pith. sign in

arxiv: 2605.16316 · v1 · pith:QJSTWXD5new · submitted 2026-05-04 · ⚛️ physics.gen-ph

Gauging Time Reversal Symmetry in Quantum Gravity: Arrow of Time from a Confinement--Deconfinement Transition

Deepak Vaid This is my paper

Pith reviewed 2026-05-21 01:05 UTC · model grok-4.3

classification ⚛️ physics.gen-ph
keywords arrow of timequantum gravityconfinement deconfinement transitionZ2 gauge theoryspin networkstime reversal symmetrysymmetry protected topological phase
0
0 comments X

The pith

The cosmological arrow of time emerges from a confinement-deconfinement transition in a Z2 gauge theory on spin networks.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

This paper proposes that the directionality of time originates in a phase transition within a model of quantum gravity. By introducing a gauge field that represents local time-reversal symmetry on the discrete geometric states, the theory splits into two phases. The confined phase describes a pre-geometric foam in which no consistent time arrow exists. The deconfined phase describes ordinary spacetime in which a single cosmological arrow is present everywhere. The transition is located by the Wilson loop, and the deconfined phase gains protection from its topological character.

Core claim

Gauging time-reversal symmetry produces an effective Z2 lattice gauge theory whose confined phase corresponds to pre-geometric quantum gravitational foam without a coherent arrow of time and whose deconfined phase corresponds to semiclassical spacetime with a uniform cosmological arrow of time. The transition between these phases is detected by the Wilson loop order parameter. The deconfined phase is a symmetry-protected topological phase that stabilizes the coherent time orientation against local perturbations. The topologically protected surface excitations of this phase are conjectured to give rise to fermionic matter degrees of freedom.

What carries the argument

A Z2 gauge field placed on spin-network states to encode local time-reversal symmetry, with its confinement-deconfinement transition serving as the mechanism for the arrow of time.

If this is right

  • The arrow of time appears precisely when the system crosses from the confined to the deconfined phase.
  • The deconfined phase matches the properties of semiclassical spacetime with a uniform cosmological arrow.
  • The topological order in the deconfined phase protects the arrow of time from being disrupted by local changes.
  • The surface states of the topological phase may account for the appearance of fermionic particles.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • Viewing the arrow of time as a collective phase property opens the possibility of similar explanations for other global spacetime features using gauge symmetries on discrete structures.
  • Explicit calculations on finite spin networks could locate the critical point of the transition and test its dependence on the network structure.
  • The added stability from topological order suggests that once established, the time arrow would survive the strong quantum fluctuations expected near the big bang.

Load-bearing premise

The assumption that gauging time-reversal symmetry can be carried out on spin networks using their correspondence to tensor networks to yield phases that map directly to gravitational regimes.

What would settle it

Computing the Wilson loop in a small spin network and finding that its expectation value changes sharply at a particular coupling value that separates the two phases.

Figures

Figures reproduced from arXiv: 2605.16316 by Deepak Vaid.

Figure 1
Figure 1. Figure 1: Topological orders: terminology, characteristics and examples [PITH_FULL_IMAGE:figures/full_fig_p005_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Single vertex of a matrix product state. Internal indices are denoted by [PITH_FULL_IMAGE:figures/full_fig_p007_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: A connection between two adjacent vertices corresponds to multiplying the associ￾ated matrices. A1 i1 A2 i2 i1 i2 α β β α [PITH_FULL_IMAGE:figures/full_fig_p008_3.png] view at source ↗
Figure 5
Figure 5. Figure 5: Diagrammatic representation of a matrix product state (MPS). The matrix [PITH_FULL_IMAGE:figures/full_fig_p008_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Tensor network state (TNS) representation of the toric code wavefunction. Each edge (green) [PITH_FULL_IMAGE:figures/full_fig_p008_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: Action of a global gauge transformation on a matrix product state. After tracing out all the [PITH_FULL_IMAGE:figures/full_fig_p010_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: Insertion of local gauge symmetry fluxes in a MPS [PITH_FULL_IMAGE:figures/full_fig_p011_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: Matrix insertion on a subregion of a 2D tensor network state. Applying the symmetry [PITH_FULL_IMAGE:figures/full_fig_p011_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: Local symmetry flux insertions in a 2D MPS [PITH_FULL_IMAGE:figures/full_fig_p012_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: States of quantum geometry represented by spin-networks [PITH_FULL_IMAGE:figures/full_fig_p012_11.png] view at source ↗
Figure 12
Figure 12. Figure 12: A spin-network as a tensor network whose “physical” degrees of freedom correspond to the [PITH_FULL_IMAGE:figures/full_fig_p014_12.png] view at source ↗
Figure 13
Figure 13. Figure 13: How macroscopic geometry emerges from quantum geometry [PITH_FULL_IMAGE:figures/full_fig_p014_13.png] view at source ↗
Figure 14
Figure 14. Figure 14: Mapping classical volume to a quantum expression involves replacing classical vectors with [PITH_FULL_IMAGE:figures/full_fig_p015_14.png] view at source ↗
Figure 15
Figure 15. Figure 15: A 4-valent vertex with the vertex intertwiner shown as a four-index tensor [PITH_FULL_IMAGE:figures/full_fig_p015_15.png] view at source ↗
Figure 16
Figure 16. Figure 16: Tensor network representation of the 4-valent intertwiner state. The “first” qubit of each state [PITH_FULL_IMAGE:figures/full_fig_p017_16.png] view at source ↗
Figure 17
Figure 17. Figure 17: The CZX state with SPT and an on-site Z2 symmetry Z (a) The controlled Z gate. This and other quantum cir￾cuits were drawn using the Quantikz package by Alas￾tair Kay [46] X Z X Z X Z X Z (b) A circuit representation of the operator UCZX 7 SPT Phase of Quantum Geometry In this section we establish the correspondence between the CZX model described in section 6 and the spin-network states of Loop Quantum G… view at source ↗
Figure 19
Figure 19. Figure 19: The “code” subspace of the vertex Hilbert space which is mapped to itself under the action of [PITH_FULL_IMAGE:figures/full_fig_p019_19.png] view at source ↗
Figure 20
Figure 20. Figure 20: Each plaquette is in an entangled state: [PITH_FULL_IMAGE:figures/full_fig_p019_20.png] view at source ↗
Figure 21
Figure 21. Figure 21: Spin network states with bipartite and multipartite entanglement between neighboring vertices [PITH_FULL_IMAGE:figures/full_fig_p022_21.png] view at source ↗
Figure 22
Figure 22. Figure 22: Action of time-reversal symmetry on classical observables [PITH_FULL_IMAGE:figures/full_fig_p034_22.png] view at source ↗
read the original abstract

The question of the origin of time's arrow is a major outstanding problem in physics. Here we present a mechanism for the emergence of a cosmological arrow of time from a confinement--deconfinement transition in a $ Z_2 $ lattice gauge theory living on the spin-network states of Loop Quantum Gravity. Following Chen and Vishwanath \cite{Chen2015Gauging}, who showed that time-reversal symmetry can be gauged on tensor network states, and using the spin-network/tensor-network correspondence \cite{Qi2013Exact,Han2016Loop}, we introduce a $ Z_2 $ gauge field on spin networks encoding a local time-reversal symmetry. The effective theory of this gauge field contains a confined phase -- corresponding to a pre-geometric ``quantum gravitational foam'' with no coherent arrow of time -- and a deconfined phase -- corresponding to semiclassical spacetime with a uniform cosmological arrow. The emergence of the arrow of time is identified with the confinement--deconfinement transition, detected by the Wilson loop order parameter. The deconfined phase is further shown to correspond to a symmetry-protected topological (SPT) phase of the CZX type, whose topological order provides additional stability of the coherent time orientation against local perturbations. We conjecture that the topologically protected surface excitations of this SPT phase give rise to fermionic matter degrees of freedom.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

3 major / 2 minor

Summary. The paper proposes a mechanism for the emergence of the cosmological arrow of time from a confinement-deconfinement transition in a Z_2 lattice gauge theory defined on spin-network states of loop quantum gravity. Using the spin-network/tensor-network correspondence and the gauging of time-reversal symmetry on tensor networks, the confined phase is identified with a pre-geometric quantum gravitational foam lacking a coherent arrow, while the deconfined phase corresponds to semiclassical spacetime with a uniform cosmological arrow; this transition is detected via the Wilson loop order parameter, and the deconfined phase is further linked to a CZX-type SPT phase whose surface excitations are conjectured to produce fermionic matter.

Significance. If the proposed identifications and derivations hold, the work would provide a novel bridge between lattice gauge theory phases, symmetry-protected topological order, and the emergence of classical spacetime and its temporal orientation within a loop quantum gravity framework. It explicitly credits the Chen-Vishwanath gauging construction and the Qi/Han correspondence as foundational inputs, and the conjecture linking SPT surface modes to fermions offers a potential falsifiable link to matter content. However, the significance is tempered by the absence of explicit calculations establishing the phase identifications or the mapping of the Wilson-loop order parameter to a uniform time orientation.

major comments (3)
  1. [Abstract] Abstract and the paragraph introducing the phase identifications: the claim that the arrow of time emerges precisely at the confinement-deconfinement transition is circular because the confined phase is defined as lacking a coherent arrow while the deconfined phase is defined as possessing a uniform cosmological arrow, with no independent benchmark (e.g., explicit computation of time-orientation observables or comparison to external data) provided to break the loop.
  2. [Section on spin-network/tensor-network correspondence] The section discussing the spin-network/tensor-network correspondence and introduction of the Z_2 gauge field: no derivation is given showing how the gauged Z_2 field acts as an independent time-orientation degree of freedom while preserving the gravitational Gauss and diffeomorphism constraints of LQG; the assumption that the Qi/Han mapping licenses direct application of Chen-Vishwanath gauging without spoiling geometry encoding is stated but not demonstrated.
  3. [Discussion of Wilson loop detection] The paragraph on the Wilson-loop order parameter and its relation to the arrow: the statement that the deconfined-phase Wilson-loop expectation value translates into a uniform time orientation on the emergent spatial geometry lacks an explicit calculation or error estimate; it is not shown why confinement erases time orientation rather than merely disordering the gauge field.
minor comments (2)
  1. [Abstract] The abstract refers to 'the effective theory of this gauge field' without specifying the lattice action or Hamiltonian; a brief equation or reference to the precise form would improve clarity.
  2. [Conjecture on fermionic matter] The conjecture that topologically protected surface excitations give rise to fermionic matter is stated without even a schematic operator mapping or dimension-counting argument; moving this to a dedicated subsection with explicit notation would help readers assess its scope.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for the careful and constructive review of our manuscript. The comments identify key areas where the logical flow and technical details can be clarified. We address each major comment below with explanations grounded in the manuscript's framework and indicate the revisions planned for the next version.

read point-by-point responses

  1. Referee: [Abstract] Abstract and the paragraph introducing the phase identifications: the claim that the arrow of time emerges precisely at the confinement-deconfinement transition is circular because the confined phase is defined as lacking a coherent arrow while the deconfined phase is defined as possessing a uniform cosmological arrow, with no independent benchmark (e.g., explicit computation of time-orientation observables or comparison to external data) provided to break the loop.

    Authors: The phases are defined independently by the standard Z_2 lattice gauge theory diagnostics, specifically the Wilson loop expectation value (area law in confinement versus perimeter law in deconfinement). The association with the arrow of time is a subsequent physical interpretation enabled by the gauged time-reversal symmetry and the spin-network/tensor-network correspondence, where the deconfined phase permits a globally consistent orientation. This is not circular but a proposed identification. We will revise the abstract and introduction to state the gauge-theoretic definitions first, followed by the interpretive link, and introduce a concrete time-orientation observable (e.g., the expectation value of a directed operator) as an independent benchmark. revision: yes

  2. Referee: [Section on spin-network/tensor-network correspondence] The section discussing the spin-network/tensor-network correspondence and introduction of the Z_2 gauge field: no derivation is given showing how the gauged Z_2 field acts as an independent time-orientation degree of freedom while preserving the gravitational Gauss and diffeomorphism constraints of LQG; the assumption that the Qi/Han mapping licenses direct application of Chen-Vishwanath gauging without spoiling geometry encoding is stated but not demonstrated.

    Authors: The Qi/Han correspondence provides an exact duality preserving the geometric encoding in SU(2) labels and intertwiners. Chen-Vishwanath gauging is applied to the time-reversal symmetry already present in the tensor-network representation without modifying the vertex operators that enforce the Gauss law and diffeomorphism constraints. The Z_2 gauge field is an additional link degree of freedom that couples to the symmetry but leaves the original constraints intact. We agree a step-by-step derivation is needed and will add a dedicated subsection in the revised manuscript demonstrating compatibility. revision: yes

  3. Referee: [Discussion of Wilson loop detection] The paragraph on the Wilson-loop order parameter and its relation to the arrow: the statement that the deconfined-phase Wilson-loop expectation value translates into a uniform time orientation on the emergent spatial geometry lacks an explicit calculation or error estimate; it is not shown why confinement erases time orientation rather than merely disordering the gauge field.

    Authors: A non-vanishing Wilson loop in the deconfined phase indicates long-range order in the gauged time-reversal field, enforcing a uniform global orientation on the emergent geometry. Confinement produces area-law decay that destroys this long-range coherence, thereby erasing a consistent arrow rather than permitting only local disorder. This follows from standard Z_2 gauge theory analysis. While the present work is primarily conceptual, we will expand the discussion with a mean-field estimate of the order parameter, a qualitative argument for erasure of coherence, and a perturbative error estimate. revision: partial

Circularity Check

1 steps flagged

Arrow of time identified with confinement-deconfinement transition via explicit phase-to-arrow correspondence

specific steps
  1. self definitional [Abstract]
    "confined phase -- corresponding to a pre-geometric ``quantum gravitational foam'' with no coherent arrow of time -- and a deconfined phase -- corresponding to semiclassical spacetime with a uniform cosmological arrow. The emergence of the arrow of time is identified with the confinement--deconfinement transition, detected by the Wilson loop order parameter."

    The phases are defined by the presence or absence of a coherent arrow of time, after which the arrow's emergence is identified with the transition between those same phases. This renders the result equivalent to the definitional mapping by construction rather than a derived consequence of the gauge theory dynamics or order parameter.

full rationale

The paper's central derivation equates the emergence of a cosmological arrow with the confinement-deconfinement transition in a Z2 gauge theory on spin networks. This step is load-bearing because the confined and deconfined phases are introduced with direct reference to the absence or presence of a coherent arrow, making the claimed emergence equivalent to the initial interpretive assignment rather than an independent derivation from the Wilson loop or SPT structure. The underlying spin-network/tensor-network correspondence and gauging procedure are imported from external citations and do not themselves encode the arrow; the reduction occurs at the identification step. No fitted parameters or self-citations are involved, so the circularity is partial and interpretive rather than total.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 2 invented entities

The proposal rests on the spin-network/tensor-network correspondence and on interpretive mappings between gauge-theory phases and gravitational regimes that are not derived from first principles within the paper.

axioms (2)
  • domain assumption The spin-network/tensor-network correspondence holds sufficiently to transplant the gauging of time-reversal symmetry from Chen and Vishwanath directly onto LQG states.
    Invoked to justify introducing the Z2 gauge field on spin networks.
  • ad hoc to paper The deconfined phase of the gauged theory corresponds to semiclassical spacetime possessing a uniform cosmological arrow of time.
    This identification supplies the physical meaning of the arrow of time.
invented entities (2)
  • Z2 gauge field encoding local time-reversal symmetry on spin networks no independent evidence
    purpose: To produce confined and deconfined phases whose transition generates the arrow of time
    Newly introduced in the paper; no independent evidence supplied.
  • CZX-type SPT phase corresponding to the deconfined regime no independent evidence
    purpose: To furnish topological protection for the coherent time orientation
    Identified by the authors; no independent evidence supplied.

pith-pipeline@v0.9.0 · 5780 in / 1817 out tokens · 48060 ms · 2026-05-21T01:05:13.893988+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

What do these tags mean?
matches
The paper's claim is directly supported by a theorem in the formal canon.
supports
The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends
The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses
The paper appears to rely on the theorem as machinery.
contradicts
The paper's claim conflicts with a theorem or certificate in the canon.
unclear
Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

102 extracted references · 102 canonical work pages · 44 internal anchors

  1. [2]

    Xiao-Liang Qi.Exact holographic mapping and emergent space-time geometry. Sept. 2013. arXiv:1309.6282(cit. on pp. 1, 3)

    work page internal anchor Pith review Pith/arXiv arXiv 2013

  2. [3]

    Muxin Han and Ling-Yan Hung.Loop Quantum Gravity, Exact Holographic Mapping, and Holographic Entanglement Entropy. Oct. 2016. arXiv:1610.02134(cit. on pp. 1, 3)

    work page internal anchor Pith review Pith/arXiv arXiv 2016

  3. [5]

    Examination of evidence for a preferred axis in the cosmic radiation anisotropy

    Kate Land and Joao Magueijo. “Examination of evidence for a preferred axis in the cosmic radiation anisotropy”. In:Physical Review Letters95.7 (2005), p. 071301 (cit. on p. 2)

    work page 2005

  4. [6]

    Time Reversal Invariance in Quantum Mechanics

    Reza Moulavi Ardakani. “Time Reversal Invariance in Quantum Mechanics”. In: (Feb. 2018). arXiv:1802.10169(cit. on p. 2)

    work page internal anchor Pith review Pith/arXiv arXiv 2018

  5. [7]

    Time, symmetry and structure: a study in the foundations of quantum theory

    Bryan W. Roberts. “Time, symmetry and structure: a study in the foundations of quantum theory”. PhD thesis. 2012, p. 112 (cit. on p. 2)

    work page 2012

  6. [8]

    Entanglement and the foundations of statistical mechanics

    S. Popescu, A. J. Short, and A. Winter. “Entanglement and the foundations of statistical mechanics”. In:Nature Physics2.11 (Nov. 2006), pp. 754–758.issn: 1745-2473.doi:10 . 1038/nphys444(cit. on pp. 2, 3)

    work page 2006

  7. [9]

    Quantum mechanical evolution towards thermal equilibrium

    Noah Linden et al. “Quantum mechanical evolution towards thermal equilibrium”. In:Phys- ical Review E - Statistical, Nonlinear, and Soft Matter Physics79.6 (2009), p. 061103.issn: 15393755.doi:10.1103/PhysRevE.79.061103. arXiv:0812.2385(cit. on p. 2)

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.1103/physreve.79.061103 2009

  8. [10]

    Many body localization and thermalization in quantum statistical mechanics

    Rahul Nandkishore and David A. Huse. “Many-Body Localization and Thermalization in Quantum Statistical Mechanics”. In:Annual Review of Condensed Matter Physics6.1 (Mar. 2015), pp. 15–38.issn: 1947-5454.doi:10.1146/annurev- conmatphys- 031214- 014726. arXiv:1404.0686(cit. on p. 2)

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.1146/annurev- 2015

  9. [11]

    John Goold et al.The role of quantum information in thermodynamics - A topical review. Apr. 2016.doi:10.1088/1751-8113/49/14/143001. arXiv:1505.07835(cit. on p. 2)

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.1088/1751-8113/49/14/143001 2016

  10. [12]

    Quantum Solution to the Arrow-of-Time Dilemma

    Lorenzo Maccone. “Quantum Solution to the Arrow-of-Time Dilemma”. In:Physical Review Letters103.8 (2009), pp. 080401+.doi:10.1103/PhysRevLett.103.080401(cit. on p. 3)

    work page doi:10.1103/physrevlett.103.080401(cit 2009

  11. [13]

    The Solution to the Problem of Time in Shape Dynamics

    Julian Barbour, Tim Koslowski, and Flavio Mercati. “The Solution to the Problem of Time in Shape Dynamics”. In:Classical and Quantum Gravity31 (Aug. 2014), p. 155001.doi: 10.1088/0264-9381/31/15/155001. arXiv:1302.6264(cit. on p. 3)

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.1088/0264-9381/31/15/155001 2014

  12. [14]

    Julian Barbour, Tim Koslowski, and Flavio Mercati.A Gravitational Origin of the Arrows of Time. Oct. 2013. arXiv:1310.5167(cit. on p. 3)

    work page internal anchor Pith review Pith/arXiv arXiv 2013

  13. [15]

    Identification of a gravitational arrow of time

    Julian Barbour, Tim Koslowski, and Flavio Mercati. “Identification of a gravitational arrow of time”. In:Physical Review Letters113.18 (Sept. 2014).issn: 10797114.doi:10.1103/ PhysRevLett.113.181101. arXiv:1409.0917(cit. on p. 3). 29 BIBLIOGRAPHY Preprint

    work page internal anchor Pith review Pith/arXiv arXiv 2014

  14. [16]

    Entanglement and the Thermodynamic Arrow of Time

    David Jennings and Terry Rudolph. “Entanglement and the thermodynamic arrow of time”. In:Physical Review E - Statistical, Nonlinear, and Soft Matter Physics81.6 (Feb. 2010). issn: 15393755.doi:10.1103/PhysRevE.81.061130. arXiv:1002.0314(cit. on p. 3)

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.1103/physreve.81.061130 2010

  15. [17]

    Comment on `Quantum resolution to the arrow of time dilemma'

    David Jennings and Terry Rudolph. “Comment on ‘Quantum resolution to the arrow of time dilemma’”. In: (Sept. 2009).doi:10.1103/PhysRevLett.104.148901. arXiv:0909.1726 (cit. on p. 3)

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.1103/physrevlett.104.148901 2009

  16. [18]

    England.Statistical Physics of Self-Replication

    Jeremy L. England.Statistical Physics of Self-Replication. Sept. 2012. arXiv:1209 . 1179 (cit. on p. 3)

    work page 2012

  17. [19]

    In: Computer Vision – ECCV 2018

    Deepak Vaid and Sundance Bilson-Thompson.LQG for the Bewildered - The Self-Dual Ap- proach Revisited. Springer Nature, Feb. 2016.isbn: 978-3-319-43184-0.doi:10.1007/978- 3-319-43184-0(cit. on pp. 3, 15, 35, 36, 39)

    work page doi:10.1007/978- 2016

  18. [20]

    How to detect an anti-spacetime

    Marios Christodoulou, Aldo Riello, and Carlo Rovelli. “How to detect an anti-spacetime”. In: (June 2012). arXiv:1206.3903(cit. on pp. 3, 35)

    work page internal anchor Pith review Pith/arXiv arXiv 2012

  19. [21]

    Discrete Symmetries in Covariant LQG

    Carlo Rovelli and Edward Wilson-Ewing. “Discrete Symmetries in Covariant LQG”. In: (May 2012). arXiv:1205.0733(cit. on pp. 3, 35)

    work page internal anchor Pith review Pith/arXiv arXiv 2012

  20. [22]

    Group Field theory and Tensor Networks: towards a Ryu-Takayanagi formula in full quantum gravity

    Goffredo Chirco, Daniele Oriti, and Mingyi Zhang. “Group Field theory and Tensor Networks: towards a Ryu-Takayanagi formula in full quantum gravity”. In:Classical and Quantum Gravity35.11 (May 2018), p. 115011.doi:10.1088/1361-6382/aabd50. arXiv:1701.01383 (cit. on p. 3)

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.1088/1361-6382/aabd50 2018

  21. [23]

    Holographic maps from spin networks to tensor networks

    E. Colafranceschi, G. Chirco, and D. Oriti. “Holographic maps from spin networks to tensor networks”. In:Journal of High Energy Physics2021.9 (2021), pp. 1–45.doi:10 . 1007 / JHEP09(2021)010. eprint:2105.06454(cit. on p. 3)

    work page arXiv 2021

  22. [24]

    Quantum geometry from superposed spin net- works

    G. Chirco, E. Colafranceschi, and D. Oriti. “Quantum geometry from superposed spin net- works”. In:Physical Review D105.4 (2022), p. 046007.doi:10.1103/PhysRevD.105.046007. eprint:2207.07625(cit. on p. 3)

    work page doi:10.1103/physrevd.105.046007 2022

  23. [25]

    Symmetry Protected Topological phases of Quantum Matter

    T. Senthil. “Symmetry-Protected Topological Phases of Quantum Matter”. In:Annual Re- view of Condensed Matter Physics6.1 (Mar. 2015), pp. 299–324.issn: 1947-5454.doi:10. 1146/annurev-conmatphys-031214-014740. arXiv:1405.4015(cit. on p. 4)

    work page internal anchor Pith review Pith/arXiv arXiv 2015

  24. [26]

    Symmetry Protected Topological Phases, Anomalies, and Cobordisms: Beyond Group Cohomology

    Anton Kapustin. “Symmetry Protected Topological Phases, Anomalies, and Cobordisms: Beyond Group Cohomology”. In: (Mar. 2014). arXiv:1403.1467(cit. on pp. 4, 21, 22)

    work page internal anchor Pith review Pith/arXiv arXiv 2014

  25. [27]

    Reflection positivity and invertible topological phases

    Daniel S. Freed and Michael J. Hopkins. “Reflection positivity and invertible topological phases”. In:Geometry & Topology25 (2021), pp. 1165–1330.doi:10.2140/gt.2021.25

    work page doi:10.2140/gt.2021.25 2021

  26. [28]

    arXiv:1604.06527 [math-ph](cit. on p. 4)

    work page arXiv

  27. [29]

    Building up spacetime with quantum entanglement

    Mark Van Raamsdonk. “Building up spacetime with quantum entanglement”. In: (May 2010). arXiv:1005.3035(cit. on pp. 4, 5)

    work page arXiv 2010

  28. [30]

    Brian Swingle.Constructing holographic spacetimes using entanglement renormalization. Sept

    work page

  29. [31]

    arXiv:1209.3304(cit. on p. 4)

    work page internal anchor Pith review Pith/arXiv arXiv

  30. [32]

    Entanglement Renormalization and Holography

    Brian Swingle. “Entanglement Renormalization and Holography”. In: (May 2009). arXiv: 0905.1317(cit. on p. 4)

    work page internal anchor Pith review Pith/arXiv arXiv 2009

  31. [33]

    Holographic Derivation of Entanglement Entropy from AdS/CFT

    Shinsei Ryu and Tadashi Takayanagi. “Holographic Derivation of Entanglement Entropy from AdS/CFT”. In: (Mar. 2006). arXiv:hep-th/0603001(cit. on p. 5)

    work page internal anchor Pith review Pith/arXiv arXiv 2006

  32. [34]

    Aspects of Holographic Entanglement Entropy

    Shinsei Ryu and Tadashi Takayanagi. “Aspects of Holographic Entanglement Entropy”. In: Journal of High Energy Physics2006.08 (June 2006), p. 045.issn: 1029-8479.doi:10.1088/ 1126-6708/2006/08/045. arXiv:hep-th/0605073(cit. on p. 5)

    work page internal anchor Pith review Pith/arXiv arXiv 2006

  33. [35]

    Holographic entanglement in spin networks: a review

    E. Colafranceschi et al. “Holographic entanglement in spin networks: a review”. In:Frontiers in Physics10 (2022), p. 101234.doi:10 . 3389 / fphy . 2022 . 101234. eprint:2202 . 05116 (cit. on p. 5)

    work page 2022

  34. [36]

    Intertwiner Entanglement on Spin Networks

    Etera R. Livine. “Intertwiner Entanglement on Spin Networks”. In:Physical Review D97 (2018), p. 026009.doi:10.1103/PhysRevD.97.026009. arXiv:1709.08511 [gr-qc](cit. on p. 5). 30 BIBLIOGRAPHY Preprint

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.1103/physrevd.97.026009 2018

  35. [37]

    Gluing polyhedra with entanglement in loop quantum gravity

    Bekir Bayta¸ s, Eugenio Bianchi, and Nelson Yokomizo. “Gluing polyhedra with entanglement in loop quantum gravity”. In:Physical Review D98 (2018), p. 026001.doi:10 . 1103 / PhysRevD.98.026001. arXiv:1805.05856 [gr-qc](cit. on p. 5)

    work page internal anchor Pith review Pith/arXiv arXiv 2018

  36. [38]

    Hand-waving and Interpretive Dance: An Introductory Course on Tensor Networks

    Jacob C. Bridgeman and Christopher T. Chubb.Hand-waving and Interpretive Dance: An Introductory Course on Tensor Networks. Apr. 2016. arXiv:1603.03039(cit. on pp. 6–8)

    work page internal anchor Pith review Pith/arXiv arXiv 2016

  37. [39]

    Bell’s theorem without inequalities

    Daniel M. Greenberger et al. “Bell’s theorem without inequalities”. In:American Journal of Physics58.12 (1990), pp. 1131–1143.doi:10.1119/1.16243(cit. on p. 7)

    work page doi:10.1119/1.16243(cit 1990

  38. [40]

    Efficient simulation of one-dimensional quantum many-body systems

    G. Vidal. “Efficient simulation of one-dimensional quantum many-body systems”. In:Physical Review Letters91.14 (Oct. 2004), p. 147902.doi:10.1103/PhysRevLett.91.147902. arXiv: quant-ph/0310089 [quant-ph](cit. on p. 7)

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.1103/physrevlett.91.147902 2004

  39. [41]

    Entanglement spectrum of a topological phase in one dimension

    Frank Pollmann et al. “Entanglement spectrum of a topological phase in one dimension”. In: Physical Review B - Condensed Matter and Materials Physics81.6 (2010).issn: 10980121. doi:10.1103/PhysRevB.81.064439. arXiv:0910.1811(cit. on p. 11)

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.1103/physrevb.81.064439 2010

  40. [42]

    String-net condensation: A physical mechanism for topological phases

    Michael A. Levin and Xiao-Gang Wen. “String-net condensation: A physical mechanism for topological phases”. In: (Apr. 2004). arXiv:cond-mat/0404617(cit. on pp. 13, 22)

    work page internal anchor Pith review Pith/arXiv arXiv 2004

  41. [43]

    Discreteness of area and volume in quantum gravity

    Carlo Rovelli and Lee Smolin. “Discreteness of area and volume in quantum gravity”. In: (Nov. 1994). arXiv:gr-qc/9411005(cit. on p. 14)

    work page internal anchor Pith review Pith/arXiv arXiv 1994

  42. [44]

    Quantum Theory of Gravity I: Area Operators

    Abhay Ashtekar and Jerzy Lewandowski. “Quantum Theory of Gravity I: Area Operators”. In: (Aug. 1996). arXiv:gr-qc/9602046(cit. on p. 15)

    work page internal anchor Pith review Pith/arXiv arXiv 1996

  43. [45]

    Quantum Theory of Geometry II: Volume operators

    Abhay Ashtekar and Jerzy Lewandowski. “Quantum Theory of Geometry II: Volume oper- ators”. In: (Nov. 1997). arXiv:gr-qc/9711031(cit. on p. 15)

    work page internal anchor Pith review Pith/arXiv arXiv 1997

  44. [46]

    Local unitary transformation, long-range quantum entanglement, wave function renormalization, and topological order

    Xie Chen, Zheng Cheng Gu, and Xiao Gang Wen. “Local unitary transformation, long-range quantum entanglement, wave function renormalization, and topological order”. In:Physical Review B - Condensed Matter and Materials Physics82.15 (2010), p. 155138.issn: 10980121. doi:10.1103/PhysRevB.82.155138. arXiv:1004.3835(cit. on pp. 16, 18–20, 22, 42)

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.1103/physrevb.82.155138 2010

  45. [47]

    Symmetry protected topological orders and the group cohomology of their symmetry group

    Xie Chen et al. “Symmetry protected topological orders and the group cohomology of their symmetry group”. In:Physical Review B - Condensed Matter and Materials Physics87.15 (June 2011), p. 155114.issn: 10980121.doi:10.1103/PhysRevB.87.155114. arXiv:http: //arxiv.org/abs/1106.4772(cit. on pp. 18, 20, 22)

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.1103/physrevb.87.155114 2011

  46. [48]

    Tutorial on the Quantikz Package

    Alastair Kay. “Tutorial on the Quantikz Package”. In: (Sept. 2018).doi:10 . 17637 / rh . 7000520. arXiv:1809.03842(cit. on p. 18)

    work page arXiv 2018

  47. [49]

    Numerical methods for EPRL spin foam transition amplitudes and Lorentzian recoupling theory

    P. Don‘a, F. Gozzini, and G. Sarno. “Numerical analysis of spin foam vertex amplitudes with large spins”. In:Physical Review D101.10 (2020), p. 106003.doi:10.1103/PhysRevD.101. 106003. eprint:1807.03066(cit. on pp. 20, 24)

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.1103/physrevd.101 2020

  48. [50]

    Asymptotics of spin foam vertex amplitudes

    P. Don‘a and G. Sarno. “Asymptotics of spin foam vertex amplitudes”. In:Physical Review D 105.12 (2022), p. 126023.doi:10.1103/PhysRevD.105.126023. eprint:2206.10273(cit. on pp. 20, 24)

    work page doi:10.1103/physrevd.105.126023 2022

  49. [51]

    The Spin Foam Approach to Quantum Gravity

    Alejandro Perez. “The Spin-Foam Approach to Quantum Gravity”. In:Living Reviews in Relativity16 (2013), p. 3.doi:10.12942/lrr-2013-3. eprint:1205.2019(cit. on pp. 20, 24)

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.12942/lrr-2013-3 2013

  50. [52]

    Black hole entropy from Quantum Geometry

    Marcin Domagala and Jerzy Lewandowski. “Black-hole entropy from quantum geometry”. In:Classical and Quantum Gravity21 (2004), pp. 5233–5243.doi:10.1088/0264-9381/21/ 22/014. arXiv:gr-qc/0407051(cit. on p. 20)

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.1088/0264-9381/21/ 2004

  51. [53]

    Black hole entropy in Loop Quantum Gravity

    Krzysztof A. Meissner. “Black-hole entropy in loop quantum gravity”. In:Classical and Quantum Gravity21 (2004), pp. 5245–5251.doi:10.1088/0264-9381/21/22/015. arXiv: gr-qc/0407052(cit. on p. 20)

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.1088/0264-9381/21/22/015 2004

  52. [54]

    Braiding statistics approach to symmetry-protected topological phases

    Michael Levin and Zheng-Cheng Gu. “Braiding statistics approach to symmetry-protected topological phases”. In:Physical Review B86 (2012), p. 115109.doi:10.1103/PhysRevB. 86.115109. arXiv:1202.3120 [cond-mat.str-el](cit. on p. 20)

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.1103/physrevb 2012

  53. [55]

    Gauging quantum states: from global to local symmetries in many-body systems

    Jutho Haegeman et al. “Gauging quantum states: From global to local symmetries in many- body systems”. In:Physical Review X5.1 (2015).issn: 21603308.doi:10.1103/PhysRevX. 5.011024. arXiv:1407.1025(cit. on p. 20). 31 BIBLIOGRAPHY Preprint

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.1103/physrevx 2015

  54. [56]

    Physics of Three-Dimensional Bosonic Topological In- sulators: Surface-Deconfined Criticality and Quantized Magnetoelectric Effect

    Ashvin Vishwanath and T. Senthil. “Physics of Three-Dimensional Bosonic Topological In- sulators: Surface-Deconfined Criticality and Quantized Magnetoelectric Effect”. In:Physical Review X3 (2013), p. 011016.doi:10 . 1103 / PhysRevX . 3 . 011016. arXiv:1209 . 3003 [cond-mat.str-el](cit. on p. 21)

    work page 2013

  55. [57]

    Quantum Orders and Symmetric Spin Liquids

    Xiao-Gang Wen. “Quantum Orders and Symmetric Spin Liquids”. In: (Jan. 2002). arXiv: cond-mat/0107071(cit. on p. 22)

    work page internal anchor Pith review Pith/arXiv arXiv 2002

  56. [58]

    Duality in Generalized Ising Models and Phase Transitions without Local Order Parameters

    Franz J. Wegner. “Duality in Generalized Ising Models and Phase Transitions without Local Order Parameters”. In:Journal of Mathematical Physics12.10 (1971), pp. 2259–2272.doi: 10.1063/1.1665530(cit. on pp. 23, 24)

    work page doi:10.1063/1.1665530(cit 1971

  57. [59]

    An introduction to lattice gauge theory and spin systems

    John B. Kogut. “An introduction to lattice gauge theory and spin systems”. In:Reviews of Modern Physics51.4 (1979), pp. 659–713.doi:10.1103/RevModPhys.51.659(cit. on pp. 23, 24)

    work page doi:10.1103/revmodphys.51.659(cit 1979

  58. [60]

    Weak Interactions at Very High-Energies: The Role of the Higgs Boson Mass

    Eduardo Fradkin and Stephen H. Shenker. “Phase diagrams of lattice gauge theories with Higgs fields”. In:Physical Review D19.12 (1979), pp. 3682–3697.doi:10.1103/PhysRevD. 19.3682(cit. on p. 24)

    work page doi:10.1103/physrevd 1979

  59. [61]

    Impossibility of spontaneously breaking local symmetries

    S. Elitzur. “Impossibility of spontaneously breaking local symmetries”. In:Physical Review D12.12 (1975), pp. 3978–3982.doi:10.1103/PhysRevD.12.3978(cit. on pp. 24, 27)

    work page doi:10.1103/physrevd.12.3978(cit 1975

  60. [62]

    Discrete gauge theories

    Mark de Wild Propitius and F. Alexander Bais. “Discrete gauge theories”. In: (Mar. 1996). arXiv:hep-th/9511201(cit. on pp. 25, 28)

    work page internal anchor Pith review Pith/arXiv arXiv 1996

  61. [63]

    Dual formulation of spin network evolution

    Fotini Markopoulou. “Dual formulation of spin network evolution”. In:Advances in Theoret- ical and Mathematical Physics1 (1997), pp. 294–314. eprint:gr-qc/9704013(cit. on pp. 25, 28)

    work page internal anchor Pith review Pith/arXiv arXiv 1997

  62. [64]

    Quantum causal histories

    Fotini Markopoulou. “Quantum causal histories”. In:Classical and Quantum Gravity17.10 (2000), pp. 2059–2072.doi:10 . 1088 / 0264 - 9381 / 17 / 10 / 302. eprint:hep - th / 9904009 (cit. on pp. 25, 28)

    work page 2000

  63. [65]

    Topological quantum memory

    Eric Dennis et al. “Topological quantum memory”. In:Journal of Mathematical Physics43.9 (2002), pp. 4452–4505.doi:10.1063/1.1499754. eprint:quant-ph/0110143(cit. on pp. 25, 28)

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.1063/1.1499754 2002

  64. [66]

    The Higgs phase as a spin glass, and the transition between varieties of confinement

    Jeff Greensite and Kazue Matsuyama. “The Higgs phase as a spin glass, and the transition between varieties of confinement”. In:Physical Review D101 (2020), p. 054508.doi:10. 1103/PhysRevD.101.054508. arXiv:2001.03068 [hep-lat](cit. on p. 27)

    work page arXiv 2020

  65. [67]

    Probing Z2 lattice gauge theories on quantum computers

    J. Mildenberger et al. “Probing Z2 lattice gauge theories on quantum computers”. In:arXiv preprint arXiv:2203.08905(2024). eprint:2203.08905(cit. on p. 28)

    work page arXiv 2024

  66. [68]

    Quantum simulation of Z2 lattice gauge theory

    L. Homeier et al. “Quantum simulation of Z2 lattice gauge theory”. In:Communications in Physics(2023). eprint:2205.08541(cit. on p. 28)

    work page arXiv 2023

  67. [69]

    Holographic tensor network models and quantum error correction: A topical review

    Alexander Jahn and Jens Eisert. “Holographic tensor network models and quantum error correction: A topical review”. In:Quantum Science and Technology6 (2021), p. 033002.doi: 10.1088/2058-9565/ac0293. arXiv:2102.02619 [quant-ph](cit. on p. 28)

    work page doi:10.1088/2058-9565/ac0293 2021

  68. [70]

    Holographic Codes from Hyper- invariant Tensor Networks

    Matthew Steinberg, Sebastian Feld, and Alexander Jahn. “Holographic Codes from Hyper- invariant Tensor Networks”. In:Nature Communications14 (2023), p. 7314.doi:10.1038/ s41467-023-42743-z. arXiv:2304.02732 [quant-ph](cit. on p. 28)

    work page arXiv 2023

  69. [71]

    Coarse graining flow of spin foam intertwiners

    Bianca Dittrich et al. “Coarse graining flow of spin foam intertwiners”. In:Physical Review D94 (2016), p. 124050.doi:10.1103/PhysRevD.94.124050. arXiv:1609.02429 [gr-qc] (cit. on p. 28)

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.1103/physrevd.94.124050 2016

  70. [72]

    Bulk Entanglement Gravity without a Boundary: Towards Finding Einstein's Equation in Hilbert Space

    ChunJun Cao and Sean M. Carroll. “Bulk Entanglement Gravity without a Boundary: To- wards Finding Einstein’s Equation in Hilbert Space”. In:Physical Review D97 (2018), p. 086003.doi:10.1103/PhysRevD.97.086003. arXiv:1712.02803 [gr-qc](cit. on p. 28)

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.1103/physrevd.97.086003 2018

  71. [73]

    The Trinity of Re- lational Quantum Dynamics

    Philipp A. H¨ ohn, Alexander R. H. Smith, and Maximilian P. E. Lock. “The Trinity of Re- lational Quantum Dynamics”. In:Physical Review D104 (2021), p. 066001.doi:10.1103/ PhysRevD.104.066001. arXiv:1912.00033 [gr-qc](cit. on p. 28). 32 Preprint

    work page arXiv 2021

  72. [74]

    Cambridge: Cam- bridge University Press, 2014.isbn: 9781107706910.doi:10.1017/CBO9781107706910(cit

    Carlo Rovelli and Francesca Vidotto.Covariant Loop Quantum Gravity. Cambridge: Cam- bridge University Press, 2014.isbn: 9781107706910.doi:10.1017/CBO9781107706910(cit. on pp. 36, 38)

    work page doi:10.1017/cbo9781107706910(cit 2014

  73. [75]

    New Variables for Classical and Quantum Gravity

    Abhay Ashtekar. “New Variables for Classical and Quantum Gravity”. In:Phys. Rev. Lett. 57.18 (Nov. 1986), pp. 2244–2247.doi:10.1103/PhysRevLett.57.2244(cit. on p. 36)

    work page doi:10.1103/physrevlett.57.2244(cit 1986

  74. [76]

    New Hamiltonian formulation of general relativity

    Abhay Ashtekar. “New Hamiltonian formulation of general relativity”. In:Phys. Rev. D36.6 (Sept. 1987), pp. 1587–1602.doi:10.1103/PhysRevD.36.1587(cit. on p. 36)

    work page doi:10.1103/physrevd.36.1587(cit 1987

  75. [77]

    Background independent quantum gravity: a sta- tus report

    Abhay Ashtekar and Jerzy Lewandowski. “Background independent quantum gravity: a sta- tus report”. In:Classical and Quantum Gravity21.15 (2004), R53–R152.doi:10.1088/0264- 9381/21/15/R01(cit. on p. 37)

    work page doi:10.1088/0264- 2004

  76. [78]

    A Manifestly Gauge-Invariant Approach to Quantum Theories of Gauge Fields

    A. Ashtekar et al. “A Manifestly Gauge-Invariant Approach to Quantum Theories of Gauge Fields”. In:Geometry of Constrained Dynamical Systems. Aug. 1994. arXiv:9408108 [hep-th] (cit. on pp. 38, 39)

    work page 1994

  77. [79]

    Invariant Perfect Tensors

    Youning Li et al. “Invariant perfect tensors”. In:New Journal of Physics19.6 (Dec. 2017). issn: 13672630.doi:10.1088/1367-2630/aa7235. arXiv:1612.04504(cit. on p. 41)

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.1088/1367-2630/aa7235 2017

  78. [80]

    Fusion rules and modular transformations in 2D conformal field theory

    Erik Verlinde. “Fusion rules and modular transformations in 2D conformal field theory”. In:Nuclear Physics B300 (Jan. 1988), pp. 360–376.issn: 0550-3213.doi:10.1016/0550- 3213(88)90603-7(cit. on p. 41). A Time Reversal of Quantum Systems In classical mechanics the action of time-reversal symmetry on physical observables can be defined as shown in Figure 2...

    work page doi:10.1016/0550- 1988

  79. [81]

    Starting with the usual Einstein-Hilbert action in the 1 st order formalism: SEH = 1 κ Z d4x√−g R[g,Γ],(D.1) whereκ= 8πG N,Ris the Ricci scalar andg= det(g) is the determinant of the 4-metric, we perform a canonical transformation [73, 74] from the four-metric and Christoffel connection being our dynamical variables{g µν,Γ δ αβ}to a new set of variables{e...

    work page

  80. [82]

    In the new variables the Einstein-Hilbert action can be written in the form: SP [e, A] = 1 2κ Z d4x ⋆ eI ∧e J ∧F KL ϵIJKL = 1 4κ Z d4xϵµναβ ϵIJKL eI µeJ ν F KL αβ (D.2) whereF KL αβ is the curvature of the spin-connectionA IJ µ

    work page

Showing first 80 references.