Abstract
We propose entropic order parameters that capture the physics of generalized symmetries and phases in QFT’s. We do it through an analysis of simple properties (additivity and Haag duality) of the net of operator algebras attached to space-time regions. We observe that different types of symmetries are associated with the breaking of these properties in regions of different non-trivial topologies. When such topologies are connected, we show that the non locally generated operators generate an Abelian symmetry group, and their commutation relations are fixed. The existence of order parameters with area law, like the Wilson loop for the confinement phase, or the ’t Hooft loop for the dual Higgs phase, is shown to imply the existence of more than one possible choice of algebras for the same underlying theory. A natural entropic order parameter arises by this non-uniqueness. We display aspects of the phases of theories with generalized symmetries in terms of these entropic order parameters. In particular, the connection between constant and area laws for dual order and disorder parameters is transparent in this approach, new constraints arising from conformal symmetry are revealed, and the algebraic origin of the Dirac quantization condition (and generalizations thereof) is described. A novel tool in this approach is the entropic certainty relation satisfied by dual relative entropies associated with complementary regions, which quantitatively relates the statistics of order and disorder parameters.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
A. M. Jaffe and E. Witten, Quantum Yang-Mills theory, Clay Mathematics Institute Millennium Prize problems, Oxford U.K. (2000).
J. Greensite, An introduction to the confinement problem, Lecture Notes Phys. 821 (2011) 1 [INSPIRE].
N. Ong, R. Bhatt and R. Bhatt, Princeton Series in Physics. Vol. 110: More is Different: Fifty Years of Condensed Matter Physics, Princeton University Press, Princeton U.S.A. (2001), https://books.google.com.ar/books?id=oledr2LiDxYC.
L. Álvarez-Gaumé and F. Zamora, Duality in quantum field theory and string theory, AIP Conf. Proc. 423 (1998) 46 [hep-th/9709180] [INSPIRE].
O. Aharony, S. S. Gubser, J. M. Maldacena, H. Ooguri and Y. Oz, Large N field theories, string theory and gravity, Phys. Rept. 323 (2000) 183 [hep-th/9905111] [INSPIRE].
R. Haag and D. Kastler, An Algebraic approach to quantum field theory, J. Math. Phys. 5 (1964) 848 [INSPIRE].
R. Haag, Local quantum physics: Fields, particles, algebras, Springer Science & Business Media, New York U.S.A. (2012).
S. Hollands and K. Sanders, Entanglement measures and their properties in quantum field theory, arXiv:1702.04924 [INSPIRE].
D. Gaiotto, A. Kapustin, N. Seiberg and B. Willett, Generalized Global Symmetries, JHEP 02 (2015) 172 [arXiv:1412.5148] [INSPIRE].
S. Doplicher, R. Haag and J. E. Roberts, Fields, observables and gauge transformations I, Commun. Math. Phys. 13 (1969) 1 [INSPIRE].
S. Doplicher, R. Haag and J. E. Roberts, Fields, observables and gauge transformations. 2., Commun. Math. Phys. 15 (1969) 173 [INSPIRE].
S. Doplicher, R. Haag and J. E. Roberts, Local observables and particle statistics. 1, Commun. Math. Phys. 23 (1971) 199 [INSPIRE].
S. Doplicher, R. Haag and J. E. Roberts, Local observables and particle statistics. 2, Commun. Math. Phys. 35 (1974) 49 [INSPIRE].
S. Doplicher and J. E. Roberts, Why there is a field algebra with a compact gauge group describing the superselection structure in particle physics, Commun. Math. Phys. 131 (1990) 51 [INSPIRE].
A. M. Polyakov, Nonhamiltonian approach to conformal quantum field theory, Zh. Eksp. Teor. Fiz. 66 (1974) 23 [INSPIRE].
S. Ferrara, A. F. Grillo and R. Gatto, Tensor representations of conformal algebra and conformally covariant operator product expansion, Annals Phys. 76 (1973) 161 [INSPIRE].
D. Poland, S. Rychkov and A. Vichi, The Conformal Bootstrap: Theory, Numerical Techniques, and Applications, Rev. Mod. Phys. 91 (2019) 015002 [arXiv:1805.04405] [INSPIRE].
D. Buchholz and K. Fredenhagen, Locality and the Structure of Particle States, Commun. Math. Phys. 84 (1982) 1 [INSPIRE].
J. Fröhlich and T. Kerler, Quantum groups, quantum categories and quantum field theory, Lecture Notes Math. 1542 (2006) 1.
S. S. Horuzhy, Mathematics and Its Applications. Vol. 19: Introduction to algebraic quantum field theory, Springer Science & Business Media, New York U.S.A. (2012).
R. Brunetti, D. Guido and R. Longo, Modular structure and duality in conformal quantum field theory, Commun. Math. Phys. 156 (1993) 201 [INSPIRE].
H. Casini, M. Huerta, J. M. Magán and D. Pontello, Entanglement entropy and superselection sectors. Part I. Global symmetries, JHEP 02 (2020) 014 [arXiv:1905.10487] [INSPIRE].
R. Longo and F. Xu, Relative Entropy in CFT, Adv. Math. 337 (2018) 139 [arXiv:1712.07283] [INSPIRE].
J. M. Magan and D. Pontello, Quantum Complementarity through Entropic Certainty Principles, Phys. Rev. A 103 (2021) 012211 [arXiv:2005.01760] [INSPIRE].
G. ’t Hooft, On the Phase Transition Towards Permanent Quark Confinement, Nucl. Phys. B 138 (1978) 1 [INSPIRE].
W. Donnelly and A. C. Wall, Entanglement entropy of electromagnetic edge modes, Phys. Rev. Lett. 114 (2015) 111603 [arXiv:1412.1895] [INSPIRE].
S. Ghosh, R. M. Soni and S. P. Trivedi, On The Entanglement Entropy For Gauge Theories, JHEP 09 (2015) 069 [arXiv:1501.02593] [INSPIRE].
R. M. Soni and S. P. Trivedi, Aspects of Entanglement Entropy for Gauge Theories, JHEP 01 (2016) 136 [arXiv:1510.07455] [INSPIRE].
R. M. Soni and S. P. Trivedi, Entanglement entropy in (3 + 1)-d free U(1) gauge theory, JHEP 02 (2017) 101 [arXiv:1608.00353] [INSPIRE].
K.-W. Huang, Central Charge and Entangled Gauge Fields, Phys. Rev. D 92 (2015) 025010 [arXiv:1412.2730] [INSPIRE].
H. Casini, M. Huerta and J. A. Rosabal, Remarks on entanglement entropy for gauge fields, Phys. Rev. D 89 (2014) 085012 [arXiv:1312.1183] [INSPIRE].
H. Casini, M. Huerta, J. M. Magán and D. Pontello, Logarithmic coefficient of the entanglement entropy of a Maxwell field, Phys. Rev. D 101 (2020) 065020 [arXiv:1911.00529] [INSPIRE].
M. Duetsch and K.-H. Rehren, Generalized free fields and the AdS-CFT correspondence, Annales Henri Poincaré 4 (2003) 613 [math-ph/0209035] [INSPIRE].
M. Bischoff, Y. Kawahigashi, R. Longo and K.-H. Rehren, Springer Series in Mathematical Physics. Vol. 3: Tensor categories and endomorphisms of von neumann algebras: with applications to quantum field theory, Springer, Berlin Germany (2015), https://www.springer.com/gp/book/9783540566236.
S. Doplicher and R. Longo, Standard and split inclusions of von Neumann algebras, Invent. Math. 75 (1984) 493.
S. Doplicher and R. Longo, Local aspects of superselection rules. II, Commun. Math. Phys. 88 (1983) 399 [INSPIRE].
P. Bueno and H. Casini, Reflected entropy, symmetries and free fermions, JHEP 05 (2020) 103 [arXiv:2003.09546] [INSPIRE].
R. Longo and K.-H. Rehren, Nets of subfactors, Rev. Math. Phys. 7 (1995) 567 [hep-th/9411077] [INSPIRE].
P. Etingof, S. Gelaki, D. Nikshych and V. Ostrik, Mathematical Surveys and Monographs. Vol. 205: Tensor categories, AMS Press, Providence U.S.A. (2016), http://www-math.mit.edu/ etingof/tenscat.pdf.
E. Witten, Dyons of Charge eθ/2π, Phys. Lett. B 86 (1979) 283 [INSPIRE].
K. G. Wilson, Confinement of Quarks, Phys. Rev. D 10 (1974) 2445 [INSPIRE].
A. Kapustin, Wilson-’t Hooft operators in four-dimensional gauge theories and s-duality, Phys. Rev. D 74 (2006) 025005 [hep-th/0501015] [INSPIRE].
T. Bröcker and T. Dieck, Graduate Texts in Mathematics. Vol. 98: Representations of Compact Lie Groups, Springer, Heidelberg Germany (2003), https://books.google.com.ar/books?id=AfBzWL5bIIQC.
J. Cornwell, Techniques of Physics. Vol. 1: Group Theory in Physics, Academic Press, San Diego U.S.A. (1997), http://www.sciencedirect.com/science/article/pii/B9780121898007500200.
G. Costa and G. Fogli, Symmetries and group theory in particle physics. An introduction to space-time and internal symmetries, Lecture Notes Phys. 823 (2012) 1 [INSPIRE].
M. Hamermesh, Group Theory and Its Application to Physical Problems, Addison Wesley Series in Physics, Dover Publications, New York U.S.A. (1989), https://books.google.com.ar/books?id=c0o9_wlCzgcC.
S. Sternberg, Lie Algebras, University Press of Florida, Gainesville U.S.A. (2009), https://books.google.com.ar/books?id=zRR_RAAACAAJ.
A. Zee, Group Theory in a Nutshell for Physicists, Princeton University Press, Princeton U.S.A. (2016), https://books.google.com.ar/books?id=FWkujgEACAAJ.
R. W. Carter, I. G. MacDonald, G. B. Segal and M. Taylor, Lectures on Lie Groups and Lie Algebras, London Mathematical Society Student Texts, Cambridge University Press, Cambridge U.K. (1995).
S. Roman, Fundamentals of Group Theory: An Advanced Approach, Birkhäuser, Boston U.S.A. (2011), https://books.google.com.ar/books?id=eWkqG0aiVsMC.
P. Goddard, J. Nuyts and D. I. Olive, Gauge Theories and Magnetic Charge, Nucl. Phys. B 125 (1977) 1.
O. Aharony, N. Seiberg and Y. Tachikawa, Reading between the lines of four-dimensional gauge theories, JHEP 08 (2013) 115 [arXiv:1305.0318] [INSPIRE].
D. Tong, Lectures on gauge theories, http://www.damtp.cam.ac.uk/user/tong/gaugetheory.html.
D. Harlow and H. Ooguri, Symmetries in quantum field theory and quantum gravity, Commun. Math. Phys. 383 (2021) 1669 [arXiv:1810.05338] [INSPIRE].
C. Bachas, Convexity of the Quarkonium Potential, Phys. Rev. D 33 (1986) 2723 [INSPIRE].
E. Witten, APS Medal for Exceptional Achievement in Research: Invited article on entanglement properties of quantum field theory, Rev. Mod. Phys. 90 (2018) 045003 [arXiv:1803.04993] [INSPIRE].
H. Casini, Wedge reflection positivity, J. Phys. A 44 (2011) 435202 [arXiv:1009.3832] [INSPIRE].
H. Casini, Entropy inequalities from reflection positivity, J. Stat. Mech. 1008 (2010) P08019 [arXiv:1004.4599] [INSPIRE].
M. Ohya and D. Petz, Quantum entropy and its use, Springer Science & Business Media, New York U.S.A. (2004).
M. Takesaki, Conditional expectations in von neumann algebras, J. Funct. Anal. 9 (1972) 306.
K. Furuya, N. Lashkari and S. Ouseph, Generalized entanglement entropy, charges, and intertwiners, JHEP 08 (2020) 046 [arXiv:2005.11389] [INSPIRE].
S. Hollands, Variational approach to relative entropies (with application to QFT), arXiv:2009.05024 [INSPIRE].
F. Xu, On relative entropy and global index, Trans. Am. Math. Soc. 373 (2020) 3515 [arXiv:1812.01119] [INSPIRE].
V. Jones, Index for subfactors, Invent. Math. 72 (1983) 1.
H. Kosaki, Extension of jones’ theory on index to arbitrary factors, J. Funct. Anal. 66 (1986) 123.
R. Longo, Index of subfactors and statistics of quantum fields. I, Commun. Math. Phys. 126 (1989) 217 [INSPIRE].
T. Teruya, Index for von neumann algebras with finite-dimensional centers, Publ. Res. Inst. Math. Sci. 28 (1992) 437.
L. Giorgetti and R. Longo, Minimal Index and Dimension for 2-C ∗ -Categories with Finite-Dimensional Centers, Commun. Math. Phys. 370 (2019) 719.
H. Umegaki, Conditional expectation in an operator algebra. I, Tohoku Math. J. 6 (1954) 177.
H. Umegaki, Conditional expectation in an operator algebra. II, Tohoku Math. J. 8 (1956) 86.
H. Umegaki, Conditional expectation in an operator algebra. III, Kodai Math. Sem. Rep. 11 (1959) 51.
H. Umegaki, Conditional expectation in an operator algebra. IV. Entropy and information, Kodai Math. Sem. Rep. 14 (1962) 59.
V. Vedral, The role of relative entropy in quantum information theory, Rev. Mod. Phys. 74 (2002) 197 [quant-ph/0102094] [INSPIRE].
P. Francesco, P. Mathieu and D. Sénéchal, Conformal field theory, Springer Science & Business Media, New York U.S.A. (2012).
J. Cardy, Some results on the mutual information of disjoint regions in higher dimensions, J. Phys. A 46 (2013) 285402 [arXiv:1304.7985] [INSPIRE].
S. Coleman, Aspects of Symmetry: Selected Erice Lectures, Cambridge University Press, Cambridge U.K. (1985), [INSPIRE].
D. Petz, Quantum information theory and quantum statistics, Springer Science & Business Media, New York U.S.A. (2007).
R. Giles, The Reconstruction of Gauge Potentials From Wilson Loops, Phys. Rev. D 24 (1981) 2160 [INSPIRE].
J. C. Baez, Spin network states in gauge theory, Adv. Math. 117 (1996) 253 [gr-qc/9411007] [INSPIRE].
R. Roth, On the conjugating representation of a finite group, Pac. J. Math. 36 (1971) 515.
J. L. Pena, S. Majid and K. Rietsch, Lie theory of finite simple groups and the roth property, Math. Proc. Camb. Philos. Soc. 163 (2017) 301.
G. Heide, J. Saxl, P. H. Tiep and A. E. Zalesski, Conjugacy action, induced representations and the Steinberg square for simple groups of Lie type, Proc. Lond. Math. Soc. 106 (2013) 908.
D. Blanco, L. Lanosa, M. Leston and G. Pérez-Nadal, Rényi mutual information inequalities from Rindler positivity, JHEP 12 (2019) 078 [arXiv:1909.03144] [INSPIRE].
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher’s Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
ArXiv ePrint: 2008.11748
Rights and permissions
Open Access . This article is distributed under the terms of the Creative Commons Attribution License (CC-BY 4.0), which permits any use, distribution and reproduction in any medium, provided the original author(s) and source are credited.
About this article
Cite this article
Casini, H., Huerta, M., Magán, J.M. et al. Entropic order parameters for the phases of QFT. J. High Energ. Phys. 2021, 277 (2021). https://doi.org/10.1007/JHEP04(2021)277
Received:
Revised:
Accepted:
Published:
DOI: https://doi.org/10.1007/JHEP04(2021)277