Appendix: Glossary and Quick Reference

This appendix provides a quick index of core terms, symbols, formulas, and references used throughout the tutorial.

A. Glossary

A

Anisotropy（各向异性）: Physical properties vary with direction. In condensed matter physics, refers to directional dependence of lattice or interactions.

Anomalous Hall Effect（反常霍尔效应）: Transverse conductance produced by spin-orbit coupling inside materials without external magnetic field. Related to topological invariants (Chern number).

B

Berry Phase（Berry相位）: Geometric phase acquired by a quantum state after adiabatic evolution around a closed path in parameter space. Formula: $\gamma ={\oint }_{\mathcal{C}}\mathbf{A}\cdot d\mathbf{R}$, where $\mathbf{A}$ is the Berry connection.

Birman-Kreĭn Formula（Birman-Kreĭn公式）: Formula connecting scattering matrix determinant with spectral shift: $\det S(\omega )=\exp \{-2\pi \mathrm{i}\xi (\omega )\}$.

Black Hole Entropy（黑洞熵）: Bekenstein-Hawking formula: $S=\frac{k_{B}A}{4{\ell }_P^2}$, where $A$ is the event horizon area, ${\ell }_P$ is the Planck length.

Brillouin Zone（布里渊区）: Unit cell in momentum space for periodic systems (such as lattices). The first Brillouin zone corresponds to the smallest periodic unit.

C

Cayley Map（Cayley映射）: Correspondence between scattering matrix $S$ and Hamiltonian $K$: $S=(I-\mathrm{i}K)(I+\mathrm{i}K{)}^{-1}$.

Chern Number（Chern数）: Topological invariant characterizing topological properties of energy bands. $\nu =\frac{1}{2\pi }{\int }_{\text{BZ}}{F}_{xy}{d}^{2}k$, where $F_{xy}$ is the Berry curvature.

Chern-Simons Term（Chern-Simons项）: Three-dimensional topological term in topological field theory: $S_{CS}=\frac{k}{4\pi }\int A\wedge dA$.

Causal Structure（因果结构）: Causal relationships between events in spacetime, determined by light cone structure. GLS theory generalizes this to dynamic causal cones.

Consciousness（意识）: Property of physical systems satisfying 5 structural conditions defined in §13.3: integration, differentiation, self-reference, intrinsic time, causal controllability.

D

Discriminant（判别子）: Set of points in parameter space causing system singularities (such as resonances). In self-referential scattering networks: $D=\{(\omega ,\vartheta ):\det (I-\mathcal{C}{S}_{ii})=0\}$.

Dirac Fermion（Dirac费米子）: Relativistic fermion satisfying Dirac equation. In condensed matter, refers to low-energy excitations with linear dispersion (such as graphene).

Discrete Time Crystal, DTC（离散时间晶体）: Time crystal in periodically driven systems, with response period being an integer multiple of the drive period (e.g., 2 times, $\pi$ spectral pairing).

E

Eigenstate Thermalization Hypothesis, ETH（本征态热化假设）: Individual eigenstates of isolated quantum systems exhibit thermal equilibrium properties. Diagonal ETH: $⟨n|O|n⟩=\bar{O}({\epsilon }_n)+\delta O$; off-diagonal ETH characterizes statistical distribution of matrix elements.

Emergent Spacetime（涌现时空）: Spacetime geometry is not fundamental but emerges from more fundamental microscopic degrees of freedom (such as qubits). Discussed in §5.

Entanglement Entropy（纠缠熵）: Measure quantifying quantum entanglement between subsystems. von Neumann entropy: $S=-\mathrm{tr}({\rho }_A\log {\rho }_A)$.

Exceptional Point, EP（例外点）: Point in non-Hermitian systems where eigenvalues and eigenvectors simultaneously degenerate. Leads to topological singularities.

F

Fermi Surface（费米面）: Constant energy surface at Fermi level in momentum space. Determines transport properties of metals.

Fisher Information（Fisher信息）: Sensitivity of quantum states to parameter changes. Quantum Fisher information: $F_Q[\rho ]=2{\sum }_{n,m}\frac{({\partial }_{\lambda }{p}_n-{\partial }_{\lambda }{p}_m{)}^2}{p_n+p_m}$ (diagonalized representation).

Floquet Engineering（Floquet工程）: Effective Hamiltonian design through periodic driving. Applied to time crystals, topological pumping, etc.

Floquet Operator（Floquet算符）: One-period evolution operator of periodically driven systems: $U(T)=\mathcal{T}\exp \{-\mathrm{i}{\int }_0^{T}H(t)dt\}$.

G

Gauge Field（规范场）: Vector field describing interactions (such as electromagnetic field $A_{\mu }$). Gauge invariance requires physical quantities to be independent of gauge choice.

Generalized Light Structure, GLS（广义光结构）: Core framework of this tutorial, unifying spacetime, gravity, and quantum field theory. Generalizes Lorentz transformations to dynamic metrics and nonlinear structures.

Gravitational Waves（引力波）: Propagating fluctuations of spacetime curvature. GLS predicted correction term: $h_{\mu \nu }^{\text{GLS}}=h_{\mu \nu }^{\text{GR}}+\delta h_{\mu \nu }[\kappa ]$.

Group Delay（群延迟）: Delay time of wave packet propagation, defined as derivative of phase with respect to frequency: ${\tau }_g(\omega )={\partial }_{\omega }\varphi (\omega )$. In §13.4, merging of group delay double peaks is a fingerprint of topological changes.

H

Hall Conductance（霍尔电导）: Transverse conductance, proportional to Chern number: ${\sigma }_{xy}=\frac{e^2}{h}\nu$ (quantized).

Hawking Radiation（Hawking辐射）: Particles radiated outward from black holes due to quantum effects, temperature: $T_H=\frac{\hslash c^3}{8\pi GM{k}_B}$.

Hilbert-Schmidt Norm（Hilbert-Schmidt范数）: HS norm of operator $A$: $\Vert A{\Vert }_2=\sqrt{\mathrm{tr}(A^{†}A)}$. Used to define convergence of trace-class operators.

I

Integrated Information（整合信息）: Core quantity in consciousness theory, measuring system indecomposability. $I_{\text{int}}({\rho }_O)={\sum }_{k=1}^{n}I(k:\overline{k}{)}_{{\rho }_O}$.

Intrinsic Time（本征时间）: Subjective time defined in §13.3, determined by quantum Fisher information: $\tau (t)={\int }_{t_0}^t\sqrt{F_Q[{\rho }_O(s)]}ds$.

J

$J$-Unitary（$J$-幺正）: Generalized unitarity in non-Hermitian systems: $S^{†}JS=J$, where $J$ is the Kreĭn metric. Preserves generalized energy conservation.

Jost Function（Jost函数）: Analytic function characterizing resonances in scattering theory. Zeros correspond to resonance states.

K

Unified Time Scale（统一时间刻度） $\kappa (\omega )$: Core concept of this tutorial, connecting four advanced topics. Definition: $\kappa (\omega )={\phi }^{\prime }(\omega )/\pi ={\rho }_{\text{rel}}(\omega )=(2\pi {)}^{-1}\mathrm{tr}Q(\omega )$. Physical meaning: inverse of information diffusion rate.

Kreĭn Angle（Kreĭn角）: Generalized phase slope in $J$-unitary systems: ${\varkappa }_j=\mathrm{Im}⟨{\psi }_j,J{S}^{-1}(\partial S){\psi }_j⟩/⟨{\psi }_j,J{\psi }_j⟩$.

L

Lagrangian: Action density in classical field theory. Field equations derived from variational principle: $\delta S=\delta \int \mathcal{L}{d}^{4}x=0$.

Liouvillian: Superoperator for open quantum systems, describing density matrix evolution: $\dot{\rho }=\mathcal{L}[\rho ]$. In dissipative time crystals, Liouvillian spectral gap guarantees long lifetime.

Lorentz Transformation（Lorentz变换）: Transformation preserving spacetime interval in special relativity. GLS theory generalizes it to field-strength-dependent dynamic transformations.

M

Majorana Fermion（Majorana费米子）: Its own antiparticle ($\psi ={\psi }^c$). Boundary state of topological superconductors, used in topological quantum computation.

Many-Body Localization, MBL（多体局域化）: Non-thermalizing phase in strongly disordered interacting systems. Preserves local conserved quantities, violates ETH. MBL time crystals utilize this effect.

Metric（度规）: Tensor $g_{\mu \nu }$ describing spacetime geometry, defining spacetime interval: $d{s}^2=g_{\mu \nu }d{x}^{\mu }d{x}^{\nu }$. In GLS, metric is a dynamic field.

N

Nevanlinna Function（Nevanlinna函数）: Analytic function from upper half complex plane to itself, satisfying positive imaginary part condition. Corresponds to causal Green’s function in physics.

No-Go Theorem（no-go定理）: Theorem excluding existence of certain types of physical systems. No-go theorems for time crystals (Bruno, Watanabe-Oshikawa) exclude equilibrium time crystals.

Noether’s Theorem（Noether定理）: Correspondence between symmetry and conservation laws. Each continuous symmetry corresponds to a conserved current.

O

Order Parameter（序参量）: Physical quantity characterizing phase transitions and ordered phases. Order parameter of time crystals: $⟨O(t+T)⟩=-⟨O(t)⟩$ (subharmonic response).

P

Page Curve（Page曲线）: Evolution of entanglement entropy with time during black hole evaporation. Increases then decreases, reflecting information conservation. Quantum chaos (ETH) explains Page curve.

Planck Scale（Planck尺度）: Scale where quantum gravity effects become significant. Planck length: ${\ell }_P=\sqrt{G\hslash /c^3}\approx 1.6\times 10^{-35}$ m.

Quasiparticle（准粒子）: Collective excitation in condensed matter systems, behaving like particles (such as phonons, magnons). Parameters include effective mass, lifetime, etc.

Q

Quantum Cellular Automaton, QCA（量子元胞自动机）: Unitary evolution on lattice, satisfying locality and reversibility. §13.1 uses QCA to model the universe.

Quantum Chaos（量子混沌）: Quantum counterpart of classical chaos. Manifested as Wigner-Dyson level statistics, ETH, etc.

Quasinormal Mode, QNM（准正模）: Decaying oscillation mode of black holes, corresponding to complex frequency $\omega ={\omega }_R+\mathrm{i}{\omega }_I$ (${\omega }_I<0$).

R

Redheffer Star Product（Redheffer星乘）: Cascade operation of scattering networks: $S^{(1)}\star S^{(2)}$. Closed-loop connection with feedback.

Renormalization Group, RG（重整化群）: Systematic method studying system behavior at different energy scales. Fixed points correspond to phase transitions.

S

Scattering Matrix, $S$-Matrix（散射矩阵）: Linear relationship between output and input states: $\mathbf{b}=S\mathbf{a}$. In unitary systems $S^{†}S=I$ (energy conservation).

Schur Complement（Schur补）: Method of eliminating partial degrees of freedom in block matrices. Used in self-referential networks for closed-loop simplification: $S^{\circlearrowleft }=S_{ee}+S_{ei}(I-\mathcal{C}{S}_{ii}{)}^{-1}\mathcal{C}{S}_{ie}$.

Self-Referential Scattering Network, SSN（自指散射网络）: Scattering system with feedback. Core object of §13.4.

Spectral Flow（谱流）: Number of times eigenvalues cross a specific value (such as $-1$) as parameters vary. ${\mathrm{Sf}}_{-1}(S)=$ number of times eigenvalues cross $-1$ (with sign).

Spectral Shift（谱位移）: Energy level shift relative to reference system. Birman-Kreĭn formula connects $\xi (\omega )$ with scattering phase.

Symplectic Geometry（辛几何）: Geometric framework for Hamiltonian systems. Symplectic form $\omega =dp\wedge dq$, symplectic transformations correspond to canonical transformations.

T

Time Crystal（时间晶体）: Physical system breaking time translation symmetry. Four types: prethermal DTC, MBL-DTC, dissipative TC, topological TC.

Topological Insulator（拓扑绝缘体）: Material insulating in bulk, conducting on surface/boundary. Protected by topological invariants (such as ${\mathbb{Z}}_2$ invariant).

Topological Invariant（拓扑不变量）: Quantity depending only on topological properties of system, invariant under continuous deformations. Examples: Chern number, winding number.

Trace-Class Operator（迹类算符）: Operator satisfying $\mathrm{tr}|A|<\infty$. $S-I$ being trace-class is a common assumption in scattering theory.

U

Unitary（幺正）: Operator satisfying $U^{†}U=I$. Preserves inner product (probability conservation). Evolution of isolated quantum systems is unitary.

Unified Field Equation（统一场方程）: Core equation of GLS theory, unifying gravity and other interactions. Detailed in Chapter 4.

W

Wigner-Dyson Statistics（Wigner-Dyson统计）: Level repulsion statistics of quantum chaotic systems. Gaussian orthogonal/unitary/symplectic ensembles (GOE/GUE/GSE).

Wigner-Smith Matrix（Wigner-Smith矩阵）: Matrix representation of scattering delay: $Q(\omega )=-\mathrm{i}{S}^{†}{\partial }_{\omega }S$ (unitary case). Trace $\mathrm{tr}Q$ gives total delay time.

Z

${\mathbb{Z}}_2$ Invariant（${\mathbb{Z}}_2$不变量）: Topological invariant taking values $\pm 1$. Classification index for time-reversal invariant topological insulators. Half-phase invariant $\nu$ of self-referential networks.

Zero Mode（零模）: Eigenstate with zero energy. Often appears at boundaries in topological systems (such as Majorana zero modes).

B. Common Symbol Table

Spacetime and Geometry

Quantum Mechanics

Unified Time Scale and Scattering

Topology and Geometric Phase

Thermodynamics and Statistical Physics

Information Theory

C. Physical Constants Table

D. Key Formulas Quick Reference

Chapters 2-4: GLS Foundations

Metric and Connection: $${\Gamma }_{\mu \nu }^{\lambda }=\frac{1}{2}{g}^{\lambda \sigma }({\partial }_{\mu }{g}_{\nu \sigma }+{\partial }_{\nu }{g}_{\mu \sigma }-{\partial }_{\sigma }{g}_{\mu \nu })$$

Ricci Tensor: $$R_{\mu \nu }={\partial }_{\lambda }{\Gamma }_{\mu \nu }^{\lambda }-{\partial }_{\nu }{\Gamma }_{\mu \lambda }^{\lambda }+{\Gamma }_{\lambda \sigma }^{\lambda }{\Gamma }_{\mu \nu }^{\sigma }-{\Gamma }_{\nu \sigma }^{\lambda }{\Gamma }_{\mu \lambda }^{\sigma }$$

Einstein Field Equation (General Relativity): $$R_{\mu \nu }-\frac{1}{2}{g}_{\mu \nu }R=\frac{8\pi G}{c^4}{T}_{\mu \nu }$$

GLS Correction: $$G_{\mu \nu }[\kappa ]=8\pi G{T}_{\mu \nu }+{\Lambda }_{\kappa }[\omega ]g_{\mu \nu }$$

Chapters 11-12: Topology and Berry Phase

Berry Connection and Curvature: $$A_{\mu }^n=\mathrm{i}⟨u_n|{\partial }_{\mu }|u_n⟩,F_{\mu \nu }^n={\partial }_{\mu }{A}_{\nu }^n-{\partial }_{\nu }{A}_{\mu }^n$$

Chern Number: $$\nu =\frac{1}{2\pi }{\int }_{\text{BZ}}{F}_{xy}{d}^{2}k$$

Quantized Hall Conductance: $${\sigma }_{xy}=\frac{e^2}{h}\nu$$

§13.1: Quantum Chaos and ETH

Diagonal ETH: $$⟨{\psi }_n|O_X|{\psi }_n⟩={\overline{O}}_X({\epsilon }_n)+O(e^{-c|\Omega |})$$

Off-Diagonal ETH: $$\mathbb{E}[|⟨{\psi }_m|O_X|{\psi }_n⟩{|}^2]\le e^{-S(\bar{\epsilon })}{g}_O(\bar{\epsilon },\omega )$$

Unified Time Scale: $$\kappa (\omega )=\frac{{\phi }^{\prime }(\omega )}{\pi }={\rho }_{\text{rel}}(\omega )=\frac{1}{2\pi }\mathrm{tr}Q(\omega )$$

§13.2: Time Crystals

Prethermal DTC Lifetime: $${\tau }_{\ast }\sim \exp \left(c\frac{\omega }{J}\right)$$

$\pi$ Spectral Pairing: $$e^{\mathrm{i}{\varphi }_n(K+\pi )}=-e^{\mathrm{i}{\varphi }_n(K)}$$

Liouvillian Spectral Gap of Dissipative Time Crystals: $${\Delta }_{\text{Liouv}}=\underset{\lambda \ne 0}{\min }|\mathrm{Re}(\lambda )|$$

§13.3: Physical Foundation of Consciousness

Integrated Information: $$I_{\text{int}}({\rho }_O)=\sum _{k=1}^{n}I(k:\overline{k}{)}_{{\rho }_O}$$

Intrinsic Time: $$\tau (t)={\int }_{t_0}^t\sqrt{F_Q[{\rho }_O(s)]}ds$$

Causal Controllability: $${\mathcal{E}}_T(t)=\underset{\pi }{\sup }I(A_t:S_{t+T})$$

Consciousness Level: $$\mathcal{C}(t)=g(F_Q,{\mathcal{E}}_T,I_{\text{int}},H_{\mathcal{P}})$$

§13.4: Self-Referential Scattering Networks

Schur Closed Form: $$S^{\circlearrowleft }=S_{ee}+S_{ei}(I-\mathcal{C}{S}_{ii}{)}^{-1}\mathcal{C}{S}_{ie}$$

Half-Phase Invariant: $${\nu }_{\sqrt{\det S^{\circlearrowleft }}}(\gamma )=\exp \left(\frac{\mathrm{i}}{2}{\oint }_{\gamma }\mathrm{d}\arg \det S^{\circlearrowleft }\right)$$

Four-Fold Equivalence: $${\nu }_{\sqrt{\det S}}=\exp \left(-\mathrm{i}\pi \oint \mathrm{d}\xi \right)=(-1{)}^{{\mathrm{Sf}}_{-1}}=(-1{)}^{I_2(\gamma ,D)}$$

${\mathbb{Z}}_2$ Composition Law: $${\nu }_{\text{net}}={\nu }_{(1)}\cdot {\nu }_{(2)}(\mathrm{mod}2)$$

E. Extended Reading Resources (by Topic)

General Relativity and Cosmology

1. Wald, R. M. General Relativity. University of Chicago Press (1984).
2. Misner, C. W., Thorne, K. S., & Wheeler, J. A. Gravitation. W. H. Freeman (1973).
3. Carroll, S. M. Spacetime and Geometry: An Introduction to General Relativity. Pearson (2003).

Quantum Field Theory

4. Peskin, M. E., & Schroeder, D. V. An Introduction to Quantum Field Theory. Westview Press (1995).
5. Weinberg, S. The Quantum Theory of Fields (Vols. 1-3). Cambridge University Press (1995-2000).
6. Srednicki, M. Quantum Field Theory. Cambridge University Press (2007).

Condensed Matter Physics and Topology

7. Altland, A., & Simons, B. D. Condensed Matter Field Theory. Cambridge University Press (2010).
8. Bernevig, B. A., & Hughes, T. L. Topological Insulators and Topological Superconductors. Princeton University Press (2013).
9. Hasan, M. Z., & Kane, C. L. “Colloquium: Topological insulators,” Rev. Mod. Phys. 82, 3045 (2010).

Quantum Information and Quantum Computation

10. Nielsen, M. A., & Chuang, I. L. Quantum Computation and Quantum Information. Cambridge University Press (2010).
11. Preskill, J. Quantum Computation lecture notes. http://theory.caltech.edu/~preskill/ph229/
12. Kitaev, A. “Fault-tolerant quantum computation by anyons,” Ann. Phys. 303, 2 (2003).

Quantum Chaos and Random Matrices

13. Haake, F. Quantum Signatures of Chaos. Springer (2010).
14. Mehta, M. L. Random Matrices. Elsevier (2004).
15. D’Alessio, L., Kafri, Y., Polkovnikov, A., & Rigol, M. “From quantum chaos and eigenstate thermalization to statistical mechanics and thermodynamics,” Adv. Phys. 65, 239 (2016).

Time Crystals

16. Wilczek, F. “Quantum Time Crystals,” Phys. Rev. Lett. 109, 160401 (2012).
17. Else, D. V., Bauer, B., & Nayak, C. “Floquet Time Crystals,” Phys. Rev. Lett. 117, 090402 (2016).
18. Yao, N. Y., et al. “Discrete Time Crystals: Rigidity, Criticality, and Realizations,” Phys. Rev. Lett. 118, 030401 (2017).

Physical Theories of Consciousness

19. Tononi, G., Boly, M., Massimini, M., & Koch, C. “Integrated information theory: from consciousness to its physical substrate,” Nat. Rev. Neurosci. 17, 450 (2016).
20. Tegmark, M. “Consciousness as a State of Matter,” Chaos Solitons Fractals 76, 238 (2015).
21. Oizumi, M., Albantakis, L., & Tononi, G. “From the phenomenology to the mechanisms of consciousness: Integrated Information Theory 3.0,” PLOS Comput. Biol. 10, e1003588 (2014).

Scattering Theory and Topology

22. Redheffer, R. “On a Certain Linear Fractional Transformation,” Pacific J. Math. 9, 871 (1959).
23. Fulga, I. C., Hassler, F., & Akhmerov, A. R. “Scattering Formula for the Topological Quantum Number,” Phys. Rev. B 85, 165409 (2012).
24. Simon, B. Trace Ideals and Their Applications. AMS (2005).

Mathematical Physics

25. Nakahara, M. Geometry, Topology and Physics. CRC Press (2003).
26. Arnold, V. I. Mathematical Methods of Classical Mechanics. Springer (1989).
27. Woodhouse, N. M. J. Geometric Quantization. Oxford University Press (1992).

F. Learning Suggestions and Usage Instructions

How to Use This Appendix:

1. While Reading: When encountering unfamiliar terms, consult §A Glossary for quick definitions.
2. While Calculating: Refer to §D Formula Quick Reference to avoid flipping through main text.
3. For Deep Learning: Based on §E Extended Reading, choose relevant textbooks or papers.
4. Symbol Confirmation: If you forget the meaning of a symbol, check §B Symbol Table.

Cross-Chapter Indexing of Terms:

Many terms appear in multiple chapters. Suggestions:

• When first learning, start understanding from the “First Appearance” chapter
• When studying in depth, track the term’s applications in other chapters

Hierarchy of Formulas:

• Foundation Formulas (Chapters 2-6): Define GLS framework, need to master
• Application Formulas (Chapters 7-12): Results for specific systems, learn as needed
• Frontier Formulas (Chapter 13): Research-level content, understand ideas only

Conclusion

This appendix aims to serve as a quick reference tool, not a replacement for detailed explanations in the main text. Suggestions:

• Beginners: First read relevant chapters thoroughly, then use this appendix for review
• Researchers: Use this appendix as a “cheat sheet” to quickly find needed definitions or formulas
• Cross-disciplinary readers: Build conceptual connections between different fields through the glossary

If you find missing terms or unclear definitions, feedback for improvement is welcome!

Last Updated: This appendix is synchronized with the tutorial main body. Version: 1.0

Acknowledgments: Thanks to all researchers who contributed to the unified theoretical framework, and readers who provided valuable feedback.