# 自旋網路

## 潘洛斯原始定義

1971年，羅傑·潘洛斯提出一種圖形表示法，其中每個線段代表一個「單元」（基本粒子或粒子的複合系統）之世界線。三條線段匯聚在一個頂點。頂點可以詮釋為一個事件；在此事件中，一個單元分裂成兩個單元，或兩個單元碰撞合而為一。當一圖表中所有的線段都在頂點會合，則此圖為「封閉自旋網路」。時間以單一方向行進，比如從圖的底部走到圖的頂部。然而在封閉自旋網路的例子，時間行進的方向對於計算不構成影響。

• 三角不等式a必須小於或等於b + cb必須小於或等於a + c，以及c必須小於或等於a + b
• 費米子守恆（Fermion conservation）：a + b + c必須是偶數。

## 參考文獻

Early papers:

• Sum of Wigner coefficients and their graphical representation, I. B. Levinson, ``Proceed. Phys-Tech Inst. Acad Sci. Lithuanian SSR 2, 17-30 (1956)
• Applications of negative dimensional tensors, Roger Penrose, in Combinatorial Mathematics and its Applications, Academic Press (1971)
• Hamiltonian formulation of Wilson's lattice gauge theories, John Kogut英语John Kogut and Leonard Susskind, Phys. Rev. D 11, 395–408 (1975)
• The lattice gauge theory approach to quantum chromodynamics, , Rev. Mod. Phys. 55, 775–836 (1983) (see the Euclidean high temperature (strong coupling) section)
• Duality in field theory and statistical systems, Robert Savit, Rev. Mod. Phys. 52, 453–487 (1980) (see the sections on Abelian gauge theories)

Modern papers:

• Spin Networks and Quantum Gravity, Carlo Rovelli and Lee Smolin, Physical Review D 53, 5743 (1995); gr-qc/9505006.
• The dual of non-Abelian lattice gauge theory, Hendryk Pfeiffer and Robert Oeckl, hep-lat/0110034.
• Exact duality transformations for sigma models and gauge theories, Hendryk Pfeiffer, hep-lat/0205013.
• Generalized Lattice Gauge Theory, Spin Foams and State Sum Invariants, Robert Oeckl, hep-th/0110259.
• Spin Networks in Gauge Theory, John C. Baez, Advances in Mathematics, Volume 117, Number 2, February 1996, pp. 253–272.
• Quantum Field Theory of Many-body Systems – from the Origin of Sound to an Origin of Light and Fermions, Xiao-Gang Wen, [1]. (Dubbed string-nets here.)
• A Spin Network Primer, Seth A. Major, American Journal of Physics, Volume 67, 1999, gr-qc/9905020.
• Pre-geometry and Spin Networks. An introduction. [2].

Books:

• Diagram Techniques in Group Theory, G. E. Stedman, Cambridge University Press, 1990
• Group Theory: Birdtracks, Lie's, and Exceptional Groups, , Princeton University Press, 2008, http://birdtracks.eu/