quantum cosmology

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## Idea

Despite what one might expect, much of [[cosmology]] is based on [[classical physics]]. Certainly the [[standard model of particle physics]] is constructed from _classical [[gravity]]_, namely an [[FRW model]] solution to [[Einstein's equations]], which serves as a fixed "background" over which one considers in a [[semiclassical approximation]] [[quantum fields]], say to model the [[cosmic microwave background radiation]].

On the one hand it is remarkable how well the [[standard model of cosmology]] built this way compares to available experimental data. On the other hand it is clear that a more precise model will need to take [[quantum field theory]] into account more properly. In particular non-classical effects of [[quantum gravity]] will play a role at least close to the initial singularity, if not also for the explanation of the [[cosmological constant]] and other aspects of the [[standard model of cosmology]] which currently don't have a conceptual explanation in the model.

Such a more comprehensively quantum mechanical discussion of [[cosmology]] goes by the term **quantum cosmology**.

However, despite the existence of this term, quantum cosmology is to a large extent only a hypothetical field of research. This is due to two major open conceptual problems of present day fundamental [[physics]]:

1. the nature of [[quantum gravity]] remains largely unsettled;

1. the very formulations of [[quantum mechanics]]/[[quantum field theory]] available all require, more or less explicitly and to lesser or greater extent, an _external classical observer_ to make sense of the predictions of the theory. This makes sense, in principle, for every subsystem of the [[observable universe]]. But a comprehensive theory of "quantum cosmology" would seem to need a notion of [[observables]] of quantum phenomena by "quantum observers" _internal_ to the quantum system which they observe, instead of by "classical observers" external to that quantum system.

It has been suggested ([Doering-Isham, section 1](#DoeringIsham)) that the second of the conceptual problems above might have a natural solution in terms of the [[internal logic]] of a [[topos]]-incarnation of the [[space of states]] of a quantum mechanical system. One proposed formalization of this is the notion of a _[[nLab:Bohr topos]]_. While these constructions may have something to them, implications, if any, for actual quantum cosmology are not clear yet.

## Related concepts

* [[no boundary proposal]]

## References

### General

* Calcagni, G.; Papantonopoulos, L.; Siopsis, G.; Tsamis, N. (Eds.), _Quantum Gravity and Quantum Cosmology_ Lecture Notes in Physics, Vol. 863

* Claus Kiefer, Barbara Sandhoefer, _Quantum cosmology_ ([arXiv:0804.0672](http://arxiv.org/abs/0804.0672v2))

* J.J.Halliwell, _Introductory Lectures on Quantum Cosmology (1990)_ ([arXiv:0909.2566](http://arxiv.org/abs/0909.2566))

* Edward Anderson, _On the Semiclassical Approach to Quantum Cosmology_ Class. Quant. Grav. 28 185008 (2011) ([arXiv:1101.4916](http://arxiv.org/abs/1101.4916))

### Applications

On [[structure formation]] during [[cosmic inflation]] in quantum [[inhomogeneous cosmology]]:

* [[Martin Bojowald]], Ding Ding, *Canonical description of cosmological backreaction*, JCAP **03** (2021) 083
([arXiv:2011.03018](https://arxiv.org/abs/2011.03018))

### Supersymmetric quantum cosmology

Quantum cosmology for [[supergravity]] and its relation to the [[E10]] [[U-duality]] group is discussed in

* {#Damour14} [[Thibault Damour]], _Quantum supersymmetric Bianchi IX  Cosmology and its hidden Kac-Moody Structure_, talk at [Gravitation, Solitons and Symmetries](http://www.lestudium-ias.com/#!registration-conference-21-may-gibbons/ci2z) ([[DamourSQC14.pdf:file]])

In regards of [[cosmic inflation]]:

* N.E. Martínez-Pérez, C. Ramírez, V.M. Vázquez Báez, *Phenomenological inflationary model in Supersymmetric Quantum Cosmology* [[arXiv:2208.04412](https://arxiv.org/abs/2208.04412)]

### On conceptual problems

A survey of conceptual problems in quantum cosmology, or issues regarded as such, is in

* Claus Kiefer, _Conceptual issues in quantum cosmology_ ([arXiv:gr-qc/9906100](http://arxiv.org/abs/gr-qc/9906100))

Discussion of the problem of internal quantum observers with an eye towards the [[internal logic]] of a [[topos]] is in the introduction of

* Andreas Doering, [[Chris Isham]], _A Topos Foundation for Theories of Physics: I. Formal Languages for Physics_, J.Math.Phys.49:053515,2008 ([arXiv:quant-ph/0703060](http://arxiv.org/abs/quant-ph/0703060))
 {#DoeringIsham}

For more on this see at _[[Bohr topos]]_.

[[!redirects supersymmetric quantum cosmology]]


quantum cryptography

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## Idea

Quantum cryptography is [[cryptography]] making use of effects of [[quantum physics]] and [[quantum information theory]]/[[quantum computation]], such as the [[no-cloning theorem]] and versions of [[quantum teleportation]].

## Related concepts

* [[Shor's algorithm]], [[Grover's algorithm]]

* [[quantum supremacy]]

* [[quantum chemistry]], [[quantum biology]]

## References

Original articles:

* [[Stephen Wiesner]], *Conjugate Coding*, circulated in the 1960s, finally published in ACM SIGACT News **15** 1 (1083) 78–88 ([original pdf](http://users.cms.caltech.edu/~vidick/teaching/120_qcrypto/wiesner.pdf), [doi:10.1145/1008908.1008920](https://doi.org/10.1145/1008908.1008920))

Introducing the notion of [[quantum key distribution]]:

* [[Charles H. Bennett]], [[Gilles Brassard]], *Quantum cryptography: Public key distribution and coin tossing*, International Conference on Computers, Systems & Signal Processing, Bangalore, India (1984) 175-179 [scan: [arXiv:2003.06557](https://arxiv.org/abs/2003.06557)]

  reprinted in: 	Theoretical Computer Science **560** 1 (2014) 7-11 [[doi:10.1016/j.tcs.2014.05.025](https://doi.org/10.1016/j.tcs.2014.05.025)]

* [[Artur K. Ekert]], *Quantum cryptography based on Bell’s theorem*, Phys. Rev. Lett. **67** 661 (1991) [[doi:10.1103/PhysRevLett.67.661](https://doi.org/10.1103/PhysRevLett.67.661)]

Textbook account with focus on [[experiment|experimental]] realization:

* [[Dirk Bouwmeester]], [[Artur Ekert]], [[Anton Zeilinger]] (eds.), *The Physics of Quantum Information -- Quantum Cryptography, Quantum Teleportation, Quantum Computation*, Springer (2020) [[doi:10.1007/978-3-662-04209-0](https://doi.org/10.1007/978-3-662-04209-0)]

See also

* Wikipedia, *[Quantum cryptography](https://en.wikipedia.org/wiki/Quantum_cryptography)*

* [[Renato Renner]], Ramona Wolf, *Quantum Advantage in Cryptography*, AIAA Journal  (2023) 1-16 [[arXiv:2206.04078](https://arxiv.org/abs/2206.04078), [doi:10.2514/1.J062267](https://doi.org/10.2514/1.J062267)]

[[!redirects quantum key distribution]]


quantum decoherence

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## Idea

In [[quantum physics]] _decoherence_ refers to the disappearance of quantum [[entanglement]] and [[superposition]] in the limit where small [[quantum mechanical systems]] are coupled to large [[thermal baths]].

This has been argued to resolve (and has been argued not to resolve) the problem with the [[interpretation of quantum mechanics|interpretation]] of [[quantum measurement]].

## Properties

[[!include quantum channels and decoherence -- section]]

## Related concepts

* [[quantum entanglement]]

* [[classical limit]]

* [[density matrix]]

* [[quantum measurement]], [[interpretation of quantum mechanics]]

* [[measurement]], [[experiment]]

* [[observable universe]]

* [[consistent histories]]

* [[einselection]]

## References

Precursor discussion:

* {#vonNeumann32} [[John von Neumann]], §VI.3 of:

  *Mathematische Grundlagen der Quantenmechanik* (German) (1932, 1971) [[doi:10.1007/978-3-642-96048-2](https://link.springer.com/book/10.1007/978-3-642-96048-2)]

  *Mathematical Foundations of Quantum Mechanics* Princeton University Press (1955) [[doi:10.1515/9781400889921](https://doi.org/10.1515/9781400889921), [Wikipedia entry](https://en.wikipedia.org/wiki/Mathematical_Foundations_of_Quantum_Mechanics)]

Original discusssion identifying quantum decoherence as interaction with an averaged environment ("bath"):

* {#Zeh70} [[H. Dieter Zeh]], *On the interpretation of measurement in quantum theory*, Found Phys **1** (1970) 69–76 [[doi:10.1007/BF00708656](https://doi.org/10.1007/BF00708656)]

* {#Zurek81} [[Wojciech Zurek]], *Pointer basis of quantum apparatus: Into what mixture does the wave packet collapse?*, Phys. Rev. D **24** (1981) 1516-1525 [[doi:10.1103/PhysRevD.24.1516](https://doi.org/10.1103/PhysRevD.24.1516)]

* {#Zurek82} [[Wojciech Zurek]], *Environment-induced superselection rules*, Phys. Rev. D **26** (1982) 1862-1880 [[doi:10.1103/PhysRevD.26.1862](https://doi.org/10.1103/PhysRevD.26.1862)]

* {#JoosZeh85} [[Erich Joos]], [[H. Dieter Zeh]], *The emergence of classical properties through interaction with the environment*, Z. Physik B -- Condensed Matter **59** (1985) 223–243 [[doi:10.1007/BF01725541](https://doi.org/10.1007/BF01725541)]

* [[Erich Joos]], *[www.decoherence.de](http://www.decoherence.de/)*

* {#Zurek03} [[Wojciech Zurek]], *Decoherence, einselection, and the quantum origins of the classical*, Rev. Mod. Phys. **75** (2003) 715-775 [[quant-ph/0105127](http://arxiv.org/abs/quant-ph/0105127), [doi:10.1103/RevModPhys.75.715](http://dx.doi.org/10.1103/RevModPhys.75.715)]

With regards to the [[measurement problem]] and [[interpretations of quantum mechanics]]:

* [[Roland Omnès]], §7 of: *[[The Interpretation of Quantum Mechanics]]*, Princeton University Press (1994) [[ISBN:9780691036694](http://press.princeton.edu/titles/5525.html)]

* [[Maximilian Schlosshauer]], *Decoherence, the measurement problem, and interpretations of quantum mechanics*,  	Rev. Mod. Phys. **76** (2004) 1267-1305 [[arXiv;quant-ph/0312059](https://arxiv.org/abs/quant-ph/0312059), [doi:10.1103/RevModPhys.76.1267](https://doi.org/10.1103/RevModPhys.76.1267)]

* Guido Bacciagaluppi, *The Role of Decoherence in Quantum Mechanics*,  Stanf. Enc. of Phil. (2020) [[web](https://plato.stanford.edu/entries/qm-decoherence/)]

Textbook accounts:

* [[Heinz-Peter Breuer]], [[Francesco Petruccione]], *Decoherence*, Chapter 4 in: *The Theory of Open Quantum Systems*, Oxford University Press (2007) [book:[doi:10.1093/acprof:oso/9780199213900.001.0001](https://doi.org/10.1093/acprof:oso/9780199213900.001.0001), chapter:[.003.04](https://doi.org/10.1093/acprof:oso/9780199213900.003.04)]

* {#Schlosshauer07} [[Maximilian Schlosshauer]], *Decoherence and the Quantum-To-Classical Transition*, The Frontiers Collection, Springer (2007) [[doi:10.1007/978-3-540-35775-9](https://doi.org/10.1007/978-3-540-35775-9)]

Further review:

* Claus Kiefer, [[Erich Joos]], *Decoherence: Concepts and Examples*, in: *Quantum Future From Volta and Como to the Present and Beyond*, Lecture Notes in Physics *517** (1999) [[arXiv:quant-ph/9803052](https://arxiv.org/abs/quant-ph/9803052), [doi:10.1007/BFb0105342](https://doi.org/10.1007/BFb0105342)]

* [[Maximilian Schlosshauer]], *Quantum Decoherence*, Phys. Rep. **831** (2019) 1-57 [[arXiv:1911.06282](https://arxiv.org/abs/1911.06282), [doi:10.1016/j.physrep.2019.10.001](https://doi.org/10.1016/j.physrep.2019.10.001)]

See also:

* Chris Nagele, Oliver Janssen, Matthew Kleban, _Decoherence: A Numerical Study_ [[arXiv:2010.04803](https://arxiv.org/abs/2010.04803)]

* Wikipedia, _[Quantum decoherence](http://en.wikipedia.org/wiki/Quantum_decoherence)_

A proposal for mathematical quantification of coherence:

* Fedor Herbut, *A quantum measure of coherence and incompatibility*, Journal of Physics A: Mathematical and General 38, 2959 (2005) ([arXiv:quant-ph/0503077](https://arxiv.org/abs/quant-ph/0503077), [arXiv:10.1088/0305-4470/38/13/010](https://iopscience.iop.org/article/10.1088/0305-4470/38/13/010))

which was rediscovered and then became famous with:

* {#BaumgratzCramerPlenio14} T. Baumgratz, M. Cramer, M. B. Plenio, *Quantifying Coherence*, Phys. Rev. Lett. **113** 140401 (2014) [[arXiv:1311.0275](https://arxiv.org/abs/1311.0275), [doi:10.1103/PhysRevLett.113.140401](https://doi.org/10.1103/PhysRevLett.113.140401)]

See also:

* [[Radhakrishnan Chandrashekar|R. Chandrashekar]], P. Manikandan, J. Segar, [[Tim Byrnes]], *Distribution of quantum coherence in multipartite systems*, Phys. Rev. Lett. **116** 150504 (2016) [[arXiv:1602.00286](https://arxiv.org/abs/1602.00286), [doi:10.1103/PhysRevLett.116.150504](https://doi.org/10.1103/PhysRevLett.116.150504)]

* Md. Manirul Ali, Po-Wen Chen, [[Radhakrishnan Chandrashekar|R. Chandrashekar]], *Detecting quantum phase localization using Arnold tongue*, Physica A: Statistical Mechanics and its Applications
**633** (2024) 129436 [[doi:10.1016/j.physa.2023.129436](https://doi.org/10.1016/j.physa.2023.129436)]

[[!redirects decoherence]]

[[!redirects quantum coherence]]
[[!redirects quantum coherences]]


quantum dilogarithm

The __quantum dilogarithm__ was discovered by physicist Anatole Kirillov and by Ludwig Fadeev and Kashaev. The latter also discovered a fundamental pentagon relation for the quantum dilogarithm, which is an analogue of the classical Roger's identities and makes a connection to the current subject of [[cluster algebra|cluster transformations]]. Cf. also classical [[dilogarithm]].

Some references:

* L.D.Fadeev, R.M.Kashaev, _Quantum dilogarithm_, Mod. Phys. Lett. A. 9 (1994) p.427–434.

* V V Bazhanov, N Yu Reshetikhin, _Remarks on quantum dilogarithm_, J. Phys. A: Math. Gen. 28 2217--2226 (1995)
[doi:10.1088/0305-4470/28/8/014](http://dx.doi.org/10.1088/0305-4470/28/8/014)

* A. N. Kirillov, _Dilogarithm identities_, [hep-th/9408113](http://arxiv.org/abs/hep-th/9408113), in: Quantum field theory, integrable models and beyond (Kyoto, 1994). Progr. Theoret. Phys. Suppl. No. __118__ (1995), 61–142, [doi](http://dx.doi.org/10.1143/PTPS.118.61); Dilogarithm identities, partitions, and spectra in conformal field theory. Algebra i Analiz __6__ (1994), no. 2, 152--175, [pdf](http://www.mathnet.ru/php/getFT.phtml?jrnid=aa&paperid=439&volume=6&year=1994&issue=2&fpage=152&what=fullt&option_lang=eng).

*  A.B. Goncharov, _The pentagon relation for the quantum dilogarithm and quantized $M_{0,5}$_, [arxiv:0706.4054](http://arxiv.org/abs/0706.4054)

*  V. V. Fock, A. B. Goncharov, _The quantum dilogarithm and representations of quantum cluster varieties_, [arxiv:math/0702397](http://arxiv.org/abs/math/0702397)

* V. V. Fock, A. B. Goncharov, _Cluster ensembles, quantization and the dilogarithm II: The intertwiner_, [arxiv:math/0702398](http://arxiv.org/abs/math/0702398)

* Ettore Aldrovandi, Leon A. Takhtajan, _Generating functional in CFT and effective action for twodimensional quantum gravity on higher genus Riemann surfaces_, Comm. Math. Phys. 188 (1997), no. 1, 29--67; _Generating functional in CFT on Riemann surfaces. II. Homological aspects_, Comm. Math. Phys. 227 (2002), no. 2, 303--348, [arXiv:math.AT/0006147](http://arxiv.org/abs/math.AT/0006147).

* R. M. Kashaev, _Quantization of Teichmuller spaces and the quantum dilogarithm_, Lett. Math. Phys., Vol. 43, No. 2, 1998

* R.M.Kashaev, _The hyperbolic volume of knots from the quantum dilogarithm_, Letters in mathematical physics __39__, n3, 1997

* Aug 9-13, 2010 -- workshop on quantum dilogarithm in Aarhus: [web](http://www.qgm.au.dk/events/quantumdilogarithm)

8-6)

* S. Alexandrov, B. Pioline, _Theta series, wall-crossing and quantum dilogarithm identities_, Lett. Math. Phys. __106__:1037 (2016) [doi](https://doi.org/10.1007/s11005-016-0857-3)

* S. L. Woronowicz, _Quantum exponential function_, Rev. Math. Phys. __12__ (2000) 873–920

* [[Bernhard Keller]], _On cluster theory and quantum dilogarithm identities_, pp. 85–116 in A. Skowronski, K. Yamagata K. (eds.), Representations of algebras and related topics, Eur. Math. Soc. 2011 [doi](https://doi.org/10.4171/101-1/3) [arXiv/1102.4148](https://arxiv.org/abs/1102.4148)

Appearance of quantum dilogarithm in computations related to some cases of [[mirror symmetry]] and in [[topological string]]s are studied in

* Rinat Kashaev, Marcos Mariño, _Operators from mirror curves and the quantum dilogarithm_, Comm. Math. Phys. 2016 (Online First) [arxiv/1501.01014](http://arxiv.org/abs/1501.01014) [doi](http://dx.doi.org/10.1007/s00220-015-2499-1)

* Rinat Kashaev, Marcos Mariño, Szabolcs Zakany, _Matrix models from operators and topological strings, 2_, Annales Henri Poincaré 2016 (Online First) [doi](http://dx.doi.org/10.1007/s00023-016-0471-z); continuation of M. Mariño, S. Zakany, _Matrix models from operators and topological strings_, [arxiv/1502.02958](http://arxiv.org/abs/1502.02958) (part I has less coverage of quantum dilogarithm aspects)


quantum electrodynamics

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# Contents
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## Idea

_Quantum electrodynamics_ ("QED") is the [[perturbative quantum field theory]] of the [[electromagnetic field]] coupled to a [[Dirac field]] via the [[electron-photon interaction]]: it describes the [[quantum theory]] of [[photons]] and [[electrons]].

## Definition

The [[Lagrangian density]] defining QED is the [[sum]] of

1. the [[free field]] contribution of the [[electromagnetic field]];

1. the [[free field]] contribution of a [[Dirac field]];

1. the [[electron-photon interaction]] term with [[coupling constant]] given by the [[fine structure constant]].

See at _[[A first idea of quantum field theory]]_ [this example](A+first+idea+of+quantum+field+theory#LagrangianQED).

The underlying [[free field theory]] admits a [[quantization]] via the corresponding [[Wick algebra]]. QED is, by definition, the [[perturbative quantum field theory]] obtained from this by finding a perturbative [[S-matrix]] for the [[electron-photon interaction]].

As a result, [[scattering amplitudes]] for [[electron-photon interactions]] in QED are then expressed in terms of [[Feynman amplitudes]]:

[[!include Feynman diagrams in causal perturbation theory -- summary]]

## Properties

* [[anomalous magnetic moment of the electron]]

* [[Lamb shift]]

* [[Casimir effect]]

* [[axial anomaly]]

## Related concepts

* [[fine structure constant]]

* [[photon propagator]]

* [[Schwinger effect]]

* [[quantum field theory]]

  * [[scalar field]]

  * [[gauge theory]]

    * **QED**

    * [[QCD]], [[quantization of Yang-Mills theory]]

## References

### Original articles

The term "quantum electrodynamics" is due to

* [[Paul Dirac]], _The quantum theory of emission and absorption of radiation_, Proc. Roy. Soc. London A 114, 243, 1927 ([spire](http://inspirehep.net/record/42586?ln=en), [pdf](http://wwwhome.lorentz.leidenuniv.nl/~boyarsky/media/Proc.R.Soc.Lond.-1927-Dirac-243-65.pdf))

This left some issues with the quantization of the radiation field. These were addressed, and the [[causal propagator]] for the [[electromagnetic field]] was first found in

* [[Pascual Jordan]], [[Wolfgang Pauli]], _Zur Quantenelektrodynamik ladungsfreier Felder_, Zeitschrift für Physik 47, 151 (1928)

A comprehensive, albeit informal, theory of QED and of [[perturbative quantum field theory]] in general was eventually developed by

* [[Schwinger-Tomonaga-Feynman-Dyson]], 1940s

For more on the history see

* [[Rudolf Haag]], _Early papers on quantum field theory (1992-1930)_ ([pdf](https://link.springer.com/content/pdf/bfm%3A978-3-642-70078-1%2F1%2F1.pdf))

and ([Scharf 95, section 0.0](#Scharf95)).

On [[quantum electrodynamics]] via [[discretized light-cone quantization]]:

* [[Andrew C. Tang]], *Discretized light cone quantization: Application to quantum electrodynamics*, PhD thesis, Stanford (1990) [[spire:296776](https://inspirehep.net/literature/296776), [pdf](https://inspirehep.net/files/5ef318e56c87e7561d36559f54895bd1), [[Tang-DLCQ.pdf:file]]]

* {#TangBrodskyPauli91} [[Andrew C. Tang]], [[Stanley J. Brodsky]], [[Hans-Christian Pauli]], *Discretized light-cone quantization: Formalism for quantum electrodynamics*, Phys. Rev. D **44** (1991) 1842 [[doi:10.1103/PhysRevD.44.1842](https://doi.org/10.1103/PhysRevD.44.1842)]

Discussion of [[vacuum stability]] in [[QED]]:

* {#Scharf96} [[Günter Scharf]], _Vacuum stability in quantum field theory_, Nuovo Cim. A109 (1996) 1605-1607 ([spire:432208](http://inspirehep.net/record/432208/))

* {#Nikolic01} [[Hrvoje Nikolić]], _Physical stability of the QED vacuum_, 2001 ([arXiv:hep-ph/0105176](https://arxiv.org/abs/hep-ph/0105176))

The rigorous construction of [[perturbative QFT|perturbative]] QED in [[causal perturbation theory]] is worked out in

* [Scharf 95](#Scharf95)

and refined to [[perturbative AQFT]] in

* {#DuetschFredenhagen98} [[Michael Dütsch]], [[Klaus Fredenhagen]], _A local (perturbative) construction of observables in gauge theores: the example of QED_ , Commun. Math. Phys. 203 (1999), no.1, 71-105,  ([arXiv:hep-th/9807078](http://xxx.uni-augsburg.de/abs/hep-th/9807078)).

and generalized to possibly non-abelian [[Yang-Mills theory]] in

* {#Hollands07} [[Stefan Hollands]], _Renormalized Quantum Yang-Mills Fields in Curved Spacetime_, Rev. Math. Phys. 20:1033-1172, 2008 ([arXiv:0705.3340](https://arxiv.org/abs/0705.3340))

The [[weak adiabatic limit]] of QED was established in

* {#BlanchardSeneor75} P. Blanchard and R. Seneor, _Green’s functions for theories with massless particles (in perturbation theory)_, Ann. Inst. H. Poincaré Sec. A
23 (2), 147–209 (1975) ([Numdam](http://www.numdam.org/item?id=AIHPA_1975__23_2_147_0))

* {#Duch17} [[Paweł Duch]], _Massless fields and adiabatic limit in quantum field theory_ ([arXiv:1709.09907](https://arxiv.org/abs/1709.09907))

The [[local net of algebras of observables]] and hence the [[algebraic adiabatic limit]] was worked out in

* [Dütsch-Fredenhagen 98](#DuetschFredenhagen98)

Further discussion of the [[adiabatic limit]] and [[infrared divergences]] includes

* Daniel Kapec, Malcolm Perry, Ana-Maria Raclariu, [[Andrew Strominger]], _Infrared Divergences in QED, Revisited_, Phys. Rev. D 96, 085002 (2017) ([arXiv:1705.04311](https://arxiv.org/abs/1705.04311))

### Review

Traditional discussion includes

* [[Radovan Dermisek]], _Quantum Electrodynamics (QED)_ ([pdf](http://www.physics.indiana.edu/~dermisek/QFT_09/qft-II-5-4p.pdf), [[DermisekQED.pdf:file]])

* [[Hitoshi Murayama]], _Quantum Electrodynamics_ ([pdf](http://hitoshi.berkeley.edu/221B-S02/QED.pdf))

* {#Zeidler} [[Eberhard Zeidler]], _Quantum field theory II: Quantum electrodynamics --  A bridge between mathematicians and physicists_, Springer (2009)

* Einan Gardi, lectures 19, 20 of _[Modern Quantum Field Theory](https://www2.ph.ed.ac.uk/~egardi/MQFT/)_, 2015 ([pdf](https://www2.ph.ed.ac.uk/~egardi/MQFT/MQFT_2015_lecture_19_20.pdf))

Mathematically rigorous discussion in [[causal perturbation theory]]/[[perturbative AQFT]] is in

* {#Scharf95} [[Günter Scharf]], _[[Finite Quantum Electrodynamics -- The Causal Approach]]_, Berlin: Springer-Verlag, 1995, 2nd edition

* {#Duetsch18} [[Michael Dütsch]], chapter 5 of _[[From classical field theory to perturbative quantum field theory]]_, 2018

  following ([Dütsch-Fredenhagen 98](#DuetschFredenhagen98))

### Phenomenology

Comparison to [[experiment]] is reviewed in

* _Tests of QED_ ([pdf](http://edu.itp.phys.ethz.ch/hs10/ppp1/PPP1_6.pdf))

Specifically via the [[Lamb shift]]:

* Savely G. Karshenboim, D. I. Mendeleev, _The Lamb Shift of Hydrogen and Low-Energy Tests of QED_ ([arXiv:hep-ph/9411356](https://arxiv.org/abs/hep-ph/9411356))

[[!redirects quantum electrodynamics]]
[[!redirects quantum electro dynamics]]
[[!redirects QED]]


quantum error correction

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## Idea

### General

*Quantum error correction* is concerned with ensuring the robustness of [[quantum computation]] against [[noise]] (as in classical [[error correction]]) and particularly against [[quantum noise]] and [[quantum decoherence]] (e.g. against [[bit flip channel|bit flip errors]]).

From [Ferris & Poulin 2013](#FerrisPoulin13):

> The basic principle of quantum error correction (QEC) is to encode information into the long-range correlations of  [[quantum entanglement|entangled]] quantum many-body  states in  such  a  way that it  cannot be accessed locally. When a local error affects the system, it leaves a detectable imprint—called the error syndrome. The decoding problem consists  in inferring the recovery with greatest probability of success given the error syndrome.  In general, this is a hard problem, but for well-chosen codes, it can be solved efficiently either exactly or heuristically.

From [Steane 06](#Steane06):

> Error correction is especially important in quantum computers, because efficient quantum algorithms make use of large scale quantum interference, which is fragile, i.e. sensitive to imprecision in the computer and to unwanted coupling between the computer and the rest of the world. This makes large scale quantum computation so difficult as to be practically impossible unless error correction methods are used. Indeed, before quantum error correction was discovered, it seemed that this fragility would forbid large scale quantum computation. The ability to correct a quantum computer efficiently without disturbing the coherence of the computation is highly non-intuitive, and was thought more or less impossible. The discovery of powerful error correction methods therefore caused much excitement, since it converted large scale quantum computation from a practical impossibility to a possibility.

From [Devitt, Nemoto & Munro 2009](#DevittNemotoMunro09):

> Starting in 1995, several papers appeared, in rapid succession, proposing codes which were appropriate to perform error correction on quantum data ([Sho95](#Shor95); [Ste96a](#Steane96); [CS96](#CalderbankShor96);  [LMPZ96](#LMPZ96);  [BSW96](#BDSW96)). This  was a key theoretical development needed to convince the general community that quantum computation was indeed a possibility. Since this initial introduction, the progress in this field has been extensive.

From [Zhu & Cross 20](#ZhuCross20):

> Although we are currently in an era of quantum computers with tens of noisy qubits, it is likely that a decisive, practical quantum advantage can only be achieved with a scalable, fault-tolerant, error-corrected quantum computer. Therefore, development of quantum error correction is one of the central themes of the next five to ten years.

[Lucero 21](#Lucero21):

> Within the decade, Google aims to build a useful, error-corrected quantum computer. $[\cdots]$ Our journey to build an error-corrected quantum computer within the decade includes several scientific milestones, including building an error-corrected logical qubit.

Specifically on [[holographic tensor network|holographic quantum error correcting codes]] (see references [below](#ReferencesViaHolographicTensorNetworks)):

From [Harlow 20](#Harlow20):

> The codes provided by AdS/CFT often come close to saturating theoretical bounds on the performance of quantum codes. It seems AdS/CFT may be a tool for discovering better quantum cryptography?

[WVSB 20](#WVSB20):

> $[$ [[holographic tensor network|holographic codes]] $]$ could be promising candidates to circumvent our results and could possibly realise a universal set of unitary implementations of logical operators.

From [CDCW 21](#CDCW21):

> There are a number of reasons to suspect that [[holographic tensor network|holographic codes]] may be of practical use for quantum computing.

> Holographic codes can admit erasure thresholds comparable to that of the widely-studied surface code, and likewise for their threshold against Pauli errors.  Their holographic structure also naturally leads to an organization of encoded qubits into a hierarchy of levels of protection from errors, which could be useful for applications which call for many qubits withvarying levels of protection. In particular, this is reminiscent of many schemes for magic state distillation -- and indeed, the concatenated codes utilized for magic state distillation share a similar hierarchical structure to holographic codes. The layered structure of holographic codes is also reminiscent of memory architectures in classical computers, where it is useful to have different levels of short- and long-term memory.  Although these codes have some notable drawbacks, in particular holographic stabilizer codes require nonlocal stabilizer generators, other codes such as concatenated codes suffer similar drawbacks and have still proven to be useful. Conversely, the stringent requirement of non-local stabilizer generators allows holographic codes to protect many more qubits than a topological code and in fact attain a finite nonzero encoding rate, which is typically not possible for topological codes. Nonetheless, many open questions remain about the usefulness of holographic codes for fault-tolerant quantum computing.

### Quantum error correcting codes

The simple but important special case of passive correction of erasures may be  handled by *quantum error correcting codes*:

Recall that a classical [[error correcting code]] on a [[finite set]] of states $L$ is, typically, a choice of [[injection]] of $L$ into some larger set, typically a [[Cartesian power]]

$$
  L \overset{code}{\hookrightarrow} P \coloneqq H \times \cdots \times H
$$

(often considered in the form of [[linear codes]], but classical nonetheless). In [[quantum physics]] the [[Cartesian product]] of sets of states is replaced by the [[tensor product of vector spaces|tensor product]] of [[Hilbert spaces]].

For $\mathcal{L}$ a given [[Hilbert space]] of [[quantum states]] ([[finite dimensional vector space|finite-dimensional]] in practice), a *quantum error correcting code* is a choice of [[linear subspace|linear embedding]] (the *code subspace*) of $\mathcal{L}$ into a larger Hilbert spaces, often an  $n$-fold [[tensor product]] of copies of some $\mathcal{H}$:

$$
  \array{
    \underset{
     {\color{blue}logical}
     \atop
     {\color{blue}qbits}
    }{
      \mathcal{L}
    }
    &
      \underoverset
        {{\color{blue}code} \atop {\color{blue} subspace}}
        {\;\;\;code\;\;\;}
        {
        \hookrightarrow
      }
    &
    \mathcal{P}
    \coloneqq
    \underset{
      \color{blue}
      physical\;qbits
    }{
    \mathcal{H}^{(1)}
      \otimes
        \cdots
      \otimes
    \mathcal{H}^{(n)}
    }
  }
  \,,
$$

such that the information in $code(\psi)$ in some of the tensor factors $\mathcal{H}^{(i)}$ can be lost without obstructing the reconstruction of $\psi$ (e.g. [Rowell & Wang 17, Sec. 3.2.3](#RowellWang17)).

(Often one demands additional properties, such as that a given set of [[linear operators]] $O$ ([[quantum observables]]) acting on $\psi \in \mathcal{H}$ are implemented on $code(\psi)$ by combinations of operators that act non-trivially only on some of the tensor factors.)

While this is superficially analogous to a classical [[error correcting code]], the crucial and subtle difference is that quantum error correction codes thus take place in a non-[[cartesian monoidal category|Cartesian]] [[symmetric monoidal category|symmetric]] [[monoidal category]]. For this reason, effects of [[quantum entanglement]] play a paramount role in quantum error correction codes.

## Example

### Specific examples

#### Bit flip codes

See at *[[bit flip code]]*.

#### A 3-qutrit code
 {#A3FoldqtritCode}

The following is a simple illustrative example from [Cleve, Gottesman & Lo 99, p. 1-2](#CleveGottesmanLo99):

Consider a [[quantum system]] with a 3-dimensional [[space of quantum states]]:

$$
  \mathcal{H}
    \;\coloneqq\;
  \mathbb{C}^3
    \;=\;
  Span\big(
    \left\vert 0 \right\rangle,\,
    \left\vert 1 \right\rangle,\,
    \left\vert 2 \right\rangle
  \big)
$$

and consider a code space inside 3 copies of this space given by the following [[linear map]]

$$
  \array{
    \mathcal{H}
    &
    \overset{
      \phantom{AAAA}
      code
      \phantom{AAAA}
    }{\longrightarrow}&
    \mathcal{H}^{\otimes 3}
    \\
    \array{
      & \alpha \left\vert 0 \right\rangle
      \\
      + & \beta \left\vert 1 \right\rangle
      \\
      + & \gamma \left\vert 2 \right\rangle
    }
    &
      \mapsto
    &
    \array{
      & \alpha
        \tfrac{1}{\sqrt{3}}
        \big(
          \left\vert 000 \right\rangle
          +
          \left\vert 111 \right\rangle
          +
          \left\vert 222 \right\rangle
        \big)
      \\
      + & \beta
        \tfrac{1}{\sqrt{3}}
        \big(
          \left\vert 012 \right\rangle
          +
          \left\vert 120 \right\rangle
          +
          \left\vert 201 \right\rangle
        \big)
      \\
      + & \gamma
        \tfrac{1}{\sqrt{3}}
        \big(
          \left\vert 021 \right\rangle
          +
          \left\vert 102 \right\rangle
          +
          \left\vert 210 \right\rangle
        \big)
    }
  }
$$

This code corrects errors consisting of the loss of one of the three copies, in the following sense:

There is a [[linear operator]] acting only on two of the three copies, explicitly given ([ADH 14 (3.6)](#ADH14)) by

$$
  \array{
    \mathcal{H} \otimes \mathcal{H}
    &
    \overset{
      \;\;\;\;\;\;
      U^{(12)}
      \;\;\;\;\;\;
    }{\longrightarrow}
    &
    \mathcal{H} \otimes \mathcal{H}
    \\
    \left\vert 00 \right\rangle &\mapsto& \left\vert 00 \right\rangle
    \\
    \left\vert 01 \right\rangle &\mapsto& \left\vert 12 \right\rangle
    \\
    \left\vert 02 \right\rangle &\mapsto& \left\vert 21 \right\rangle
    \\
    \left\vert 10 \right\rangle &\mapsto& \left\vert 22 \right\rangle
    \\
    \left\vert 11 \right\rangle &\mapsto& \left\vert 01 \right\rangle
    \\
    \left\vert 12 \right\rangle &\mapsto& \left\vert 10 \right\rangle
    \\
    \left\vert 20 \right\rangle &\mapsto& \left\vert 11 \right\rangle
    \\
    \left\vert 21 \right\rangle &\mapsto& \left\vert 20 \right\rangle
    \\
    \left\vert 22 \right\rangle &\mapsto& \left\vert 02 \right\rangle
  }
$$

such that

$$
  \big( U^{(12)} \otimes id^{(3)} \big)
  \circ
  code
  \big(
    \left\vert \psi \right\rangle
  \big)
  \;\; = \;\;
  \left\vert \psi \right\rangle
  \,
  \otimes
  \,
  \tfrac{1}{\sqrt{3}}
  \left(
    \left\vert 00 \right\rangle
    +
    \left\vert 11 \right\rangle
    +
    \left\vert 22 \right\rangle
  \right)
$$

#### The HaPPY code

The [[HaPPY code]] is a [[quantum error correction code]] (a class of such codes really, indexed by a "cutoff" [[natural number]]) which is thought to exhibit characteristic properties akin to the encoding of [[bulk]]-[[quantum states]] by [[boundary field theory|boundary]]-states expected in the [[AdS/CFT correspondence]]. In particular, the HaPPY code (or rather the [[tensor network]] that defines it) exhibits a discretized form of the [[Ryu-Takayanagi formula]] for [[holographic entanglement entropy]].

Concretely, the the HaPPY [[code subspace]] is the [[image]] of the [[linear map]] formed by:

\begin{imagefromfile}
    "file_name": "HaPPYCodeTesselation.jpg",
    "float": "right",
    "width": 440,
    "unit": "px",
    "margin": {
        "top": -40,
        "bottom": 20,
        "right": 0,
        "left": 10
    },
    "caption": "From [Harlow 18](#Harlow18)"
\end{imagefromfile}

1. picking a [[perfect tensor]] $T$ of [[rank]] 6;

1. picking a finite cutoff of the [[pentagon|pentagonal]] [[tesselation]] of the [[hyperbolic plane]];

1. regarding its [[Poincaré duality|Poincaré dual]] [[graph]] as a [[tensor network]] ([[string diagram]] in [[finite-dimensional vector spaces]]) by

   1. assigning $T$ to each [[vertex]] at the center of the [[pentagons]] (show in blue), with 5 of its indices contracted with its neighbours in the hyperbolic plane,

   1. and its 6th uncontracted index remaining as an input (shown in red);

1. regarding the uncontracted edges at the cutoff boundary as output (shown in white)

and thus as a [[linear map]] from the [[tensor product]] over the [[bulk]]-[[vertices]] to the [[tensor product]] over the edges sticking out over the boundary.

#### Majorana dimer codes

See at *[[Majorana dimer code]]*.

### Classes of examples

* [[bit flip code]]

* [[Æ codes]]

* [[stabilizer code]]

  * [[HaPPY code]]

  * [[Majorana dimer code]]

## Related concepts

* [[error correcting code]]

* [[dynamic lifting]]

* [[repeat-until-success computing]]

* [[quantum entanglement]]

* [[entanglement entropy]], [[holographic entanglement entropy]]

* [[quantum computing]], [[topological quantum computing]]

* [[quantum secret sharing]]

* [[tensor network]]

* [[quantum relation]]

## References

### General

The idea of quantum encryption based on the [[no-cloning theorem]] originates with:

* [[Stephen Wiesner]], *Conjugate Coding*, circulated in the 1960s, finally published in ACM SIGACT News **15** 1 (1983) 78–88 ([original pdf](http://users.cms.caltech.edu/~vidick/teaching/120_qcrypto/wiesner.pdf), [doi:10.1145/1008908.1008920](https://doi.org/10.1145/1008908.1008920))

Original articles on quantum error correcting codes:

* {#Shor95} [[Peter W. Shor]], *Scheme for reducing decoherence in quantum computer memory*, Phys. Rev. A 52, R2493(R) 1995 ([doi:10.1103/PhysRevA.52.R2493](https://doi.org/10.1103/PhysRevA.52.R2493))

* {#Steane96} [[Andrew M. Steane]], *Error Correcting Codes in Quantum Theory*, Phys. Rev. Lett. 77, 793 1996 ([doi:10.1103/PhysRevLett.77.793](https://doi.org/10.1103/PhysRevLett.77.793))

* {#CalderbankShor96} [[Robert Calderbank]], [[Peter W. Shor]], *Good Quantum Error-Correcting Codes Exist*, Phys. Rev. A, Vol. 54, No. 2, pp. 1098-1106, 1996 ([doi:10.1103/PhysRevA.54.1098](https://doi.org/10.1103/PhysRevA.54.1098))

* {#LMPZ96} [[Raymond Laflamme]], Cesar Miquel, Juan Pablo Paz, [[Wojciech H. Zurek]], *Perfect Quantum Error Correction Code*, Phys. Rev. Lett., 77:198, 1996 ([arXiv:quant-ph/9602019](https://arxiv.org/abs/quant-ph/9602019))

* {#BDSW96} Charles H. Bennett, David P. DiVincenzo, John A. Smolin, William K. Wootters, *Mixed State Entanglement and Quantum Error Correction*, Phys. Rev. A54:3824-3851, 1996 ([doi:10.1103/PhysRevA.54.3824](https://doi.org/10.1103/PhysRevA.54.3824))

* [[Andrew M. Steane]], *Multiple Particle Interference and Quantum Error Correction*, Proc. Roy. Soc. Lond. A452 (1996) 2551 ([arXiv:quant-ph/9601029](https://arxiv.org/abs/quant-ph/9601029))

* [[Emanuel Knill]], [[Raymond Laflamme]], *A Theory of Quantum Error-Correcting Codes*, Phys. Rev. Lett. 84:2525-2528, 2000 ([arXiv:quant-ph/9604034](https://arxiv.org/abs/quant-ph/9604034))

Realization that quantum error correcting codes could make quantum computation practically feasible:

* [[Emanuel Knill]], [[Raymond Laflamme]], [[Wojciech H. Zurek]], *Resilient Quantum Computation: Error Models and Thresholds*, Proceedings of the Royal Society A **454** 1969 (1998) [[arXiv:quant-ph/9702058](https://arxiv.org/abs/quant-ph/9702058)]

* [[John Preskill]], *Fault-tolerant quantum computation*, in: *Introduction to Quantum Computation and Information*, World Scientific (1998) $[$[arXiv:quant-ph/9712048](https://arxiv.org/abs/quant-ph/9712048), [doi:10.1142/3724](https://doi.org/10.1142/3724)$]$

* [[John Preskill]], *Reliable Quantum Computers*, Proc. Roy. Soc. Lond. A **454** (1998) 385-410 $[$[arXiv:quant-ph/9705031](https://arxiv.org/abs/quant-ph/9705031), [doi:10.1098/rspa.1998.0167](https://doi.org/10.1098/rspa.1998.0167)$]$

* [[Daniel Gottesman]], *Theory of fault-tolerant quantum computation*, Phys. Rev. A **57** (1998) 127-137 $[$[doi:10.1103/PhysRevA.57.127](https://doi.org/10.1103/PhysRevA.57.127)$]$

* [[Dorit Aharonov]], [[Michael Ben-Or]], *Fault-Tolerant Quantum Computation With Constant Error Rate*, SIAM J. Comput., **38** 4 (2008) 1207–1282 [[arXiv:quant-ph/9906129](https://arxiv.org/abs/quant-ph/9906129), [doi:10.1007/978-3-642-32512-0_31](https://doi.org/10.1007/978-3-642-32512-0_31)]

* [[Daniel Gottesman]], *Fault-Tolerant Quantum Computation*,  Physics in Canada **63** 4  (2007) 183-189 $[$[arXiv:quant-ph/0701112](https://arxiv.org/abs/quant-ph/0701112)$]$

* Dorit Aharonov, Michael Ben-Or, *Fault-Tolerant Quantum Computation With Constant Error Rate*, SIAM J. Comput. **38** 4 (2008) 1207–1282 ([arXiv:quant-ph/9906129](https://arxiv.org/abs/quant-ph/9906129), [doi:10.1137/S0097539799359385](https://doi.org/10.1137/S0097539799359385))

* Panos Aliferis, [[Daniel Gottesman]], [[John Preskill]], *Quantum accuracy threshold for concatenated distance-3 codes*, Quant. Inf. Comput. 6 (2006) 97-165 ([arXiv:quant-ph/0504218](https://arxiv.org/abs/quant-ph/0504218))

* Earl T. Campbell, Barbara M. Terhal, Christophe Vuillot: *Roads towards fault-tolerant universal quantum computation*, Nature **549** (2017) 172–179 $[$[doi:10.1038/nature23460](https://doi.org/10.1038/nature23460)$]$

[[operator algebra|Operator algebraic]] formulation of quantum error correction:

* [[David Kribs]], [[Raymond Laflamme]], [[David Poulin]], *A Unified and Generalized Approach to Quantum Error Correction*, Phys. Rev. Lett. 94, 180501 (2005) ([arXiv:quant-ph/0412076](https://arxiv.org/abs/quant-ph/0412076))

* [[David W. Kribs]], [[Raymond Laflamme]], [[David Poulin]], [[Maia Lesosky]], *Operator quantum error correction*, Quant. Inf. & Comp., 6 (2006), 383-399 ([arXiv:quant-ph/0504189](https://arxiv.org/abs/quant-ph/0504189))

on [[quantum observables]]:

* {#BenyKempfKribs06} Cédric Bény, Achim Kempf, [[David W. Kribs]], _Generalization of Quantum Error Correction via the Heisenberg Picture_, Phys. Rev. Lett. 98, 100502 – Published 7 March 2007 ([doi:10.1103/PhysRevLett.98.100502](https://doi.org/10.1103/PhysRevLett.98.100502), [arXiv:quant-ph/0608071](https://arxiv.org/abs/quant-ph/0608071))

* Cédric Bény, Achim Kempf, [[David W. Kribs]], _Quantum Error Correction of Observables_, Phys. Rev. A 76, 042303 (2007) ([arXiv:0705.1574](https://arxiv.org/abs/0705.1574))

Introduction and survey:

* [[Alexei Kitaev]],  *Quantum computations: algorithms and error correction*, Russian Mathematical Surveys, **52** 6 (1997) [[doi:10.1070/RM1997v052n06ABEH002155](https://iopscience.iop.org/article/10.1070/RM1997v052n06ABEH002155), pdf]

* [[Michael A. Nielsen]], [[Isaac L. Chuang]], Chapter 10 of: *Quantum computation and quantum information*, Cambridge University Press (2000) [[doi:10.1017/CBO9780511976667](https://doi.org/10.1017/CBO9780511976667), [pdf](http://csis.pace.edu/~ctappert/cs837-19spring/QC-textbook.pdf), [[NielsenChuangQuantumComputation.pdf:file]]]

* Simeon Ball, Aina Centelles, Felix Huber, *Quantum error-correcting codes and their geometries* ([arXiv:2007.05992](https://arxiv.org/abs/2007.05992))

* {#Steane06} [[Andrew M. Steane]], *A Tutorial on Quantum Error Correction*, in: Proceedings of the International School of Physics “Enrico Fermi”, course CLXII, "Quantum Computers, Algorithms and Chaos"  (2006) [[pdf](https://www2.physics.ox.ac.uk/sites/default/files/ErrorCorrectionSteane06.pdf), [[SteaneTutorialQuantumErrorCorrection.pdf:file]]]

* [[Isaac L. Chuang]], *Quantum error correction*, Chapter 7 in: *Quantum Machines: Measurement and Control of Engineered Quantum Systems* Lecture Notes of the Les Houches Summer School **96** (2011) 273–320  [[doi:10.1093/acprof:oso/9780199681181.003.0007](https://doi.org/10.1093/acprof:oso/9780199681181.003.0007)]

* {#DevittNemotoMunro09} [[Simon J. Devitt]], [[Kae Nemoto]], [[William J. Munro]], *Quantum Error Correction for Beginners*, Rep. Prog. Phys. **76** (2013) 076001 [[arXiv:0905.2794](https://arxiv.org/abs/0905.2794), [doi:10.1088/0034-4885/76/7/076001](https://iopscience.iop.org/article/10.1088/0034-4885/76/7/076001)]

* {#RowellWang17} [[Eric Rowell]], [[Zhenghan Wang]], Section 3.2.3 of: _Mathematics of Topological Quantum Computing_, Bull. Amer. Math. Soc. 55 (2018), 183-238 ([arXiv:1705.06206](https://arxiv.org/abs/1705.06206), [doi:10.1090/bull/1605](https://doi.org/10.1090/bull/1605))

* [[Scott Aaronson]], Chapter 27 of: *Introduction to Quantum Information Science* (2018) [[pdf](https://www.scottaaronson.com/qclec.pdf), [webpage](https://www.scottaaronson.com/cs378/)]

* Joschka Roffe, *Quantum Error Correction: An Introductory Guide*, Contemporary Physics 2019 ([arXiv:1907.11157](https://arxiv.org/abs/1907.11157))

* [[Bei Zeng]], [[Xie Chen]], [[Duan-Lu Zhou]], [[Xiao-Gang Wen]]:

  Sec. 3 of: *[[Quantum Information Meets Quantum Matter]] -- From Quantum Entanglement to Topological Phases of Many-Body Systems*, Quantum Science and Technology (QST), Springer (2019) $[$[arXiv:1508.02595](https://arxiv.org/abs/1508.02595), [doi:10.1007/978-1-4939-9084-9](https://doi.org/10.1007/978-1-4939-9084-9)$]$

* {#ZhuCross20} Guanyu Zhu, Andrew Cross, *[Hardware-aware approach for fault-tolerant quantum computation](https://www.ibm.com/blogs/research/2020/09/hardware-aware-quantum/)*, IBM Research Blog, September 2, 2020

See also:

* Wikipedia, _[Quantum error correction](https://en.m.wikipedia.org/wiki/Quantum_error_correction)_

* {#AlbertECZ} [[Victor V. Albert]] et al.,  *[errorcorrectionzoo.org](https://errorcorrectionzoo.org)*

In the context of [[quantum secret sharing]]:

* {#CleveGottesmanLo99} Richard Cleve, [[Daniel Gottesman]], Hoi-Kwong Lo, *How to share a quantum secret*, Phys. Rev. Lett. 83 (1999) 648-651 ([arXiv:quant-ph/9901025](https://arxiv.org/abs/quant-ph/9901025))

In the context of quantum communication:

* [[Matthias Christandl]], Alexander Müller-Hermes, *Fault-tolerant Coding for Quantum Communication*,
IEEE Transactions on Information Theory, [[arXiv:2009.07161](https://arxiv.org/abs/2009.07161)  [doi:10.1109/TIT.2022.3169438](https://ieeexplore.ieee.org/document/9761242)]

On (in-)compatibility of quantum error correction with universality of quantum gates ([[Eastin-Knill theorem]]):

* [[Bryan Eastin]], [[Emanuel Knill]], *Restrictions on Transversal Encoded Quantum Gate Sets*,  Phys. Rev. Lett. 102, 110502 (2009) ([arXiv:0811.4262](https://arxiv.org/abs/0811.4262))

* {#WVSB20} Paul Webster, Michael Vasmer, Thomas R. Scruby, Stephen D. Bartlett, *Universal Fault-Tolerant Quantum Computing with Stabiliser Codes* ([arXiv:2012.05260](https://arxiv.org/abs/2012.05260))

For [[quantum annealing]] processes:

* Kristen L. Pudenz, Tameem Albash, Daniel A. Lidar: *Error-corrected quantum annealing with hundreds of qubits*, Nature Communications **5** 3243 (2014) [[doi:10.1038/ncomms4243](https://doi.org/10.1038/ncomms4243)]

On continuous variable quantum codes:

* [[Samuel L. Braunstein]], Peter van Loock, §IV.C in: *Quantum information with continuous variables*, Rev. Mod. Phys. **77** 2 (2005) 513 [[arXiv:quant-ph/0410100](https://arxiv.org/abs/quant-ph/0410100), [doi:10.1103/RevModPhys.77.513](https://doi.org/10.1103/RevModPhys.77.513)]

* [[Samuel L. Braunstein]], Arun K. Pati, *Quantum Information with Continuous Variables*, Springer (2003) [[doi:10.1007/978-94-015-1258-9](https://doi.org/10.1007/978-94-015-1258-9)]

* Allan D. C. Tosta, Thiago O. Maciel, [[Leandro Aolita]], *Grand Unification of continuous-variable codes* [[arXiv:2206.01751](https://arxiv.org/abs/2206.01751)]

In relation to the [[SYK model]]:

* Shao-Kai Jian, Chunxiao Liu, Xiao Chen, [[Brian Swingle]], Pengfei Zhang, *Quantum error as an emergent magnetic field* ([arXiv:2106.09635](https://arxiv.org/abs/2106.09635))

In relation to continuous symmetries:

* Zi-Wen Liu, Sisi Zhou, *Approximate symmetries and quantum error correction* ([arXiv:2111.06355](https://arxiv.org/abs/2111.06355))

Further developments:

* Mackenzie H. Shaw, [[Andrew C. Doherty]], Arne L. Grimsmo, *Stabilizer subsystem decompositions for single- and multi-mode Gottesman-Kitaev-Preskill codes* [[arXiv:2210.14919](https://arxiv.org/abs/2210.14919)]

On relevant classical control mechanisms:

* Hideo Mabuchi, *Continuous quantum error correction as classical hybrid control*, New J. Phys. **11** (2009) 105044
[[doi:10.1088/1367-2630/11/10/105030](https://iopscience.iop.org/article/10.1088/1367-2630/11/10/105030)]

* {#PalerHerrDevitt19} Alexandru Paler, Daniel Herr, [[Simon J. Devitt]], *Really Small Shoe Boxes - On Realistic Quantum Resource Estimation*, Computer **52** 6 (2019) [[arXiv:1902.08104](https://arxiv.org/abs/1902.08104), [doi:10.1109/MC.2019.2908621](https://doi.org/10.1109/MC.2019.2908621)]

In relation to the [[Penrose tiling]]:

* Zhi Li, [[Latham Boyle]], *The Penrose Tiling is a Quantum Error-Correcting Code* [[arXiv:2311.13040](https://arxiv.org/abs/2311.13040)]

### Experimental realization

Realization of [[quantum error correction]] in [[experiment]], hence in actual [[quantum computers]]:

* D. G. Cory, M. D. Price, W. Maas, [[Emanuel Knill]], [[Raymond Laflamme]], [[Wojchiek H. Zurek]], T. F. Havel, and S. S. Somaroo, *Experimental Quantum Error Correction*, Phys. Rev. Lett. 81, 2152 (1998) ([doi:10.1103/PhysRevLett.81.2152](https://journals.aps.org/prl/abstract/10.1103/PhysRevLett.81.2152))

* Daniel Nigg, Markus Mueller, Esteban A. Martinez, Philipp Schindler, Markus Hennrich, Thomas Monz, Miguel A. Martin-Delgado, Rainer Blatt,

  *Experimental Quantum Computations on a Topologically Encoded Qubit*, Science 18 Jul 2014: Vol. 345, Issue 6194, pp. 302-305 ([arXiv:1403.5426](https://arxiv.org/abs/1403.5426), [doi:10.1126/science.1253742](https://science.sciencemag.org/content/345/6194/302))

  > (via methods from [[topological quantum computation]])

* Adrian Cho, *[The biggest flipping challenge in quantum computing](https://www.sciencemag.org/news/2020/07/biggest-flipping-challenge-quantum-computing)*, Science News Jul. 9, 2020

  > (exposition)

* Y. Ma et al. *Error-transparent operations on a logical qubit protected by quantum error correction*, Nature Physics volume 16, pages 827–831 (2020) ([doi:10.1038/s41567-020-0893-x](https://doi.org/10.1038/s41567-020-0893-x))

* Ming Gong et al. *Experimental exploration of five-qubit quantum error correcting code with superconducting qubits*, National Science Review, nwab011 (2021) ([doi:10.1093/nsr/nwab011](https://doi.org/10.1093/nsr/nwab011))

  survey in:

   EurekaAlert, Science China Press: *Demonstration of the universal quantum error correcting code with superconducting qubits*, March 2021 ([2021-03/scp-dot031521](https://www.eurekalert.org/pub_releases/2021-03/scp-dot031521.php))

* {#Lucero21} Erik Lucero (Google), *[Unveiling our new Quantum AI campus](https://blog.google/technology/ai/unveiling-our-new-quantum-ai-campus/)*,  May 18, 2021

  > (outlook)

* Th. Monz et al., *Demonstration of fault-tolerant universal quantum gate operations*, Nature **605** (2022) 675–680 [[doi:10.1038/s41586-022-04721-1](https://doi.org/10.1038/s41586-022-04721-1)]

* {#MüllerWallraffEtAl22} [[Markus Müller]], [[Andreas Wallraff]] et al., *Realizing Repeated Quantum Error Correction in a Distance-Three Surface Code*, Nature **605** (2022) 669–674 [[arXiv:2112.03708](https://arxiv.org/abs/2112.03708), [doi:10.1038/s41586-022-04566-8](https://doi.org/10.1038/s41586-022-04566-8)]

* [[Markus Müller]] et al., *Demonstration of fault-tolerant universal quantum gate operations*, Nature **605** (2022) 675-680 [[doi:10.1038/s41586-022-04721-1](https://doi.org/10.1038/s41586-022-04721-1), [arXiv:2111.12654](https://arxiv.org/abs/2111.12654)]

### Via holographic tensor networks
 {#ReferencesViaHolographicTensorNetworks}

First quantum error correcting codes associated with planar bulk/boundary systems:

* S. B. Bravyi, [[Alexei Kitaev]], *Quantum codes on a lattice with boundary* ([arXiv:quant-ph/9811052](https://arxiv.org/abs/quant-ph/9811052))

* [[Michael Freedman]], David A. Meyer, *Projective Plane and Planar Quantum Codes*, Found. Comput. Math. 1, 325–332 (2001) ([arXiv:quant-ph/9810055](https://arxiv.org/abs/quant-ph/9810055), [doi:10.1007/s102080010013](https://doi.org/10.1007/s102080010013))

First suggestion relating quantum error correction to [[black hole entropy]]:

* {#VerlindeVerlinde12} [[Erik Verlinde]], [[Herman Verlinde]], *Black Hole Entanglement and Quantum Error Correction*, J. High Energ. Phys. 2013, 107 (2013) ([arXiv:1211.6913](https://arxiv.org/abs/1211.6913),  doi:10.1007/JHEP10(2013)107)

Introducing the idea of quantum error correcting codes given by [[tensor network states]]:

* {#FerrisPoulin13} Andrew J. Ferris, [[David Poulin]], *Tensor Networks and Quantum Error Correction*, Phys. Rev. Lett. 113, 030501 (2014) ([arXiv:1312.4578](https://arxiv.org/abs/1312.4578))

* Dave Bacon, Steven T. Flammia, Aram W. Harrow, Jonathan Shi, *Sparse Quantum Codes from Quantum Circuits*, Proc. of STOC '15, pp. 327-334 (2015); IEEE Transactions on Information Theory, vol 63, no 4, pp 2464-2479, April 2017 ([arXiv:1411.3334](https://arxiv.org/abs/1411.3334))

following observations in

* {#Swingle09} [[Brian Swingle]], _Entanglement Renormalization and Holography_, Phys. Rev. D 86, 065007 (2012) ([arXiv:0905.1317](https://arxiv.org/abs/0905.1317))

* {#Swingle12} [[Brian Swingle]], _Constructing holographic spacetimes using entanglement renormalization_ ([arXiv:1209.3304](https://arxiv.org/abs/1209.3304), [spire:1185813](http://inspirehep.net/record/1185813))

Interpretation of [[holographic tensor networks]] encoding [[holographic entanglement entropy]] in models for [[AdS2-CFT1 duality]] as [[quantum error correcting codes]]:

* {#ADH14} [[Ahmed Almheiri]], [[Xi Dong]], [[Daniel Harlow]], _Bulk Locality and Quantum Error Correction in AdS/CFT_, JHEP 1504:163,2015 ([arXiv:1411.7041](https://arxiv.org/abs/1411.7041), doi:10.1007/JHEP04(2015)163)

  > (using [Bény-Kempf-Kribs 06](#BenyKempfKribs06))

with precursor observations in

* [[Beni Yoshida]], *Information storage capacity of discrete spin systems*, Annals of Physics 338, 134 (2013) ([arXiv:1111.3275](https://arxiv.org/abs/1111.3275))

  > (focus on classical [[error correcting codes]])

* Jose I. Latorre, German Sierra, *Holographic codes* ([arXiv:1502.06618](https://arxiv.org/abs/1502.06618))

and concrete implementation by the [[HaPPY code]]:

* {#PYHP15} [[Fernando Pastawski]], [[Beni Yoshida]], [[Daniel Harlow]], [[John Preskill]], _Holographic quantum error-correcting codes: Toy models for the bulk/boundary correspondence_, JHEP 06 (2015) 149 ([arXiv:1503.06237](https://arxiv.org/abs/1503.06237))

and possibly more generally:

* [[Daniel Harlow]], *The Ryu–Takayanagi Formula from Quantum Error Correction*, Comm. Math. Phys. **354** (2017) 865–912 [[doi:10.1007/s00220-017-2904-z](https://doi.org/10.1007/s00220-017-2904-z), [arXiv:1607.03901](https://arxiv.org/abs/1607.03901)]

Introduction of the more general [[Majorana dimer code]]:

* [[Alexander Jahn]], [[Marek Gluza]], [[Fernando Pastawski]], [[Jens Eisert]], _Majorana dimers and holographic quantum error-correcting codes_, Phys. Rev. Research 1, 033079 (2019) ([arXiv:1905.03268](https://arxiv.org/abs/1905.03268))

Exposition and review:

* [[John Preskill]], *Is spacetime a quantum error-correcting code?*, talk at KITP 2015 ([pdf](http://theory.caltech.edu/~preskill/talks/Preskill-KITP-holographic-2015.pdf), [[Preskill_SpacetimeQEC.pdf:file]])

* {#Harlow18} [[Daniel Harlow]], _TASI Lectures on the Emergence of Bulk Physics in AdS/CFT_, PoS TASI2017 (2018) 002 ([arXiv:1802.01040](https://arxiv.org/abs/1802.01040), [doi:10.22323/1.305.0002](https://doi.org/10.22323/1.305.0002))

* [[Pratik Rath]], *Aspects of Holography And Quantum Error Correction*, 2020 ([pdf](https://digitalassets.lib.berkeley.edu/etd/ucb/text/Rath_berkeley_0028E_19810.pdf), [[RathHolographyAndQEC.pdf:file]])

* [[Melanie Swan]], [[Renato P dos Santos]], [[Frank Witte]], *The AdS/CFT Correspondence and Holographic Codes* ([doi:10.1142/9781786348210_0013](https://doi.org/10.1142/9781786348210_0013), [doi:10.1142/9781786348210_0014](https://www.worldscientific.com/doi/abs/10.1142/9781786348210_0014)), Part 5 in: Between Science and Economics, Volume 2: *Quantum Computing Physics, Blockchains, and Deep Learning Smart Networks*, World Scientific 2020 ([doi:10.1142/q0243](https://doi.org/10.1142/q0243))

* {#Harlow20} [[Daniel Harlow]], *Computation and Holography*, talk at [Snowmass Computational Frontier Workshop 2020](https://indico.fnal.gov/event/43829/) ([pdf](https://indico.fnal.gov/event/43829/contributions/193566/attachments/132763/163346/snowmasscomp.pdf), [[HarlowComputationHolography.pdf:file]])

* [[Alexander Jahn]], [[Jens Eisert]], _Holographic tensor network models and quantum error correction: A topical review_ ([arXiv:2102.02619](https://arxiv.org/abs/2102.02619))

* Tanay Kibe, Prabha Mandayam, Ayan Mukhopadhyay, *Holographic spacetime, black holes and quantum error correcting codes: A review* ([arXiv:2110.14669](https://arxiv.org/abs/2110.14669))

Further discussion of holographic quantum error correcting codes:

* Henrique Lazari, Reginaldo Palazzo Jr., *Geometrically uniform hyperbolic codes*, Comput. Appl. Math. vol.24 no.2 Petrópolis 2005 ([doi:10.1590/S0101-82052005000200002](https://doi.org/10.1590/S0101-82052005000200002))

* Enrico M. Brehm, Benedikt Richter, *Classical Holographic Codes*, Phys. Rev. D 96, 066005 (2017) ([arXiv:1609.03560](https://arxiv.org/abs/1609.03560))

* [[Fernando Pastawski]], [[John Preskill]], *Code properties from holographic geometries*, Phys. Rev. X 7, 021022 (2017) ([arXiv:1612.00017](https://arxiv.org/abs/1612.00017))

* [[Robert J. Harris]], [[Nathan A. McMahon]], [[Gavin K. Brennen]], [[Thomas M. Stace]], *Calderbank-Steane-Shor Holographic Quantum Error Correcting Codes*, Phys. Rev. A 98, 052301 (2018) ([arXiv:1806.06472](https://arxiv.org/abs/1806.06472))

* Tamara Kohler, Toby Cubitt, *Toy Models of Holographic Duality between local Hamiltonians*, 	J. High Energy Phys. 2019:17 (2019) ([arXiv:1810.08992](https://arxiv.org/abs/1810.08992))

* Tobias J. Osborne, Deniz E. Stiegemann, *Dynamics for holographic codes*, J. High Energ. Phys. 2020, 154 (2020) ([arXiv:1706.08823](https://arxiv.org/abs/1706.08823))

* Martina Gschwendtner, Robert König, Burak Şahinoğlu & Eugene Tang, *Quantum error-detection at low energies*, Journal of High Energy Physics volume 2019, Article number: 21 (2019) ([arXiv:1902.02115](https://arxiv.org/abs/1902.02115))

* [[Nathan A. McMahon]], [[Gavin K. Brennen]], [[Thomas M. Stace]], [[Robert J. Harris]], Elliot Coupe, *Decoding Holographic Codes with an Integer Optimisation Decoder* ([arXiv:2008.10206](https://arxiv.org/abs/2008.10206))

* [[Terry Farrelly]], [[Robert J. Harris]], [[Nathan A. McMahon]], [[Thomas M. Stace]], *Tensor-network codes* ([arXiv:2009.10329](https://arxiv.org/abs/2009.10329))

* ChunJun Cao, Brad Lackey, *Approximate Bacon-Shor Code and Holography* ([arXiv:2010.05960](https://arxiv.org/abs/2010.05960))

* {#CDCW21} Sam Cree, Kfir Dolev, Vladimir Calvera, Dominic J. Williamson, *Fault-tolerant logical gates in holographic stabilizer codes are severely restricted* ([arXiv:2103.13404](https://arxiv.org/abs/2103.13404))

* Robert de Mello Koch, Eunice Gandote, Nirina Hasina Tahiridimbisoa, Hendrik J.R. Van Zyl, *Quantum Error Correction and Holographic Information from Bilocal Holography* ([arXiv:2106.00349](https://arxiv.org/abs/2106.00349))

* [[Chris Akers]], [[Geoff Penington]], *Quantum minimal surfaces from quantum error correction* ([arXiv:2109.14618](https://arxiv.org/abs/2109.14618))

* ChunJun Cao, Brad Lackey, *Quantum Lego: Building Quantum Error Correction Codes from Tensor Networks*, PRX Quantum **3** 020332 (2022) $[$[arXiv:2109.08158](https://arxiv.org/abs/2109.08158), [doi:10.1103/PRXQuantum.3.020332](https://doi.org/10.1103/PRXQuantum.3.020332)$]$

* Jason Pollack, Patrick Rall, Andrea Rocchetto, *Understanding holographic error correction via unique algebras and atomic examples*, Journal of High Energy Physics, 56 (2022) $[$[arXiv:2110.14691](https://arxiv.org/abs/2110.14691)$]$

* Dmitry S. Ageev, *Exploring uberholography* [[arXiv:2208.07387](https://arxiv.org/abs/2208.07387)]

* Matthew Steinberg, Sebastian Feld, [[Alexander Jahn]], *Holographic Codes from Hyperinvariant Tensor Networks* [[arXiv:2304.02732](https://arxiv.org/abs/2304.02732)]

Understanding in terms of the [[eigenstate thermalization hypothesis]]:

* [[Ning Bao]], Newton Cheng, *Eigenstate Thermalization Hypothesis and Approximate Quantum Error Correction*, JHEP 08 (2019) 152 ([arXiv:1906.03669](https://arxiv.org/abs/1906.03669))

In relation to [[holographic Renyi entropy]]:

* [[Chris Akers]], [[Pratik Rath]], *Holographic Renyi entropy from quantum error correction*,  J. High Energ. Phys. 2019, 52 (2019) ([arXiv:1811.05171](https://arxiv.org/abs/1811.05171))

From [[tesselations]] of higher-dimensional [[hyperbolic space]]:

* Vivien Londe, Anthony Leverrier, *Golden codes: quantum LDPC codes built from regular tessellations of hyperbolic 4-manifolds* ([arXiv:1712.08578](https://arxiv.org/abs/1712.08578))

In view of [[black hole thermodynamics]]:

* [Verlinde & Verlinde 2012](#VerlindeVerlinde12)

* [[Ahmed Almheiri]], _Holographic Quantum Error Correction and the Projected Black Hole Interior_ ([arXiv:1810.02055](https://arxiv.org/abs/1810.02055))

* Isaac H. Kim, Eugene Tang, [[John Preskill]], *The ghost in the radiation: Robust encodings of the black hole interior*, JHEP 2020, 31 (2020) ([arXiv:2003.05451](https://arxiv.org/abs/2003.05451))

* [[Chris Akers]], [[Netta Engelhardt]], [[Daniel Harlow]], [[Geoff Penington]], [[Shreya Vardhan]], *The black hole interior from non-isometric codes and complexity* [[arXiv:2207.06536](https://arxiv.org/abs/2207.06536)]

* [[Daniel Harlow]], *A theory of the black hole interior*, talk at *[Strings 2022](https://indico.cern.ch/event/1085701) [[indico:4940817/](https://indico.cern.ch/event/1085701/contributions/4940817/)]

Discussion of [[gauge symmetry]] of [[holographic tensor networks]] and their [[quantum error correcting codes]]:

* Kfir Dolev, Vladimir Calvera, Sam Cree, Dominic J. Williamson,
*Gauging the bulk: generalized gauging maps and holographic codes* ([arXiv:2108.11402](https://arxiv.org/abs/2108.11402))

Relation of quantum error correcting codes to the [[Monster vertex operator algebra]] and more generally to [[2d SCFT]] and [[string theory]]:

* [[Jeffrey A. Harvey]], [[Gregory W. Moore]], *Moonshine, Superconformal Symmetry, and Quantum Error Correction*,  J. High Energ. Phys. **2020**, 146 (2020). ([arXiv:2003.13700](https://arxiv.org/abs/2003.13700))

In relation to the [[large N limit]]:

* Alexey Milekhin, *Quantum error correction and large $N$* ([arXiv:2008.12869](https://arxiv.org/abs/2008.12869))

In relation to [[renormalization group flow]]:

* Keiichiro Furuya, Nima Lashkari, Mudassir Moosa, *Renormalization group and approximate error correction* ([arXiv:2112.05099](https://arxiv.org/abs/2112.05099))

In relation to fixed-point [[path integrals]]:

* Andreas Bauer, _Topological error correcting processes from fixed-point path integrals_ ([arXiv:2303.16405](https://arxiv.org/abs/2303.16405))

Musings on possible implications on relations between [[quantum gravity]] and [[quantum information]]:

* {#SimonsFoundationItFromQBit} [[Simons Foundation]], *[It from Qubit: Simons Collaboration on Quantum Fields, Gravity and Information](https://www.simonsfoundation.org/mathematics-physical-sciences/it-from-qubit/)*

* Iulia Georgescu, *Strings and qubits*, Nature Reviews Physics volume 1, page 477 (2019) ([doi:s42254-019-0087-6](https://www.nature.com/articles/s42254-019-0087-6))

* Natalie Wolchover, *[How Space and Time Could Be a Quantum Error-Correcting Code](https://www.quantamagazine.org/how-space-and-time-could-be-a-quantum-error-correcting-code-20190103/)*, Quanta Magazine, Jan. 3 2019

* [[Tom Banks]], *Holographic Space-time and Quantum Information* ([arXiv:2001.08205](https://arxiv.org/abs/2001.08205))

* ChunJun Cao, *From Quantum Codes to Gravity: A Journey of Gravitizing Quantum Mechanics* ([arXiv:2112.00199](https://arxiv.org/abs/2112.00199))

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quantum field > history

* [[field (physics)]]


quantum field theory

> This entry provides some broad pointers. For a detailed introduction see _[[geometry of physics -- perturbative quantum field theory]]_.

+-- {: .rightHandSide}
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#### Quantum systems
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[[!include quantum systems -- contents]]
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#### Algebraic Quantum Field Theory
+--{: .hide}
[[!include AQFT and operator algebra contents]]
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#### Functorial Quantum field theory
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[[!include functorial quantum field theory - contents]]
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#Contents#
* table of contents
{:toc}

## Idea
 {#Idea}

_Quantum field theory_ is the general framework for the description of the fundamental processes in [[physics]] as understood today. These are carried by configurations of [[field (physics)|fields]] under the generalized rules of [[quantum mechanics]], therefore the name. Notably the [[standard model of particle physics]] is a quantum field theory and has been the main motivation for the development of the concept in general.

Historically quantum field theory grew out of attempts to combine [[classical field theory]] in the context of [[special relativity]] with [[quantum mechanics]]. While some aspects of it are understood in exceeding detail, the overall picture of what quantum field theory actually is used to be quite mysterious. There are two main approaches for axiomatizing and formalizing the notion:

* algebraic quantum field theory: [[AQFT]] -- this encodes a quantum field theory as an assignment of [[operator algebras]] "of [[quantum observables]]" to patches of [[spacetime]];

* functorial quantum field theory: [[FQFT]] -- this encodes a quantum field theory as an assignment of _[[spaces of quantum states]]_ to patches of [[codimension]] 1, and of maps between spaces of states -- the time evolution operator -- to [[cobordisms]] between such patches.

Both these approaches aim at capturing the notion of a _full_ [[nonperturbative quantum field theory]]. On the other hand, much activity in physics is concerned "only" with [[perturbative quantum field theory]]. This is a priori to be thought of as just an approximation to a full quantum field theory akin to the approximation of a [[smooth function]] by its [[Taylor series]] around some point. But since [[nonperturbative quantum field theory]] has remained largely out of reach, and since [[perturbative quantum field theory]] is already so rich, the latter is often understood by default when referring to the topic,.

Recent times have seen major progress in understanding these axiomatizations and connecting them to the structures studied in [[physics]] (see the references below), but still there is a large number of interesting phenomena in quantum field theory that physicists handle semi-rigorously and that are waiting for a fully formal understanding.

## Locally covariant perturbative quantum field theory
 {#LocallyCoveriantPerturbativeQuantumFieldTheory}

The formulation of [[quantum field theory]] has many aspects and perspectives. Two almost complementary threads are the following:

**1) [[perturbation theory]] by means of [[formal geometry|formal]] [[Feynman diagram]] expansions of ([[Wick rotation|Wick rotated]]) [[path integrals]]**

   This is the approach predominant in [[phenomenology]].

   It produces the [[observable]] [[numbers]] which are checked to great precision in [[experiments]] starting from the early development of [[quantum electrodynamics]], fully established with the success of [[quantum chromodynamics]] and recently culminating in the [[Higgs field]] physics seen at the [[LHC]] experiment, confirming the [[standard model of particle physics]].

   While many of the mathematical intricacies of this approach have found solutions over the decades, most of these rely on global properties of [[Minkowski spacetime]] such as translation invariance and existence of an invariant [[vacuum]] [[quantum state]], hence on a consistent concept of [[particles]], none of which generalizes robustly to [[quantum field theory on curved spacetimes]] of relevance in [[cosmology]], [[black hole radiation]] or the [[instanton in QCD|instanton vacuum of QCD]].

**2) [[algebraic quantum field theory]] by means of [[local nets]] of [[C*-algebras]] of [[observables]]**

   This is the approach predominant in [[mathematical physics]].

   It produces the structural [[theorems]] of quantum field theory, such as the [[PCT theorem]] and the [[spin-statistics theorem]] and it seamlessly generalizes to [[QFT on curved spacetimes]].

   In its insistence on [[C*-algebras]] its ambition is to describe the full [[non-perturbative quantum field theory|non-perturbative]] quantum field theory. But as a matter of fact not a single relevant example ([[interaction|interacting]] QFT in spacetime dimensions 4 or greater) is known. Indeed, the construction of the motivating example, [[quantization of Yang-Mills theory]], is one of the open "millenium problems".

$\,$

But more recently a synthesis of these two threads has been developed:

**3) [[locally covariant perturbative quantum field theory]] by means of [[local nets]] of [[formal power series]] [[algebras of observables]]**

This rests on the observation ([Il'in-Slavnov 78](#IlinSlavnov78), [Brunetti-Fredenhagen 00](#BrunettiFredenhagen00)) that the formulation of [[renormalization]] in [[causal perturbation theory]] ([Epstein-Glaser 73](#EpsteinGlaser73), [Scharf 95](#Scharf95)) produces a [[local net]] of [[formal power series]] [[algebras of observables]].

This formulation hence allows to construct the usual examples of [[perturbative quantum field theories]] in a rigorous fashion, and then to extend them to [[quantum field theory on curved spacetimes]], such as in formulation of perturbative [[quantum gravity]] ([Brunetti-Fredenhagen-Rejzner 13](#BrunettiFredenhagenRejzner13)).

There remains just one problem:

When applied to [[gauge theory]] on [[quantum field theory on curved spacetime|on curved spacetime]] the usual axioms on a [[local net]] break: either they enforce all [[characteristic classes]] of the [[gauge field]] ("[[instanton sectors]]") to be trivial, or else the axioms encoding locality are broken (see [Schenkel 14](#Schenkel14), [Schreiber 14](https://ncatlab.org/schreiber/show/Higher+field+bundles+for+gauge+fields)).

This remaining problem is solved by passing from plain [[algebras of observables]] to [[homotopical algebra|homotopical algebras]] ([[higher algebra|higher algebras]]), and hence to a formulation of **[[homotopical algebraic quantum field theory]]** (see [Schenkel 17](#Schenkel17)). This is still in the making.

## Higher categories in quantum field theory

(See also [[higher category theory and physics]] and ([SatiSchreiber](#SatiSchreiber))).

Even though _quantum field theory_ has been around for decades and has been very successful both as a phenomenological model in experimental physics as well as a source of deep mathematical structures and theorems, from a mathematical perspective it is still to a large extent mysterious, though recently much progress is being made.

There are essentially two alternative approaches for formalizing quantum field theory and making it accessible to mathematical treatment:

 * _algebraic quantum field theory_: [[AQFT]]

 * _functorial quantum field theory_: [[FQFT]].

(Other structures which are used to define quantum field theories, such as [[vertex operator algebra]]s are now more or less understood to be special cases of these two approaches. See there for details.)

Both [[AQFT]] and [[FQFT]] involve the language of [[category theory]] and  [[higher category theory]]. In fact, a couple of important higher categorical structures were motivated from and first considered in the context of quantum field theory. For instance

* [[John Roberts]] first conceived the idea of $\infty$-categorical [[nonabelian cohomology]] in the context of [[AQFT]].

* the description of 2-dimensional [[CFT]] and 3-dimensional [[TFT]] in terms of [[modular tensor categories]] provides a major application of the theory of [[monoidal categories]];

* the notion of the [[(∞,n)-category of cobordisms]], which is thought to play a role analogous to, and as fundamental as, that of the [[sphere spectrum]] was motivated from [[FQFT]];

* In early 1990s [[A-∞ categories]] first appeared in works of [[Maxim Kontsevich|Kontsevich]] and Fukaya on the categorical description of twisted [[sigma model]]s what is then used in the formulation of [[homological mirror symmetry]].

* the [[cobordism hypothesis]] was formulated by [[John Baez]] and [[Jim Dolan]] in the context of extended [[topological quantum field theory|topological]] [[FQFT]].

* various structures involving [[(∞,1)-operad]]s, such as [[topological chiral homology]] and [[blob homology]] are motivated by, and find their application in, the algebraic description of quantum field theory;

* the description of higher background [[gauge field]]s very much motivated the development and study of [[differential cohomology]], which like all notions of [[cohomology]] is intrinsically about [[(∞,1)-topos]] theory.

There are some indications that such [[higher category theory|higher categorical structures]], such as those appearing in [[groupoidification]], are essential for clarifying some of the mysteries of quantum field theory, such as the [[path integral]]. While this is far from being clarified, this is what motivates research in higher categorical structures in QFT.

Ours is the age to figure this out.

## Examples

* [[free field theory]]

* [$\phi^4$ theory](http://ncatlab.org/nlab/show/phi%5E4+theory)

* [[topological quantum field theory]]

* [[gauge theory]]

  * [[electromagnetism]]: [[QED]],

  * [[Yang-Mills theory]]: [[QCD]]

* [[conformal field theory]]

## Related concepts

* [[relativistic quantum field theory]]

* [[Euclidean quantum field theory]], [[thermal quantum field theory]]

* [[scattering amplitude]]

* [[perturbation theory]]

  * [[Feynman diagram]],

  * [[Feynman rule]], [[worldline formalism]]

  * [[renormalization]]

* [[non-perturbative quantum field theory]]

* [[AQFT]], [[FQFT]],

  * [[local quantum field theory]], [[extended quantum field theory]]

  * [[topological quantum field theory]]

* [[constructive quantum field theory]]

* [[prequantum field theory]]

  * [[Lagrangian]], [[extended Lagrangian]], [[action functional]]

* [[geometry of physics -- perturbative quantum field theory]]

* [[QFT on non-commutative spacetime]]

## References

The early history of the subject:

* [[Rudolf Haag]], _Early papers on quantum field theory (1929-1930)_ ([pdf](https://link.springer.com/content/pdf/bfm%3A978-3-642-70078-1%2F1%2F1.pdf))

* [Scharf 95, section 0.0](#Scharf95)

Introductions:

* _[[geometry of physics -- perturbative quantum field theory]]_

* [[Arnold Neumaier]], chapter B of: _[Theoretical Physics FAQ](http://www.mat.univie.ac.at/~neum/physfaq/physics-faq.html)_

* Gerald B. Folland, _Quantum field theory: A tourist guide for mathematicians_, Math. Surveys and Monographs __149__ ([ZMATH] (http://www.zentralblatt-math.org/zmath/en/advanced/?q=an:1155.81003&format=complete))

* [[Richard Borcherds]], A. Barnard, _Lectures on QFT_, ([arxiv:math-ph/0204014](http://arxiv.org/abs/math-ph/0204014))

* [[David Tong]], *Lectures on Quantum Field Theory*, Cambridge (2006-7) [[pdf](https://www.damtp.cam.ac.uk/user/tong/qft/qft.pdf), [web](https://www.damtp.cam.ac.uk/user/tong/qft.html)]

* [[Matthew Schwartz]], *Quantum Field Theory and the Standard Model*, Cambridge University Press (2014) [[ISBN:9781107034730](https://www.cambridge.org/de/universitypress/subjects/physics/theoretical-physics-and-mathematical-physics/quantum-field-theory-and-standard-model?format=HB&isbn=9781107034730), [doi:10.1017/9781139540940](https://doi.org/10.1017/9781139540940), [pdf](https://indico.ihep.ac.cn/event/11134/attachments/3783/4308/QFT-Schwartz.pdf)]

A standard textbook written from the perspective of [[effective field theory]]:

* {#Weinberg95} [[Steven Weinberg]] *The Quantum Theory of Fields Vol 1: Foundations*, Cambridge University Press (1995) [[doi:10.1017/CBO9781139644167](https://doi.org/10.1017/CBO9781139644167)]

* {#Weinberg96} [[Steven Weinberg]] *The Quantum Theory of Fields Vol 2: Applications*, Cambridge University Press (1996) [[doi:10.1017/CBO9781139644174](https://doi.org/10.1017/CBO9781139644174)]

Application in/to [[solid state physics]]:

* [[Eduardo Fradkin]], *Field Theories of Condensed Matter Physics*, Cambridge University Press (2013) [ISBN: 9781139015509, [doi:10.1017/CBO9781139015509](https://doi.org/10.1017/CBO9781139015509)]

An extensive compilation of material on QFT aiming for mathematical precision is

* [[Eberhard Zeidler]], _Quantum field theory. A bridge between mathematicians and physicists. I: Basics in mathematics and physics._ ([ZMATH entry] (http://www.zentralblatt-math.org/zmath/en/advanced/?q=an:1124.81002&format=complete)), _II: Quantum electrodynamics_ ([ZMATH entry] (http://www.zentralblatt-math.org/zmath/en/advanced/?q=an:1155.81005&format=complete)), _III: Gauge theory_ ([ZMATH entry](http://www.zentralblatt-math.org/zmath/en/advanced/?q=an:1228.81005&format=complete)).

Survey of the mathematically rigorous discussion in [[locally covariant perturbative quantum field theory]] is in

* {#FredenhagenRejzner12} [[Klaus Fredenhagen]], [[Katarzyna Rejzner]], _Perturbative algebraic quantum field theory_, In _Mathematical Aspects of Quantum Field Theories_, Springer 2016 ([arXiv:1208.1428](https://arxiv.org/abs/1208.1428))

* {#FredenhagenRejzner15} [[Klaus Fredenhagen]], [[Katarzyna Rejzner]], _Perturbative Construction of Models of Algebraic Quantum Field Theory_ ([arXiv:1503.07814](https://arxiv.org/abs/1503.07814))

based on the [[causal perturbation theory]] of

* {#EpsteinGlaser73} [[Henri Epstein]] and [[Vladimir Glaser]], _[[The Role of locality in perturbation theory]]_, Annales Poincaré Phys. Theor. A 19 (1973) 211.

* {#IlinSlavnov78} V. A. Il'in and D. S. Slavnov, _Observable algebras in the S-matrix approach_, Theor. Math. Phys. 36 (1978) 32.

* {#BrunettiFredenhagen00} [[Romeo Brunetti]], [[Klaus Fredenhagen]], _Microlocal Analysis and Interacting Quantum Field Theories: Renormalization on Physical Backgrounds_, Commun. Math. Phys. 208 : 623-661,2000 ([math-ph/9903028](https://arxiv.org/abs/math-ph/9903028))

and developed as [[causal perturbation theory]] for [[gauge theory]] such as [[quantum electrodynamics]] in

* {#Scharf95} [[Günter Scharf]], _[[Finite Quantum Electrodynamics -- The Causal Approach]]_, Berlin: Springer-Verlag, 1995, 2nd edition

and for [[quantum chromodynamics]] and perturbative [[quantum gravity]] in

* {#Scharf01} [[Günter Scharf]], _[[Quantum Gauge Theories -- A True Ghost Story]]_, Wiley 2001

Discussion of perturbative [[quantum gravity]] in this perspective is in

* {#BrunettiFredenhagenRejzner13} [[Romeo Brunetti]], [[Klaus Fredenhagen]], [[Katarzyna Rejzner]], _Quantum gravity from the point of view of locally covariant quantum field theory_ ([arXiv:1306.1058](https://arxiv.org/abs/1306.1058))

and survey of the generalization to [[gauge theory]] via [[homotopical algebraic quantum field theory]] is in

* {#Schenkel14} [[Alexander Schenkel]], _On the problem of gauge theories
in locally covariant QFT_, talk at _[Operator and Geometric Analysis on Quantum Theory](http://www.science.unitn.it/~moretti/convegno/convegno.html)_ Trento, 2014 ([[SchenkelTrento2014.pdf:file]]) (with further emphasis on this point in the companion talk [Schreiber 14](field+bundle#Schreiber14))

* {#Schenkel17} [[Alexander Schenkel]], _Towards Homotopical Algebraic Quantum Field Theory_, talk at _[Foundational and Structural Aspects of Gauge Theories](https://indico.mitp.uni-mainz.de/event/76/overview)_,
Mainz Institute for Theoretical Physics, 29 May – 2 June 2017. ([pdf](http://aschenkel.eu/Mainz17.pdf))

A discussion of aspects of QFT with an eye towards applications in [[string theory]] and _aimed_ at mathematicians (though requiring more of a physicist's mindset than many pure mathematicians will find themselves in) is

* [[Pierre Deligne]], P. Etingof, [[Dan Freed]], L. Jeffrey, D. Kazhdan, J. Morgan, D.R. Morrison and [[Edward Witten]], eds.  _[[Quantum Fields and Strings]], A course for mathematicians_, 2 vols. Amer. Math. Soc. Providence 1999. ([web version](http://www.math.ias.edu/qft))

Aspects of [[topology]] and [[differential geometry]]  (e.g. connections to [[index theorems]] and [[moduli spaces]]) are emphasized in

* Charles Nash, _Differential topology and quantum field theory_, Acad. Press 1991.

An indication of the modern foundational picture of quantum mechanics is attempted in

* {#SatiSchreiber} [[Hisham Sati]], [[Urs Schreiber]], _Mathematical foundations of quantum field and perturbative string theory_ Proceedings of Symposia in Pure Mathematics, AMS (web).

See also

* [[Albert Schwarz]], _Quantum field theory and topology_, Grundlehren der Math. Wissen. __307__, Springer 1993.

* [[Graeme Segal]], _[[Three Roles of Quantum Field Theory]]_ , Felix Klein lectures, Bonn (2011)

For further references see _[[FQFT]]_ and _[[AQFT]]_.

See also

* [[Tom Banks]], _Modern quantum field theory, A concise introduction_ ([pdf](http://web.phys.ntnu.no/~mika/banks.pdf))

[[!redirects quantum field theories]]

[[!redirects QFT]]
[[!redirects QFTs]]


Quantum Fields and Strings

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This entry collects linked keywords for the book

* [[Pierre Deligne]], [[Pavel Etingof]], [[Dan Freed]], [[Lisa Jeffrey]], [[David Kazhdan]], [[John Morgan]], [[David R. Morrison]] and [[Edward Witten]], eds.

  _Quantum Fields and Strings, A course for mathematicians_, 2 vols. Amer. Math. Soc. Providence 1999. ([web version](http://www.math.ias.edu/qft))

on [[quantum field theory]] and [[string theory]].

Parts of this appear separately elsewhere, for instance

* [[Pierre Deligne]], [[John Morgan]], _Notes on supersymmetry_ ([[NotesOnNotesOnSupersymmetry.pdf:file]])

* [[Pierre Deligne]], [[Daniel Freed]], _Supersolutions_ ([arXiv:hep-th/9901094](http://arxiv.org/abs/hep-th/9901094)).

  (see also at _[[signs in supergeometry]]_)

on fundamental [[supergeometry]] needed for describing [[fermion]] [[particles]] (and [[superstrings]]). See also

* [[Daniel Freed]], _[[Five lectures on supersymmetry]]_, AMS 1999

While advertized as "A course for mathematicians", experience shows that it is not really suited for pure mathematicians without previous exposition to and tolerance for physics, particularly beyond the first chapters (which show strong ambition to be mathematically precise) towards the following lectures (which are mainly standard lectures of theoretical physicists). But it is much better than the average physics text.

More in detail: this is a long collection of (in parts) long lectures by many top string theorists and also by some genuine top mathematicians. Correspondingly it covers a lot of ground, while still being introductory. Especially towards the beginning there is a strong effort towards trying to formalize or at least systematize much of the standard lore. But one can see that eventually the task of doing that throughout had been overwhelming. Nevertheless, this is probably the best source that there is out there. If you only ever touch a single book on string theory, touch this one.

\linebreak

See also at _[[string theory FAQ]]_

\linebreak

#Contents#
* table of contents
{:toc}

## Volume I

### Part 1: Classical fields and Supersymmetry

* [[classical field theory]]

* [[supersymmetry]]

#### Classical field theory

##### Chapter 1. Classical mechanics

* [[classical mechanics]]

##### Chapter 2. Lagrangian theory of classical fields

* [[Lagrangian]]

* [[variational calculus]]

* [[Euler-Lagrange equations]]

##### Chapter 3. Free field theories

##### Chapter 4. Gauge theory

* [[gauge theory]]

* [[electromagnetism]]

* [[Yang-Mills theory]]

##### Chapter 5. $\sigma$-Models and coupled gauge theories

* [[sigma-model]]

##### Chapter 6. Topological terms

* [[Chern-Simons form]]

* [[Chern-Simons theory]]

* [[WZW model]]

* [[Deligne cohomology]]

##### Chapter 7. Wick rotation

* [[Wick rotation]]

### Part 2: Formal Aspects of QFT

## Volume II

### Part 3: Conformal field theory and strings

#### Lectures on Conformal Field Theory

##### Lecture 1. Simple functional integrals

* [[path integral]]

##### Lecture 2. Axiomatic approaches to conformal field theory

* [[CFT]]

  * [[vertex operator algebra]]

  * [[FQFT]]

##### Lecture 3. $\sigma$-Models

* [[sigma-models]]

##### Lecture 4. Constructive conformal field theory

* [[WZW model]]

#### Perturbative String Theory

##### Lecture 1. Point varticles and strings

* [[particle]]

* [[string]]

##### Lecture 2. Spectrum of the free bosonic string

##### Lecture 3. String amplitudes and moduli space of curves

##### Lecture 4. Fadeev-Popov Ghost -- BRST Quantization

* [[BV-BRST formalism]]

##### Lecture 5. Moduli dependence of determinants and Green functions

##### Lecture 6. Strings on general manifolds

* [[Killing spinor]]

* [[gravity]], [[Kalb-Ramond field]], [[dilaton]]

##### Lecture 7. Free superstrings

* [[type II string theory]]

* [[type I string theory]]

##### Lecture 8. Heterotic strings

* [[heterotic string theory]]

##### Lecture 9. Superstring perturbation theory

* [[perturbation theory]]

##### Lecture 10. Supersymmetry and supergravity

* [[supersymmetry]]

* [[supergravity]]

* [[Green-Schwarz superstring]]

#### Super Space Description of Super Gravity

* [[supergravity]]

#### Notes on 2d Conformal Field Theory and String Theory

##### Chpater 0. Introduction

* [[vertex operator algebra]]

* [[D-module]]

##### Chapter 1. Chiral algebra

* [[chiral algebra]]

* [[conformal block]]

* [[correlation function]]

##### Chapter 2. CFT data

* [[CFT]]

##### Chapter 3. Examples

* [[Kac-Moody algebra]]

* [[dilaton]]

* [[bc-system]]

##### Chapter 4. BRST  and string amplitudes

* [[BRST complex]]

##### Chapter 5. Further constructons

* [[Ran space]]

##### Chapter 6. The free bosonic theory

* [[canonical line bundle]]

#### Kaluza-Klein Compactifications, Supersymmetry, and Calabi Yau Spaces

##### Lecture 1. Compactifications to dimension four

* [[Kaluza-Klein mechanism]]

* [[spontaneous symmetry breaking]]

##### Lecture 2. Supersymmetry and Calabi-Yau manifolds

* [[supersymmetry and Calabi-Yau manifolds]]

### Part 4: Dynamical Aspects of QFT

#### Dynamics of Quantum Field Theory

##### Lecture 1. Symmetry breaking

* [[spontaneous symmetry breaking]]

##### Lecture 2. Gauge symmetry breaking and more infrared behaviour

##### Lecture 3. BRST quantization of gauge theories

* [[gauge theory]]

* [[BRST complex]]

* [[BV-BRST complex]]

##### Lecture 4. Infrared behaviour of the S-matrix of the 2-dimensional $\sigma$-model  with target space $S^{N-1}$

* [[sigma-model]]

* [[S-matrix]]

##### Lecture 5. The large $N$ limit of the $\sigma$-model into Grassmannians

##### Lecture 6. The Bose-Fermi correspondence and its applications

##### Lecture 7. Two-dimensional gauge theory of bosons, the Wilson line operator and confinement

* [[Wilson line]]

* [[confinement]]

##### Lecture 8. Abelian duality

##### Lecture 9. Solitons

* [[soliton]]

##### Lecture 10. Wilson loops, 't Hooft loops and 't Hooft's  picture of confinement

* [[Wilson loop]]

* [[confinement]]

##### Lecture 11. Quantum gauge theories in two dimensions and intersection theory on moduli space

* [[Landau-Ginzburg model]]

##### Lecture 12. Supersymmetric field theories

* [[supersymmetry]]

* [[BPS state]]

##### Lecture 13. $N=2$ SUSY theories in dimension two: part I

* [[N=2 D=4 super Yang-Mills theory]]

* [[topological string]]

##### Lecture 14. $N=2$ SUSY theories in dimension two: part II, Chiral rings and twisted theories

* [[chiral ring]]

##### Lecture 15. The Landau-Ginzburg description of $N = 2$ minimal models; Quantum cohomology and Kähler manifolds

* [[Landau-Ginzburg model]]

* [[quantum cohomology]]

##### Lecture 16. Four-dimensional gauge theories

* [[N=1 D=4 super Yang-Mills theory]]

##### Lecture 17. $N=2$ supersymmetric Yang-Mills theories in dimension four: part 1

* [[N=2 D=4 super Yang-Mills theory]]

##### Lecture 18. $N=2$ supersymmetric Yang-Mills theories in dimension four: part 2

##### Lecture 19. $N=2$ supersymmetric Yang-Mills theories in dimension four: part 3, Topological applications

* [[topologically twisted D=4 super Yang-Mills theory]]

#### Dynamics of $N = 1$ Supersymmetric Field Theories in Four Dimensions

* [[N=1 D=4 super Yang-Mills theory]]

category: reference


quantum flag manifold

__Quantum flag manifolds__ (or varieties) are certain [[quantum homogeneous space]]s of duals of Drinfel'd-Jimbo [[quantum group]]s or their relatives, viewed as noncommutative quotients with respect to (co)action of the corresponding quantum Borel subgroup.

Usual [[flag variety|flag varieties]] are projective varieties, hence they have a (commutative) graded coordinate ring. Thus most of the approaches to quantum flag varieties, define the quantum flag variety via its noncommutative graded algebra of functions (cf. the book by Joseph).

* [[Yan Soibelman| Yan S. Soĭbelʹman]], _О квантовом многообразии флагов_, Функц. анализ и его прил. __26__ (1992), no. 3, 90--92, [pdf](http://www.mathnet.ru/php/getFT.phtml?jrnid=faa&paperid=809&volume=26&year=1992&issue=3&fpage=90&what=fullt&option_lang=eng) (Russian);  translation _On the quantum flag manifold_, Funct. Anal. Appl. __26__ (1992), no. 3, 225--227 [doi](http://dx.doi.org/10.1007/BF01075642)

* V. A. Lunts, A. L. Rosenberg, _Localization for quantum groups_, Selecta Math. (N.S.) __5__ (1999), no. 1, pp. 123--159 ([doi](http://dx.doi.org/10.1007/s000290050044)).

* [[Zoran Škoda]], _Localizations for construction of quantum coset spaces_, in "Noncommutative geometry and Quantum groups", W.Pusz,  P.M. Hajac, eds. Banach Center Publications __61__, pp. 265--298, Warszawa 2003, [math.QA/0301090](http://arxiv.org/abs/math.QA/0301090).

* [[Ulrich Krähmer]], _Dirac operators on quantum flag manifolds_, Lett. Math. Phys. 67/1 (2004) 49-59, [MR2005b:58009](http://www.ams.org/mathscinet-getitem?mr=2005b:58009), [doi](http://dx.doi.org/10.1023/B:MATH.0000027748.64886.23), [math.QA/0305071](http://arxiv.org/abs/math/0305071)

* A. Joseph, Quantum groups and their primitive ideals, Springer 1995.
* E. Taft, J. Towber, Quantum deformation of flag schemes and Grassmann schemes q-deformation of the shape-algebra for GL(n), J. Algebra __142__ (1991), 1-36, []()

* Quantum multilinear algebra, Mitsuyasu Hashimoto and Takahiro Hayashi
Source: Tohoku Math. J. (2) Volume 44, Number 4 (1992), 471-521, [euclid](http://projecteuclid.org/euclid.tmj/1178227246)

[[!redirects quantum flag manifold]]
[[!redirects quantum flag manifolds]]
[[!redirects quantum flag variety]]
[[!redirects quantum flag varieties]]


quantum fluctuation

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#Contents#
* table of contents
{:toc}

## Idea

The [[theory (physics)|theory]] of [[quantum physics]] ([[quantum mechanics]], [[quantum field theory]]) is at its heart [[probability theory|probabilistic]] (see at _[[hidden variable theory]]_). Any [[quantum observable]] in a given [[quantum state]] has a [[probability distribution]] with some finite width around its mean value. A [[measurement]] will find the observable with given [[probability]] in any one of its possible values, and hence it seems as if it "fluctuates". This intrinsic randomness in [[quantum physics]] is referred to as _quantum fluctuation_. (See also at _[[measurement problem]]_.)

In particular the above holds for the [[vacuum]] state of any [[quantum system]]. Hence even when a quantum system is a little "excited" as possible, it still quantum fluctuation. These are therefore also called _vacuum fluctuations_.

## Related concepts

[[!include states and observables -- content]]

## References

See also:

* Wikipedia, _[Quantum fluctuation](https://en.wikipedia.org/wiki/Quantum_fluctuation)_

[[!redirects quantum fluctuations]]

[[!redirects vacuum fluctuation]]
[[!redirects vacuum fluctuations]]


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This page collects material related to the textbook

* [[Günter Scharf]]

  \linebreak

  **Quantum Gauge Theories -- A True Ghost Story**

  \linebreak

  Wiley 2001

which applies [[causal perturbation theory]] to rigorously construct quantum [[Yang-Mills theory]] ([[quantum chromodynamics]] and [[electroweak field|electroweak theory]]) and [[quantum gravity]] as [[perturbative quantum field theories]].

In later editions the title has been changed. The most recent, third edition is

* _Gauge Field Theories: Spin One and Spin Two: 100 Years After General Relativity_, [Dover](http://store.doverpublications.com/0486805247.html), 2016,

with extra chapter 6: _Non-geometric general relativity_, on [[inhomogeneous cosmology]] that is based (it seems) on the article

* {#Scharf13} [[Günter Scharf]], _Inhomogeneous cosmology in the cosmic rest frame without dark stuff_ ([arXiv:1312.2695](https://arxiv.org/abs/1312.2695))

The analogous discussion of [[quantum electrodynamics]] via [[causal perturbation theory]] is in

* [[Günter Scharf]],  _[[Finite Quantum Electrodynamics -- The Causal Approach]]_,  Springer 1995

## Related textbooks and lecture notes

* {#Rejzner16} [[Katarzyna Rejzner]], _[[Perturbative Algebraic Quantum Field Theory]]_, Mathematical Physics Studies, Springer 2016 ([web](https://link.springer.com/book/10.1007%2F978-3-319-25901-7))

* _[[geometry of physics -- perturbative quantum field theory]]_

* [[Michael Dütsch]], _[[From classical field theory to perturbative quantum field theory]]_, 2018

category: reference