Abstract
Analogies and equivalences provided by research in condensed matter and quantum information may give unexpected insights into the structure of quantum spacetime for fundamental physics. Several examples and implications for quantum gravity phenomology are discussed.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Similar content being viewed by others
Notes
- 1.
According to an intriguing result in Topology, four-dimensional spacetime is by far the most daunting case: there is an uncountable infinity of distinct ways to equip the open 4-manifold \(\mathbb {R}^4\) with smooth structures, while in all other dimensions a unique single way exists! Somehow the dimensionality of our physical world is exactly big enough to allow wild mathematics, but small enough to tame the wildness—extra dimensions would give more wiggle room.
- 2.
Just like observables in basic quantum theory such as position x and momentum \(p_x:=-i\hbar \partial _x\) are not mere numbers but operators acting on the (Hilbert) quantum state space, fields like the electromagnetic potential \(A_\mu \) turn into field operators acting on the (Fock) state space in QFT.
- 3.
The central postulate of Thermodynamics states that for constant energy all microstates (pure states: exact quantum states \( |n\rangle \)) are equally probable in the resulting macrostate (mixed state: probability distribution p(n) over microstates) described by a so-called density matrix \(\rho \), that allows the derivation of those for other macrostates, e.g. for constant temperature \(\rho =\exp (-\beta H)/Z\) (thermal equilibrium) with \(\beta :=1/kT\) and partition function \(Z:=Tr \exp (-\beta H)\). All thermodynamic quantities can be extracted from this density matrix (or equivalently Z), most importantly the entropy \(S:=-k Tr \rho \ln \rho \) quantifying the ignorance (missing information) about the exact microstate due to the probability distribution given by the thermal macrostate (providing only a coarse-grained picture). Emergent thermodynamic observables include temperature \(T:=1/\partial _E S\) and pressure \(P:=T\partial _V S\)
- 4.
For \(E > Re(m)\) an unstable particle forms with Im(m) determining its lifetime, for \(E < Re(m)\) only a resonance as evidence of pole existence shows up in the scattering cross section.
- 5.
For experts: the asymptotic Bogoliubov coefficients depend on the acceleration (with respect to Killing time) just outside the horizon, resulting in a non-zero particle flux with blackbody spectrum at later times (not created in the immediate vicinity of the horizon as is often misrepresented in popular accounts). It has recently been argued that its origin could be traced to some form of “quantum atmosphere” hovering at a distance.
- 6.
The subscript BH can stand for either Black Holes or Bekenstein-Hawking who derived the formula touching on all areas of physics by using speed of light c from relativity, quantum of action \(\hbar \), gravity constant G and thermodynamical constant k.
- 7.
The Hamiltonian H vanishes because the energy-momentum tensor \(T^{\mu \nu }\propto \delta S / \delta g_{\mu \nu }\) is obtained by the variation of the action \(S:=\int dt \mathcal {L}\) (with topological Lagrangian density \(\mathcal {L}\)) with respect to g, so \(H=T^{00}=0\)—leading to a significant degeneracy.
- 8.
One of those being topological quantum computing: since local perturbations do not transform between the multiple ground states, decoherence is much easier to avoid. The only unwanted effect of the environment could be exciting the system to such high energies (surpassing an energy gap \(\triangle \)) that it is no longer invariant under diffeomorphisms and TQFT does not apply anymore, but this possibility is suppressed exponentially with the energy gap (\(e^{-\triangle /T}\), though the necessary excitation gap is not sufficient for the formation of a topological phase). The information is distributed among decentralized quasiparticles (Majorana zero modes are the leading candidates for those qubits) and computations are performed as operations in a non-local manner. Topological invariance does not imply trivial low-energy physics, as exemplified by the fractional statistics of anyons.
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2018 Springer International Publishing AG
About this chapter
Cite this chapter
Seltmann, M. (2018). Spacetime Structure: Analogy in Condensed Matter and Quantum Information. In: Hossenfelder, S. (eds) Experimental Search for Quantum Gravity. FIAS Interdisciplinary Science Series. Springer, Cham. https://doi.org/10.1007/978-3-319-64537-7_14
Download citation
DOI: https://doi.org/10.1007/978-3-319-64537-7_14
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-64536-0
Online ISBN: 978-3-319-64537-7
eBook Packages: Physics and AstronomyPhysics and Astronomy (R0)