in curved space-times, Brunetti et al. (2003), Sanders (2020) construct a functor that corresponds to a category of Lorentzian manifolds (Loc) a category of C* algebra (Alg) that encodes the QFT systems. In this approach, the local structure as well as the principle of general covariance is purely encoded in the morphisms of the Loc category whereas the metric structure, is contained in the objects. This might be an implication that the metrical structure is a property that can not be reduced to mere relations in a category. This is not surprising after all since the background metric is essentially linked with the gravitational field that should be treated as a physical systems on its own and be represented in terms of algebraic relation similar to the other fields in the theory. Encoding algebraically the metric structure of spacetime in the category of quantum processes is the grand goal of any theory of quantum gravity, which yet to be constructed.
\(\sigma _i\) denote the Pauli matrices.
Here \(\text {Hom}(X;Y)\) is a set—an object of the category of sets, \(\textbf{Set}\).
It is important to note that here the dual object does not represent the measuring apparatus itself, but rather codifies the effect of a measuring process i.e. the mapping from a quantum state \(\vert \psi \rangle \) to a complex number: \(\langle \psi \vert ...\rangle : \vert \psi \rangle \rightarrow \mathbb {C}\). This is already implied in Dirac notation of quantum measurement: a state that is represented as a ket state is measured by its projection onto a bra state—the dual state.
These symmetry groups form the irreducible unitary representations of the Poincaré group.
This point has been indicated by the paragraph 2.0124 in the Tractatus: on the one hand space and time shape the state of affairs, and on the other, all the possible states of affairs are given once one have access to all the objects. In this way, a balance in the dependence relation between space and objects is reached. As Fogelin puts it (1987): “By establishing a systematic parity between two fundamental principles of atomism (matter and void or being and non-being), Wittgenstein gives this position its most coherent articulation.”
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Acknowledgements
The authors are grateful to Silvia DeBianchi, Marco Forgione and Alvaro Mozota for helpful discussions.
Funding
This research has received funding and support from the European Union’s Horizon 2020 research and innovation programme under Grant Agreement No. 758145 and the project CHRONOS (PID2019-108762GB-I00) of the Spanish Ministry of Science and Innovation.
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Hadad Farshi, F. On knots and temporality: a relational view of time. Synthese 202, 95 (2023). https://doi.org/10.1007/s11229-023-04321-0
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DOI: https://doi.org/10.1007/s11229-023-04321-0