quantum Gauss decomposition

Generators of the quantized function algebra $G$ of FRT-type, for example the quantum linear groups $GL_q(n,k)$, $SL_q(n,k)$ are forming a matrix $T$ of generators with entries in $G$ and satisfying the matrix identity for the coproduct $\Delta T = T\otimes T$; in fact, $G$ is a [[matrix Hopf algebra]] with basis $T$. Using quasideterminants, one can decompose the matrix $T$ as $wUA$ where $w$ is a permutation matrix, $U$ upper triangualr unidiagonal and $A$ lower triangular; for this one needs to enlarge $G$ to include the inverses which are needed to find the solution. This can be done for example in the quotient skewfield of $G$, but this is suboptimal. Nevertheless in some cases (and some super-analogues) the simple formulas for this decomposition in terms of quantum minors have been studied by Tolstoy, Kulish, Dobrev and others and many times rediscovered by many people and sometimes called quantum Gauss decomposition.

However, as in the classical case, more true meaning can be given to the decomposition $T = wUA$. The following should be true for most [[matrix Hopf algebras]] (except that the Ore localizations should be replaced by Cohn localizations for big examples) and is proved for quantum linear groups. Instead working in the quotient field one can define $n!$ Ore sets $S_w$ generated by certain "flag" quantum minors, so that the equation $T=wUA$ has a solution in the localized algebra $G_w :=S_w^{-1}G$. One defines the quotient Hopf algebra $B$ from $G$ by letting the Hopf ideal generated by the entries of $T$ above diagonal to zero; let the projection be $p:G\to B$. This projection induces a right $B$-coaction $(id\otimes p)\circ\Delta_G:G\to G\otimes B$ on $G$ ("quantum subgroup acts on a quantum group") which has a compatibility property that it uniquely extends to $G_w$ as an algebra map (this geometrically means that $Spec G_w$ is a $Spec B$-invariant subset). The coinvariants of this extended coaction are so-called [[localized coinvariant]]s and they provide a patch on the noncommutative coset space; namely the category of modules over the algebra of localized coinvariants is a localization of the category of quasicoherent sheaves on the quantum coset space; the algebra of localized coinvariants is the smallest invariant subalgebra of $G_w$ containing the entries of matrix $U$ from $g=wUA$, here the invariance is with respect to the natural action of $B$ provided in a natural way using the Gauss decomposition as sketched below. The quantum Gauss decomposition provides the local trivialization of the quantum fiber bundle which could be symbolically denoted $Spec(G)\to Spec(G)/Spec(B)$.  Namely, $G_w$ is isomorphic as a right $B$-comodule algebra to a [[smash product algebra]] $(G_w)^{co H}\sharp B$ with the canonical $B$-coaction, where the isomorphism is induced by the unique homomorphism of algebras (it is nontrivial that such exists!) $\gamma_w:B\hookrightarrow G_w$ such that $b^i_j$ which is the projection of the $(i,j)$-entry of $T$ in $B$ maps to the $(i,j)$-th entry $a^i_j$ of the matrix $A=A_w$ defined by the decomposition $T=wUA$; the left action of $B$ on $(G_w)^{co B}$ is given by $b\triangleright u = \sum\gamma(b_{(1)})u\gamma(S_B b_{(2)})$ where $S_B:B\to B^{cop}_{op}$ is the antipode of $B$. Thus the true meaning of quantum Gauss decomposition is the local trivialization (trivial bundle=smash product) of quantum principal bundle realized by a right $B$-comodule algebra $G$. Locality is in the sense of the cover by coaction-compatible local trivializations. This result is sketched in

* Zoran Škoda, _Localizations for construction of quantum coset spaces_, [math.QA/0301090](http://front.math.ucdavis.edu/math.QA/0301090), in "Noncommutative geometry and Quantum groups", W.Pusz,  P.M. Hajac, eds. Banach Center Publications vol.61, pp. 265--298, Warszawa 2003.

and the Ore condition for not only $S_w$, but any (multiplicative set) of quantum minors is proved in

* Zoran Škoda, _Every quantum minor generates an Ore set_, International Math. Res. Notices 2008, rnn063-8; [math.QA/0604610](http://arxiv.org/abs/math.QA/0604610).

In the case of $SL_q(2)$ the local trivialization has been applied to compute the resolution of unity formula for the $SU_q(2)$-coherent states in

* Zoran Škoda, _Coherent states for Hopf algebras_,  Letters in Mathematical Physics 81, N.1, pp. 1-17, July 2007. (earlier arXiv version: [math.QA/0303357](http://arxiv.org/abs/math.QA/0303357) )

The general picture of actions, coset spaces, compatibility with localizations, localized coinvariants and equivariance in noncommutative algebraic geometry is outlined in

* Zoran Škoda, _Some equivariant constructions in noncommutative algebraic geometry_, Georgian Mathematical Journal 16 (2009), No. 1, 183--202, [arXiv:0811.4770](http://arxiv.org/abs/0811.4770)

The analogous results for the case of multiparametric $GL_{P,Q}(n)$ can be reduced to the one-parametric case using the twisting due Artin and Tate, and reinterpreted for a usage in quantum minor calculations in

* Zoran Škoda, _A simple algorithm for extending the identities for quantum minors to the multiparametric case_ [arXiv;0801.4965](http://arxiv.org/abs/0801.4965).

See also [[Gauss decomposition]].


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## Idea

The _quantum geometric Langlands correspondence_ is a conjectured equivalence between the [[derived category]] of certain twisted [[D-module]]s on the [[moduli stack]] of $G$-[[principal bundle]]s over some complex curve for some [[reductive group]] $G$, and that of $D$-modules with a dual twist on the stack of ${}^L G$-bundles, for ${}^L G$ the [[Langlands dual group]].

In a certain limit of the twisting parameter the $D$-modules on one side of this correspondence become equivalent to just $\mathcal{O}$-modules, but on the moduli space of [[local system]]s. This limit of the quantum correspondence reproduces the statement of the (ordinary) [[geometric Langlands correspondence]].

Slightly more precisely, writing $Bun_G^\circ$ for the connected component of the moduli stack of $G$-bundles, there is a [[line bundle]] denoted $\mathcal{L}^{\otimes k}$ on $Bun_G^\circ$ and dually a line bundle $\hat \mathcal{L}^{\otimes k}$ on $Bun_{{}^L G}^\circ$ and these serve to defined sheaves $\mathcal{D}^{(k,\lambda)}$ of twisted differential operators on $Bun_{G}^\circ$ and dually sheaves $\hat \mathcal{G}^{(k,\lambda)}$ of twisted differential operators on $Bun_{{}^L G}^\circ$, where the parameters lie in

$$
  (k, \lambda) \in \mathbb{CP}^1 \times \mathbb{C}
  \,.
$$

Writing then $\mathcal{D}^{k,\lambda} Mod(Bun_{G})$ and $\hat \mathcal{D}^{k,\lambda} Mod(Bun_{{}^L G})$, respectively, for the derived categories of modules over these sheaves, the conjectured statement is:

**Quantum geometric Langlands correspondence**

There is an equivalence

$$
  \mathcal{D}^{k,\lambda} Mod(Bun_{G})
  \simeq
  \hat \mathcal{D}^{\frac{1}{k},\lambda} Mod(Bun_{{}^L G})
  \,.
$$

(In fact $k$ and $\frac{1}{k}$ here should further be shifted by the [[dual Coxeter number]] of $G$ and ${}^L G$, respectively.)

## Embedding in string theory

Where the plain [[geometric Langlands correspondence]] is meant to be a shadow of the [[6d (2,0)-superconformal field theory]], the quantum version is meant to correspondingly relate to 6d [[little string theory]] ([Aganagic-Frenkel-Okounkov 17](#AganagicFrenkelOkounkov17)) (which turns into the 6d QFT in the limit that the strings shrink to points).

## Examples

### Limiting cases

The twisted sheaves of differential operators $\mathcal{D}^{k,\lambda}$ have the following limits:

* For $\lambda \neq 0$ and $k \neq \infty$ this is the sheaf of differential operators acting on $\mathcal{L}^{\otimes k}$;

* for $\lambda \neq 0$ and $k = \infty$ this is the pushforward $p_*(\mathcal{O}_{Loc_G})$ of the sheaf of $\mathcal{O}$-modules along the canonical $P : Loc_G \to Bun_G$ that sends a local system to its underlying bundle;

* for $\lambda = 0$ and $k$ arbitrary this is $p_*(\mathcal{O}_{T^* Bun_G^\circ})$.

### Abelian case

In the case where $G$ is abelian, the quantum correspondence is given by a [[Fourier-Mukai transform]] and has been constructed in ([Polishuk-Rothenstein](#PolishukRothenstein))

## Related concepts

* [[S-duality]]

  * [[electro-magnetic duality]]

  * [[geometric Langlands correspondence]]

  * **quantum geometric Langlands correspondence**

## References

The statement of the quantum geometric Langlands correspondence is surveyed on [page 70-71](http://arxiv.org/PS_cache/hep-th/pdf/0512/0512172v1.pdf#page=71) of

* [[Edward Frenkel]], _Lectures on the Langlands Program and Conformal Field Theory_ ([arXiv:hep-th/0512172](http://arxiv.org/abs/hep-th/0512172)).

with more details in:

* [[Dennis Gaitsgory]], *Quantum Langlands Correspondence* [[arXiv:1601.05279](https://arxiv.org/abs/1601.05279)]

The construction of the correspondence in the abelian case, where it is given by a [[Fourier-Mukai transform]], is given in

* {#PolishukRothenstein} A. Polishchuk and M. Rothstein, _Fourier transform for D-algebras_ , DukeMath. J. 109 (2001) 123–146.

An interpretation of the quantum Langlands correspondence in terms of the [[B-model]] [[topological string]] is given in

* [[Anton Kapustin]], _A Note on Quantum Geometric Langlands Duality, Gauge Theory, and Quantization of the Moduli Space of Flat Connections_ ([arXiv:0811.3264](http://arxiv.org/abs/0811.3264))

A general quantum geometric Langlands correspondence is produced in

* {#AganagicFrenkelOkounkov17} [[Mina Aganagic]], [[Edward Frenkel]], [[Andrei Okounkov]], Trans. Moscow Math. Soc. (2018) 1-83 , _Quantum $q$-Langlands Correspondence_ [[arXiv:1701.03146](https://arxiv.org/abs/1701.03146), [doi:10.1090/mosc/278]( https://doi.org/10.1090/mosc/278) ]

see also

* [[David Ben-Zvi]], [[Adrien Brochier]], David Jordan, *Quantum character varieties and braided module categories*,  Sel. Math. New Ser. **24** (2018) 4711–4748 [[doi:10.1007/s00029-018-0426-y](https://doi.org/10.1007/s00029-018-0426-y), [arXiv:1606.04769](https://arxiv.org/abs/1606.04769)]

Relation to [[semi-topological 4d Chern-Simons theory]]:

* [[Meer Ashwinkumar]], [[Meng-Chwan Tan]], _Unifying Lattice Models, Links and Quantum Geometric Langlands via Branes in String Theory_ ([arXiv:1910.01134](https://arxiv.org/abs/1910.01134))

See also

* Peter Koroteev, Daniel S. Sage, [[Anton Zeitlin]], _$(SL(N),q$)-opers, the $q$-Langlands correspondence, and quantum/classical duality_ ([arXiv:1811.09937](https://arxiv.org/abs/1811.09937))

* [[Davide Gaiotto]], [[Jörg Teschner]], *Quantum Analytic Langlands Correspondence* [[arXiv:2402.00494](https://arxiv.org/abs/2402.00494)]

[[!redirects quantum geometric Langlands duality]]


quantum gravity

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* table of contents
{:toc}

## Idea

A theory of _quantum gravity_ (QG) is supposed to be a [[quantum field theory]] (QFT) -- or something similar -- that corresponds to the [[classical field theory]] of [[gravity]], possibly by [[quantization]].

The key technical problem with quantum gravity as a [[non-perturbative quantum field theory]] is that all known constructions of QFTs quantize [[field (physics)|fields]] defined _on_ a given [[spacetime]] [[background field|background]], while, in contrast, the field of [[gravity]] _is_ that spacetime.

Since even for field theories that are defined on space-time backgrounds, such as [[Yang-Mills theory]], [[non-perturbative quantum field theory|non-perturbative]] quantization is a big open problem away from toy examples (for Yang-Mills theory it has been dubbed one of the _Millennium Problems_, see at _[[quantization of Yang-Mills theory]]_) this means that non-perturbative quantum gravity is an even wider open problem.

However, the [[Einstein-Hilbert action]] that defines gravity may be expanded around any of its solutions to the [[Einstein equations]] and the resulting [[perturbation]] of the field of gravity around this background behaves, to low order, like an ordinary [[field (physics)|field]] on a fixed spacetime background. This allows to treat _perturbative quantum gravity_ (pQG) as a [[perturbative quantum field theory]], see the references [below](#ReferencesAsAnEffectiveFieldTheory).

While the [[Einstein-Hilbert action]] (or any of its variants) that defines the dynamics of gravity makes perfect sense at low energies as the definition of an [[effective quantum field theory]] (see there for details), it is not expected to be a non-effective theory at arbitrary high energies, because it is [[renormalizable interaction|not renormalizable]]: all [[non-renormalizable QFTs]] that have appeared in practice turned out to be [[effective QFT]] approximations (see there) to renormalizable [[UV-completions]].

The idea has been explored that gravity is simply to be regarded as a [[classical field theory]], but trying to couple such a classical field theory to a [[stress-energy tensor]] of a matter [[quantum field theory]] leads to conceptual problems that break the very foundation of what is known about theoretical physics. The technical problem is this: the [[equations of motion]] of classical Einstein [[gravity]] ([[Einstein's equations]]) assert that the [[Einstein tensor]] of the [[field (physics)]] of [[gravity]] equals the [[energy-momentum tensor]] of all other [[force]] and [[matter]] [[field (physics)|fields]]. Here into the [[energy-momentum tensor]] enters the expression for [[energy]] and [[momentum]] of classical matter fields. However, according to the [[standard model of particle physics]], these matter fields are not described by [[classical field theory]]. This then is the technical problem: there is no consistent way in sight to modify [[Einstein's equations]] so as to have a quantum EM tensor in it. The simple idea of taking the [[expectation value]] of the quantum matter EM-tensor leads to all kinds of theoretical problems.

Thus, this idea seems to be very unlikely. At the same time, it is clear that present experiments cannot _directly_ probe the effects of that supposed [[UV-completion]] of Einstein-gravity.

Approaches to a full quantization of gravity therefore roughly fall into two different strategies

1. One assumes that the Einstein-Hilbert action is indeed the [[effective QFT]] that approximates a "[[UV-completion]]", a more fundamental theory valid at all energies. This is the approach taken for instance in [[string theory]].

1. One assumes that by some other fact that has been overlooked, one can make sense of a non-perturbative quantization of the EH action at arbitrary energies after all. This is for instance the case in speculations that EH-gravity has a [[UV fixed point]].

Accordingly, there are various proposals for what quantum gravity might be, but a comprehensive theory is missing to date. This is in stark contrast to the situation for the other fundamental [[force]]s of the [[standard model of particle physics]]: these are [[Yang-Mills field]]s, whose nature as QFTs is, while subtle, well understood. Indeed, the theoretical and experimental sucess of [[Yang-Mills theory]] is the main reason for the belief that QFT is the right framework for fundamental physics, and that also the field of gravity must have a description in this form.

Indeed, a model of fundamental physics where Yang-Mills fields are quantized but gravity is not is incomplete and arguably even inconsistent. Central motivating examples for this come from [[black hole]]-physics:

1. The study of quantum fields on the classical gravitational background of a [[black hole]] [[spacetime]] suggest that in a full theory of quantum gravity black holes could be shown to emit radiation and to eventually even decay in some way or other. This expected decay process cannot be described without a theory of quantum gravity.

1. Related to black hole radiation is a whole series of observations in classical [[general relativity]] that seem to indicate that black holes need to be thought of as systems possessing physical microstates with an [[entropy]] that cannot be accounted for by classical general relativity. This has led to the expectation that these microstates are states in quantum theory of gravity and that their understanding is necessary for understanding the physics of black holes.

1. More generally, appearances of unphysical singularities in [[general relativity]], such as in [[black hole]]s or [[cosmology|cosmological]] singularities such as the [[big bang]], have been argued to be artefacts of a non-quantum description of gravity. Various toy models of quantum gravity have been proposed where notably the big-bang singularity is just a classical approximation to a non-singular and well-defined quantum dynamics.

An evident definition of quantum gravity would seem to simply be the [[quantization]] of the [[Einstein-Hilbert action]] by the rules [[perturbation theory]] and [[renormalization]]. This straightforward approach however fails, as it can be demonstrated that this [[action functional]] is not [[renormalization|renormalizable]].

More recently it has been argued that there is evidence that the EH-action has a nontrivial [[UV-fixed point]] after all. If true this would mean that a quantization of gravity in standard QFT is possible.

Another possibility is that this is only an apparent problem, and that standard gravity is nothing but an [[effective quantum field theory]]. (See there for references on gravity).

But most approaches to the problem have taken the non-renormalizability of classical gravity to be an indication that the standard concepts of QFT might need refinement in order to include gravity.

A common idea is that the assumption that the space of configurations in gravity is really that of smooth [[spacetime]]s, i.e. of [[smooth manifold]]s, is wrong. A plethora of variants of the notion of smooth manifold have been proposed as ingredients for a theory of quantum gravity. Notably concepts in [[noncommutative geometry]]. For instance [[Alain Connes]]'s [[spectral action]] generalizes the [[Einstein-Hilbert action]] functional from [[Riemannian manifold]]s to [[spectral triple]]s. However, none of the modifications of the spaces of configuratations of gravity proposed so far have been shown to support action functionals that can actually be quantized, in one sense or another.

Accordingly there are suggestions to modify instead the principles of perturbative quantum field theory. In [[string theory]] the fundamental assumption is that the Feynman perturbation series over 1-dimensional graphs of traditional perturbative QFT is to be refined by a similar series over 2-dimensional surfaces. It has been shown that this _string perturbation series_ needs no renormalization and that it does describe Yang-Mills fields and a gravitational field. This led to huge hopes and large activity in the study of string theory as a theory of quantum gravity. But various central questions remain open and the state of the theory remains somewhat inconclusive.

While perturbative string theory did demonstrate that there is a variant of the perturbation series for gravity which is renormalized, it can, by definition of perturbation theory, be just an approximation to a more fundamental non-perturbatively defined theory. Various candidates for the non-perturbative theory to which the string perturbation series might be an approximation have been proposed. One that enjoyed a large popularity in the 1990s was called _matrix mechanics_ but the high hopes connected with it were supported mostly by toy example computations. More recently the [[AdS/CFT correspondence]] is widely thought of as providing a non-perturbative definition of string theory and of quantum gravity. This correspondence asserts that observables of a string theory on an asymptotically [[anti-de Sitter spacetime]] may be matched equivalently to certain observables in an auxiliary super [[Yang-Mills theory]]. If true, this would allow to take the well understood non-perturbative definition of Yang-Mills theory and deduce from it a theory of quantum gravity.

Still another suggestion has been that ordinary quantum field theory might be seen to apply to the ordinary configuration space and action functional of general relativity, if only a suitably adapted parameterization of the configuration space is chosen. Notably one observes that [[Riemannian manifold]]s may be encoded in terms of [[connection on a bundle|connections on the tangent bundle]] -- the [[Levi-Civita connection]] -- this being manifestly analogous to the nature of the fields in [[Yang-Mills theory]]. (See [[first-order formulation of gravity]]).

It has been suggested that therefore quantum gravity may have a description in terms of the [[holonomy]]-observables of these connections,  their [[parallel transport]] around loops. Research in this direction has therefore become known as _loop quantum gravity_ . However, little progress has been made in understanding the configuration space of smooth connections in terms of loop observables. In most of the LQG-literature instead it is assumed from the outset that it makes sense to pass to _generalized connections_ : while a smooth connection is the same as a smooth [[parallel transport]]-[[functor]] on the [[path groupoid]] $\mathbf{P}_1(X) \to \mathbf{B}G$, the "generalized connections" in this context are taken to be non-smooth and non-continuous such functors on sub-categories of paths. It is is not clear how this configuration space relates to that of ordinary gravity.

A plethora of further suggestions has developed out of this, which in the literature often still go under the name "LQG", even though they may abandon the description of configuration spaces of metrics in terms of loop observables. There is for instance the approach of _spin foams_ or _[[causal dynamical triangulations]]_, and that of _[[causets]]_ , to name just a few.

The problem with discriminating between all these proposals is the combination of two problems.

1. There exist insufficient theoretical tools for analyzing in detail the consequences of the various proposals.

1. There are few to no experiments available that would be able to observe the phenomena that a theory of quantum gravity is supposed to describe.

As a result, the field of fundamental physics today finds itself in a somewhat awkward position where the need for theoretical progress concerning quantum gravity for conceptual reasons is widely appreciated, but where disagreements about the viability of various proposals finds no conclusive resolution and divides the community.

## Effects

### First quantum correction to Newtonian $1/r$-potential
 {#FirstQuantumCorrectionToNewtonianPotential}

The first order quantum correction to the classical gravitational potential

$$
  V_{cl}(r) \;=\; - \frac{G m_1 m_2}{r}
$$

between two bodies of [[mass]] $m_1$, $m_2$, respectively, may be computed unambiguously in [[perturbative quantum field theory|perturbative]] quantum gravity ([Donoghue 95, Section 9](#Donoghue95)):

The result is a term of the form

$$
  V_{qu}(r)
  \;=\;
  V_{cl}(r)
  \left(
    1
    - \frac{135 + 2 N_\nu}{ 30 \pi^2}
    \frac{G \hbar}{r^2 c^2}
    +
    \text{higher order}
  \right)
$$

(where $N_\nu$ is the [[natural number|number]] of [[mass|massless]] [[neutrino]]-species, $G$ is the [[gravitational constant]], $c$ is the [[speed of light]] and $\hbar$ is [[Planck's constant]]).

([Donoghue 95, (65)](#Donoghue95)

Here for $r = 1$ [[fm]] we have that the quantum correction is by a tiny factor of order

$$
  \frac{G \hbar}{r^2 c^2}\vert_{r = 1 fm}
  \;\sim\;
  10^{-38}
  \,.
$$

### Vacuum energy and Cosmological constant

The [[renormalization]] freedom in [[perturbative QFT|perturbative]] [[quantization]] of gravity induces freedom in the choice of [[vacuum expectation value]] of the [[stress-energy tensor]] and hence in the [[cosmological constant]].

For details see [there](cosmological+constant#InPerturbativeQuantumGravity).

### Anomalous magnetic moment of the leptons
 {#CorrectionToAnomalousMagneticMomentOfTheLeptons}

The corrections of [[loop order|1-loop]] [[perturbative QFT|perturbative]] quantum gravity to the [[anomalous magnetic moment]] of the [[electron]] and the [[muon]] have been computed in ([Berends-Gastman 75](#BerendsGastman75)) and found to be finite, without need for [[renormalization]]. These [[Feynman diagrams]] contribute:

<img src="https://ncatlab.org/nlab/files/GravityVertexCorrectionsForQED.png" width="750">

## Special cases

* [[2d quantum gravity]]

* [[3d quantum gravity]]

## Related concepts

* [[soft graviton theorem]]:

* [[asymptotic safety]]

* [[weak gravity conjecture]]

* [[Witten conjecture]]

* [[piecewise flat spacetime]]

## References

### General

Early mentioning of potential breakdown of classical geometry at microscopic scales:

* [[Bernhard Riemann]], *[[Über die Hypothesen, welche der Geometrie zu Grunde liegen]]*, Göttingen (1845) [[doi:10.1007/978-3-642-35121-1](https://doi.org/10.1007/978-3-642-35121-1)]

  Engl. transl: [[William Clifford]]: *[[On the hypotheses which underlie geometry]]*, Nature **VIII** (1873) 183-184 [[doi:10.1007/978-3-319-26042-6](https://doi.org/10.1007/978-3-319-26042-6)]

  > [[§III.3](https://ncatlab.org/nlab/files/Riemann-OnTheHypotheses.pdf#page=11)] *Now it seems that the empirical notions on which the metric determinations of Space are based, the concept of a solid body and a light ray, lose their validity in the infinitely small; it is therefore quite definitely conceivable that the metric relations of Space in the infinitely small do not conform to the hypotheses of geometry; and in fact, one ought to assume this as soon as it permits a simpler way of explaining phenomena.*

Original discussion of the [[Wheeler-DeWitt equation]]:

* {#DeWitt67} [[Bryce DeWitt]], (5.5) in:  *Quantum Theory of Gravity. I. The Canonical Theory*, Phys. Rev. **160** (1967) 1113 [[doi:10.1103/PhysRev.160.1113](https://doi.org/10.1103/PhysRev.160.1113)]

For further early discussion of the subject see pointers in:

* [[Cécile DeWitt-Morette]], *The Pursuit of Quantum Gravity -- Memoirs of Bryce DeWitt from 1946 to 2004*, Springer (2011) [[doi:10.1007/978-3-642-14270-3_4](https://doi.org/10.1007/978-3-642-14270-3_4)]

Some of the standard lore about necessity of and the problems of quantum gravity are discussed in reviews such as

* [[Steve Carlip]], _Quantum Gravity: a Progress Report_ ([arXiv:gr-qc/0108040](http://arxiv.org/abs/gr-qc/0108040))

* [[Richard Woodard]], _How Far Are We from the Quantum Theory of Gravity?_ ([arXiv:0907.4238](http://arxiv.org/abs/0907.4238))

* [[Hermann Nicolai]], _Quantum Gravity: the view from particle physics_ ([arXiv:1301.5481](http://arxiv.org/abs/1301.5481))

* [[Steven Carlip]], Dah-Wei Chiou, Wei-Tou Ni, Richard Woodard, _Quantum Gravity: A Brief History of Ideas and Some Prospects_, Int. J. Mod. Phys. D  ([arXiv:1507.08194](http://arxiv.org/abs/1507.08194))

Exposition from the point of view of [[worldline formalism]] is in

* {#Witten11} [[Edward Witten]], _Quantum Gravity_, Solomon Lefschetz Memorial Lecture Series, November 2011 ([video](https://www.youtube.com/watch?v=uRdCJMYc2Ds#t=15))

Textbook account:

* Claus Kiefer, _Quantum Gravity_, Oxford University Press 2007 ([doi:10.1093/acprof:oso/9780199585205.001.0001](https://www.oxfordscholarship.com/view/10.1093/acprof:oso/9780199585205.001.0001/acprof-9780199585205))

### As a perturbative quantum field theory
 {#ReferencesAsAnEffectiveFieldTheory}

As a [[perturbative quantum field theory]], quantum gravity exists (usually thought of as an [[effective quantum field theory]]):

* {#Feynman63} [[Richard Feynman]], _Quantum theory of gravitation_, Acta Phys. Polon. **24** (1963) 697-722 [[spire:9148](http://inspirehep.net/record/9148/)]

* [[Martinus Veltman]], _Quantum theory of Gravitation_, In Les Houches 1975, Proceedings, Methods In Field Theory, (Amsterdam 1976) 265-327 [[[VeltmanQuantumnGravity76.pdf:file]], [spire](http://inspirehep.net/record/105735?ln=en)]

* [[John Donoghue]], *General relativity as an effective field theory: The leading quantum corrections*, Phys. Rev. D **50** (1994) 3874-3888 [[arXiv:gr-qc/9405057](https://arxiv.org/abs/gr-qc/9405057)]

* Vlad-Mihai Mandric, Tim R. Morris, Dalius Stulga, *Off-shell divergences in quantum gravity* [[arXiv:2308.07382](https://arxiv.org/abs/2308.07382)]

Introduction, review and outlook:

* {#Donoghue95} [[John Donoghue]], *Introduction to the Effective Field Theory Description of Gravity* [[arXiv:gr-qc/9512024](http://arxiv.org/abs/gr-qc/9512024), [spire](http://inspirehep.net/record/403510)]

* [[Zvi Bern]], _Perturbative Quantum Gravity and its Relation to Gauge Theory_, Living Rev. Relativ. **5** 1 (2002) 5 [[arXiv:gr-qc/0206071](https://arxiv.org/abs/gr-qc/0206071), [doi:10.12942/lrr-2002-5](https://dx.doi.org/10.12942%2Flrr-2002-5)]

* [[Cliff Burgess]], *Quantum Gravity in Everyday Life: General Relativity as an Effective Field Theory*, Living Rev. Rel. **7** 5 (2004) [[arXiv:gr-qc/0311082](https://arxiv.org/abs/gr-qc/0311082), [doi:10.12942/lrr-2004-5](https://doi.org/10.12942/lrr-2004-5)]

* [[Richard Woodard]], _Perturbative Quantum Gravity Comes of Age_,  International Journal of Modern Physics DVol. 23, No. 09, 1430020 (2014) ([arXiv:1407.4748](https://arxiv.org/abs/1407.4748))

* [[Ilya L. Shapiro]], *The background information about perturbative quantum gravity*, in: *[[Handbook of Quantum Gravity]]*, Springer (2023) [[arXiv:2210.12319](https://arxiv.org/abs/2210.12319)]

* [[John F. Donoghue]], *Quantum General Relativity and Effective Field Theory*, in: *[[Handbook of Quantum Gravity]]*, Springer (2023) [[arXiv:2211.09902](https://arxiv.org/abs/2211.09902)]

* [[John F. Donoghue]], *General Relativity as a Perturbative Quantum Field Theory*, lecture series at *[Basics of Quantum Gravity](https://indico.cern.ch/event/1279592/)*, ISQG (2023) [[web](https://blogs.umass.edu/grqft/)]

Aspects of [[gauge invariance]]:

* [[Petr M. Lavrov]], [[Ilya Shapiro]], *Gauge invariant renormalizability of quantum gravity*, in *[[Handbook of Quantum Gravity]]*, Springer (2023) [[arXiv:2210.09271](https://arxiv.org/abs/2210.09271)]

A brief survey of the relevant mathematical issues with more pointers to the literature is in

* [[Igor Khavkine]], _Gravity: an exercise in quantization_, talk at _Quantum Gravity in Perspective_, Munich, 31 May 2013 ([pdf slides](http://www.science.unitn.it/~khavkine/khavkine-munich.pdf), [video](https://www.youtube.com/watch?v=G5O9RKmXyfM#t=2444))

The rigorous construction via [[causal perturbation theory]] of quantum gravity  as a [[perturbative quantum field theory]] around [[Minkowski spacetime]] is spelled out in

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

The generalization of this to perturbation about [[AQFT on curved spacetimes|curved spacetimes]] is discussed via [[locally covariant perturbative algebraic quantum field theory]] in

* {#BrunettiFredenhagenRejzner13} [[Romeo Brunetti]], [[Klaus Fredenhagen]], [[Katarzyna Rejzner]], _Quantum gravity from the point of view of locally covariant quantum field theory_, Communications in Mathematical Physics August 2016, Volume 345, Issue 3, pp 741–779 ([arXiv:1306.1058](http://arxiv.org/abs/1306.1058))

and with applications to [[cosmology]] in

* {#BFHPR16} [[Romeo Brunetti]], [[Klaus Fredenhagen]], Thomas-Paul Hack, [[Nicola Pinamonti]], [[Katarzyna Rejzner]], _Cosmological perturbation theory and quantum gravity_, Journal of High Energy Physics August 2016:32 ([arXiv:1605.02573](https://arxiv.org/abs/1605.02573))

Regarding the [[cosmological constant]]:

* [[John Donoghue]], *The cosmological constant and the use of cutoffs* ([arXiv:2009.00728](https://arxiv.org/abs/2009.00728))

* David Prinz, *On Perturbative Quantum Gravity with a Cosmological Constant* [[arXiv:2303.14160](https://arxiv.org/abs/2303.14160)]

Corrections at [[loop order|1-loop]] from [[quantum gravity]] to the [[anomalous magnetic moment]] of the [[leptons]] are discussed in

* {#BerendsGastman75} F. A. Berends, R. Gastmans, _Quantum gravity and the electron and muon anomalous magnetic moment_, Phys. Lett. B55 Issue 3 Feb 1975 311-312 (<a href="doi.org/10.1016/0370-2693(75)90608-5">doi:10.1016/0370-2693(75)90608-5</a>)

and adapted to [[supergravity]] in

* F. del Aguila, A. Culatti, R. Munoz-Tapia, M. Perez-Victoria, _Supergravity corrections to $(g-2)_l$ in differential renormalization_, Nuclear Physics  B  504  (1997)  532-550 ([arXiv:hep-ph/9702342](https://arxiv.org/abs/hep-ph/9702342))

See also:

* F. T. Brandt, J. Frenkel, D. G. C. McKeon, *Quantum gravity in a general background gauge* [[arXiv:2208.13004](https://arxiv.org/abs/2208.13004)]

Concrete discussion of [[graviton]] [[scattering amplitudes]]:

* S. Abreu, F. Febres Cordero, H. Ita, M. Jaquier, B. Page, M. S. Ruf, V. Sotnikov, _The Two-Loop Four-Graviton Scattering Amplitudes_ ([arXiv:2002.12374](https://arxiv.org/abs/2002.12374))

With an emphasis on the [[S-matrix]]-perspective:

* [[Pierre Vanhove]], *S-matrix approach to general gravity and beyond* ([arXiv:2104.10148](https://arxiv.org/abs/2104.10148))

On [[perturbative quantum gravity]] via [[string-localized quantum fields]]:

* Christian Gaß, [[José M. Gracia-Bondía]], Karl-Henning Rehren, *Quantum general covariance* [[arXiv:2304.09843](https://arxiv.org/abs/2304.09843)]

For more discussion of perturbative [[supergravity]] see at

* _[N=8 perturbative quantum gravity](4-dimensional+supergravity#PerturbativeQuantumSupergravityN8)_

* _[[KLT relations]]_

* _[[string theory results applied elsewhere]]_.

### Non-renormalizablity
 {#ReferencesNonRenormalizability}

That pure [[Einstein gravity]] is not perturbatively [[renormalizable interaction|renormalizable]] at [[loop order|2-loops]] was shown in

* {#GoroffSagnotti85} M. H. Goroff, A. Sagnotti, _Quantum Gravity At Two Loops_, Phys. Lett. B 160 (1985) 81.

* {#vandeVen92} A. E. M. van de Ven, _Two loop quantum gravity_, Nucl. Phys. B 378 (1992) 309.

For generic matter couplings this applies already at [[loop order|1-loop]]:

* [[Gerard 't Hooft]], [[Martinus Veltman]], Annales Poincare Phys. Theor. A 20 , 69 (1974)

* [[Gerard 't Hooft]], Nucl. Phys. B 62, 444 (1973);

* [[Stanley Deser]], [[Peter van Nieuwenhuizen]], Phys. Rev. D 10, 401 (1974);

* [[Stanley Deser]], H. S. Tsao and [[Peter van Nieuwenhuizen]], Phys. Rev. D 10 , 3337 (1974)

If the matter coupling is such as to make the theory [[supersymmetry|supersymmetric]] ([[supergravity]]), the divergence appears only in higher [[loop order]].

That it appears earliest at 3-loops was established in

*  M. T. Grisaru, Phys. Lett. B 66, 75 (1977);

* E. T. Tomboulis, Phys. Lett. B67, 417 (1977);

* [[Stanley Deser]], J. H. Kay and [[Kellogg Stelle]], Phys. Rev. Lett. 38, 527 (1977);

* [[Renata Kallosh]], Phys. Lett. B99, 122 (1981);

* [[Paul Howe]], [[Kellogg Stelle]] and [[Paul Townsend]], Nucl. Phys. B 236, 125 (1984).

Specifically for [[N=8 d=4 supergravity]] there is evidence that it might be finite at all loops. See the references [there](4-dimensional+supergravity#PerturbativeQuantumSupergravityN8).

### From string field theory

Derivation of [[graviton]]-[[scattering amplitudes]] from [[string field theory]]:

* [[Taejin Lee]], _Gravitational Scattering Amplitudes and Closed String Field Theory in the Proper-Time Gauge_, EPJ Web of Conferences 168, 07004 (2018) ([doi:10.1051/epjconf/201816807004](https://doi.org/10.1051/epjconf/201816807004))

* [[Taejin Lee]], _Four-Graviton Scattering and String Path Integral in the Proper-time gauge_ ([arXiv:1806.02702](https://arxiv.org/abs/1806.02702))

### Local Gauging of global symmetries

It is being argued that after embedding into consistent [[quantum gravity]], all [[global symmetries]] must become [[local symmetries]].

A substantiation of this argument via [[AdS/CFT]] is given in

* [[Daniel Harlow]], [[Hirosi Ooguri]], _Symmetries in quantum field theory and quantum gravity_ ([arXiv:1810.05338](https://arxiv.org/abs/1810.05338))

* [[Daniel Harlow]], [[Hirosi Ooguri]], _Constraints on symmetry from holography_, Phys. Rev. Lett. 122, 191601 (2019) ([arXiv:1810.05337](https://arxiv.org/abs/1810.05337))

See also:

* Sylvain Fichet, Prashant Saraswat, _Approximate Symmetries and Gravity_ ([arXiv:1909.02002](https://arxiv.org/abs/1909.02002))

[[!include 3d gravity and Chern-Simons theory -- references]]

[[!redirects perturbative quantum gravity]]

[[!redirects pQG]]


quantum group

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#Contents#
* table of contents
{:toc}

## Idea

The notion of _quantum group_ refers to various objects which are deformations of ([[algebras of functions]] on) [[groups]], but still have very similar properties to (algebras of functions on) groups, and in particular to [[semisimple Lie groups]]. Most important are the [[Hopf algebras]] _deforming_ the function algebras on [[semisimple Lie groups]] or to the enveloping algebras of Kac-Moody Lie algebras.

## Overview

It is a common experience in [[representation theory]] that a number of mathematical structures behaves very similarly to [[algebraic group|algebraic]] or [[Lie group|Lie]] groups. After the impetus of the theory of quantum [[integrable systems]], mainly the work of Leningrad's school of mathematical physics around 1980, several mathematicians (including Drinfeld, Manin, Woronowicz, Jimbo, Faddeev--Reshetikhin--Takhtajan) found, in different formalisms, major series of examples which are mostly noncommutative noncocommutative [[Hopf algebras]] and which deform enveloping algebras of (semisimple) Lie algebras, or algebras of functions on the corresponding algebraic groups. These deformations $G_q$ depend on a parameter $q$ (sometimes one prefers a formal parameter $h$ with $q = e^{h}$), which may be taken as belonging to the [[ground field]], but also being formal (transcendental over the ground field). A peculiar case is when the parameter $q$ of the deformation is an $l$-th root of unity; the remaining cases are usually called generic $q$.

The [[representation theory]] for these 'quantum' examples is highly developed; in fact many phenomena in the representation theory of semisimple Lie algebras (e.g. canonical bases) were discovered first as a limiting case of constructions in the quantum case, which become degenerate in the classical case (the principle that quantization removes degeneracy). While representations for generic $q$ parallel classical ones, the theory at roots of unity is peculiar and related to the representation theory of affine Lie algebras; the quantum groups at roots of unity as algebras have big centers.

Nowadays, both the class of examples and the class of formalisms has been extended a lot, hence the term 'quantum group' is not a fixed notion but rather a collective term for a rather author-dependent class of group-like objects, most often subclasses or extensions of the concept of Hopf algebras which are sometimes required to belong to families of deformations of their classical counterparts. One of the common features is that if we forget the group-like features, the examples belong to the class of noncommutative spaces (see [[noncommutative geometry]]).

Mathematically better defined are notions (sometimes equated by various authors with the class of quantum groups) like [[quasitriangular Hopf algebras]], [[quantum matrix groups]] (quantum linear groups, more general FRT-algebras and Majid's $A(R)$ where $R$ is a [[quantum Yang-Baxter matrix|quantum Yang-Baxter equation]]), quantized enveloping algebras, quantum function algebras, compact matrix pseudogroups, Kac algebras, Yangians etc. The representations of quasitriangular Hopf algebras form [[braided monoidal categories]], which are in main examples related to the mathematics of Iwahori--Hecke algebras, braid groups, knot theory, finite group Chern--Simons theory and Wess--Zumino--Novikov--Witten theory of [[CFT]]. One should note that in the classical limit quantum function algebras give not simply (functions on) algebraic (or Lie) groups but also a compatible (= multiplicative) Poisson structure giving rise to Poisson--Lie or Poisson algebraic groups.

There is an extensive geometric theory of homogeneous spaces for quantum groups and [[fiber bundles]] whose structure groups are quantum groups.

## Properties

### Tannaka duality

[[!include structure on algebras and their module categories - table]]

## Related concepts

* [[Hopf algebra]], [[bialgebra]], [[gebra]], [[braided monoidal category]], [[noncommutative algebraic geometry]], [[noncommutative geometry]], [[Hopf-Galois extension]], [[matrix bialgebra]], [[Knizhnik-Zamolodchikov equation]], [[Tannaka duality]], [[Yangian]], [[Yang-Baxter equation]], [[classical Yang-Baxter equation]], [[quantum Yang-Baxter equation]], [[dynamical Yang-Baxter equation]], [[quantum linear group]], [[quantized function algebra]], [[quantized enveloping algebra]]

* [[Poisson Lie group]]

* [[quantum 2-group]]

* [[quantum topology]]

## References

* {#Drinfeld87} [[Vladimir Drinfeld]], _Quantum groups_, in: A. Gleason (ed.) *[Proceedings of the](https://archive.org/details/proceedingsofint0002inte_v5c3/mode/2up) [1986 International Congress of Mathematics](https://inspirehep.net/conferences/966215?ui-citation-summary=true)* **1** (1987) 798-820 [[pdf](https://www.mathunion.org/fileadmin/ICM/Proceedings/ICM1986.1/ICM1986.1.ocr.pdf)]

  expanded version:

  Journal of Soviet Mathematics **41** (1988) 898–915 [[doi:10.1007/BF01247086](https://doi.org/10.1007/BF01247086)]

* {#Tjin92} [[Tjark Tjin]], *An introduction to quantized Lie groups and algebras*, Int. J. Mod. Phys. A **7** (1992) 6175-6213 [[arXiv:hep-th/9111043](https://arxiv.org/abs/hep-th/9111043), [doi:10.1142/S0217751X92002805](https://doi.org/10.1142/S0217751X92002805)]

* [[Christian Kassel]], _Quantum groups_, Graduate Texts in Mathematics __155__, Springer 1995 ([doi:10.1007/978-1-4612-0783-2](https://link.springer.com/book/10.1007/978-1-4612-0783-2), [webpage](http://www-irma.u-strasbg.fr/~kassel/QGbk.html), [errata pdf](http://www-irma.u-strasbg.fr/~kassel/QGerrata030399.pdf))

* [[Shahn Majid]], _Foundations of quantum group theory_, Cambridge University Press 1995, 2000.

* [[Yuri Manin|Yu. I. Manin]], _Quantum groups and non-commutative geometry_, CRM, Montreal 1988.

* B. Parshall, J.Wang, _Quantum linear groups_, Mem. Amer. Math. Soc. 89(1991), No. 439, vi+157 pp.

* N. Yu. Reshetikhin, L. A. Takhtajan, L. D. Faddeev, _Quantization of Lie groups and Lie algebras_, Algebra i analiz __1__, 178 (1989) (Russian), English translation in Leningrad Math. J. 1.

* [[Arun Ram]], _A survey of quantum groups: background, motivation, and results_, in: Geometric analysis and Lie theory in mathematics and physics, A. Carey and M. Murray eds., Australian Math. Soc. Lecture Notes Series __11__, Cambridge Univ. Press 1997, pp. 20-104. [pdf](http://www.ms.unimelb.edu.au/~ram/Publications/1997AustMSLectNotesv11p20.pdf)

* [[Pavel Etingof]], O. Schiffmann, _Lectures on Quantum Groups_, Lectures in Math. Phys., International Press (1998).

* [[Pavel Etingof]], [[Igor Frenkel]], _Lectures on representation theory and Knizhnik-Zamolodchikov equations_

* A. U. Klymik, K. Schmuedgen, _Quantum groups and their representations_, Springer 1997.

* A. Joseph, _Quantum groups and their primitive ideals_, Springer 1995.

* [[Ross Street]], _Quantum groups : a path to current algebra_, Cambridge Univ. Press 2007

* L. I. Korogodski, Ya. S. Soibelman, _Algebras of functions on quantum groups I_, Math. Surveys and Monographs 56, AMS 1998.

* A. Varchenko, _Hypergeometric functions and representation theory of Lie algebras and quantum groups_, Advanced Series in Mathematical Physics, Vol. 21, World Scientific (1995)

* [[George Lusztig]], _Introduction to quantum groups_

* V. Chari, A. Pressley, _A guide to quantum groups_, Camb. Univ. Press 1994

* Bangming Deng, Jie Du, [[Brian Parshall]], Jianpan Wang, _Finite dimensional algebras and quantum groups_, Mathematical Surveys and Monographs __150__, Amer. Math. Soc. 2008. xxvi+759 pp. [MR2009i:17023)](http://www.ams.org/mathscinet-getitem?mr=2457938)

* [[Tom Bridgeland]], _Quantum groups via Hall algebras of complexes_, Annals of Mathematics __177__:2 (2013) 739-759 (21 pages)

* [[Richard Borcherds]], [[Mark Haiman]], [[Theo Johnson-Freyd]], [[Nicolai Reshetikhin]], [[Vera Serganova]], *Berkeley Lectures on Lie Groups and Quantum Groups*, 2020 ([pdf](http://categorified.net/LieQuantumGroups.pdf))

In relation to [[hypergeometric functions]] and the [[Knizhnik-Zamolodchikov equation]]:

* [[Alexander Varchenko]], _Multidimensional hypergeometric functions and representation theory of Lie algebras and quantum groups_, Adv. Ser. in Math. Phys. __21__, World Sci. Publ. 1995. x+371 pp. ([doi:10.1142/2467](https://doi.org/10.1142/2467))

* V. Tarasov, [[Alexander Varchenko]], _Geometry of $q$-hypergeometric functions, quantum affine algebras and elliptic quantum groups_, Astérisque __246__ (1997), vi+135 pp. ([arXiv:q-alg/9703044](https://arxiv.org/abs/q-alg/9703044), [numdam:AST_1997__246__R1_0](http://www.numdam.org/item/AST_1997__246__R1_0))

In the generality of [[noncartesian internal categories]]:

* [[Marcelo Aguiar]],  *Internal categories and quantum groups*, PhD thesis, Cornell 1997 ([pdf](http://pi.math.cornell.edu/~maguiar/thesis2.pdf), [[Aguiar_InternalCategoriesAndQuantumGroups.pdf:file]])

See also:

* MathOverflow: [q.gr. as relative Drinfeld double](http://mathoverflow.net/questions/20683/quantum-group-as-relative-drinfeld-double), [why Drinfeld-Jimbo q.gr.](http://mathoverflow.net/questions/5538/why-drinfeld-jimbo-type-quantum-groups), [Lusztig perverse sheaves on quiver varieties](http://mathoverflow.net/questions/14361/what-do-the-local-systems-in-lusztigs-perverse-sheaves-on-quiver-varieties-look), [canonical bases for extended q.env.algebras](http://mathoverflow.net/questions/8110/canonical-basis-for-the-extended-quantum-enveloping-algebras), [groups-qgroups-and-... (on elliptic case)](http://mathoverflow.net/questions/58040/groups-quantum-groups-and-fill-in-the-blank), [all posts with quantum group tag](http://mathoverflow.net/questions/tagged/quantum-group)

category: noncommutative geometry
[[!redirects quantum groups]]


quantum group Fourier transform

__Quantum group Fourier transform__ refers to several variants of Fourier transforms attached to Hopf algebras or their analytic versions.

The usual Fourier transform is between functions on locally compact abelian group $G$ and functions on its [[Pontrjagin dual]] locally compact abelian group $\hat{G}$ (the group of continuous characters on $G$ with the topology of uniform convergence on compact sets), cf. wikipedia, [Pontrjagin duality](http://en.wikipedia.org/wiki/Pontryagin_duality).

In noncommutative case, the role of Pontrjagin duality is played by a version of [[Tannaka-Krein theorem|Tannaka-Krein duality]].

To avoid analytic details for starters, and still to introduce elements of the general story, one can look at the case when $G$ is finite. Then, over a field $k$, the group Hopf algebra $k \hat{G}$ is isomorphic with the [[dual Hopf algebra]] $(k G)^*$ of the group Hopf algebra $k G$ which is itself isomorphic as a Hopf algebra to the function algebra $k(G)$. The composition is giving the Fourier transform $k \hat{G}\cong k(G)$, which is a linear map from the convolution algebra of $G$ to the function algebra on $G$. In such a situation one has a discrete version of Haar measure. Define

$$
\mathcal{F}(h)(u) := \sum_{\chi\in \hat{G}} h (\chi)\chi(u)
$$
$$
\mathcal{F}^{-1}(\phi)(\chi) := \frac{1}{|G|}\sum_{u\in G}\chi(u^{-1})\phi(u)
$$
$$
\Lambda = \sum_{u\in G} u \in k G
$$
$$
\Lambda^* = \sum_{\chi} \chi
$$

where one assumes that $|G|$ is invertible in $k$. It holds that

$\chi \Lambda^* = \epsilon(\chi)\Lambda^*$ for all $\chi\in k \hat{G}$ and $\Lambda \phi = \Lambda \epsilon(\phi)$ for all $\phi\in k G$. In other words, $\Lambda$ is a right integral _in_ $k G$ and $\Lambda^*$ is a left integral in $(k G)^*$.

One can write

$$
\mathcal{F}(h) = \Lambda_{(1)}\langle h, \Lambda_{(2)}\rangle,
$$
for $h\in k \hat{G} = (k G)^*$ (notice the usage of left [[coregular action]]) and

$$
\mathcal{F}^{-1}(\phi) = \frac{\Lambda^*_{(1)}\langle \Lambda^*_{(2)},S \phi\rangle}{\langle \Lambda^*, \Lambda \rangle}
$$

for $\phi\in k G$ and where $S: k G\to (k G)^{op,cop}$ is the antipode.
These formulas make sense for more general [[dual bialgebra|Hopf algebras in duality]] provided there are appropriate analogues of $\Lambda$ and $\Lambda^*$ and $\langle \Lambda^*, \Lambda\rangle$ is invertible in $k$. That generalization is called the __quantum group Fourier transform__.

They can also be related to the [[fundamental operator]] in Hopf algebra $H$, see under [[multiplicative unitary]].

### Literature

* V. Lyubashenko, _Modular transformations and tensor categories_, J. Pure Appl. Algebra __98__ (1995) 279–327 doi

> For an abelian tensor category we investigate a Hopf algebra F in it, the “algebra of
functions” or “automorphisms of the identity functor”. We show the existence of the object of integrals for any Hopf algebra in a rigid abelian category. If some assumptions of finiteness and non-degeneracy are satisfied, the Hopf algebra F has an integral and there are morphisms $S, T : F \to F$, called modular transformations. They yield a representation of a modular group. The properties of $S$ are similar to those of the Fourier transform.

* V. Lyubashenko, S. Majid, _Fourier transform identities in quantum mechanics and the quantum line_, Phys. Lett. B284 (1992) 66-70 doi; _Braided groups and quantum Fourier transform_, J. Algebra 166 506-28 [doi](https://doi.org/10.1006/jabr.1994.1165)
* T H Koornwinder, B J Schroers, J K Slingerland, F A Bais, _Fourier transform and the Verlinde formula for the quantum double of a finite group_,  J. Phys. A32: 48 [doi](https://doi.org/10.1088/0305-4470/32/48/313)
* [[Shahn Majid]], _Foundations of quantum group theory_, 1995, 2nd. ed 2000
* M. Enock, J. M. Schwartz, _Kac algebras and duality of locally compact groups_,  Springer-Verlag, 1992, , x+257 pp. [gBooks](http://books.google.com/books/about/Kac_algebras_and_duality_of_locally_comp.html?id=U6e6aD1gj3oC), [MR94e:46001](http://www.ams.org/mathscinet-getitem?mr=1215933)
* Massoud Amini, _Tannaka–Krein duality for compact groupoids I, Representation theory_, Advances in Mathematics __214__, n. 1, 2007, 78-91 [doi](http://dx.doi.org/10.1016/j.aim.2006.09.015)
* Laurent Freidel, [[nlab:Shahn Majid]], _Noncommutative harmonic analysis, sampling theory and the Duflo map in $2+1$ quantum gravity_, Classical Quantum Gravity __25__ (2008), no. 4, 045006 [MR2009f:83058](http://www.ams.org/mathscinet-getitem?mr=2388191), [doi](http://dx.doi.org/10.1088/0264-9381/25/4/045006)
* [[zoranskoda:ncFourier]]


quantum hadrodynamics

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#Contents#
* table of contents
{:toc}

## Idea

Quantum hadrodynamics is (or should be) an [[effective field theory]] that describes [[confinement|confined]] [[hadron]] [[bound states]] in [[nuclear physics]], in particular the [[interaction]] between [[mesons]] and [[baryons]] and specifically with [[nucleons]].

In this sense [[chiral perturbation theory]] is the archetype of quantum hadrodynamics, but the latter term is commonly used more specifically for the [[Walecka model]] and its generalizations (QHD-I [Walecka 74](#Walecka74) and  QHD-II [Serot 79](#Serot79), [Serot-Walecka 92](#SerotWalecka92), [Serot-Walecka 97](#SerotWalecka97)) to [[baryon chiral perturbation theory]], where the [[baryons]] and specifically the [[nucleons]] appear as explicit [[effective field theory|effective]] [[field (physics)|fields]] (as opposed to as emergent [[Skyrmion]] fields).

## Related concepts

* [[Walecka model]]

* [[hadron supersymmetry]], [[hadron Kaluza-Klein theory]]

  * [[flavor branes]]

* [[leptonic decay]]

[[!include effective field theories of nuclear physics -- contents]]

## References

See also the references at _[[chiral perturbation theory]]_

[[!include quantum hadrodynamics -- references]]

[[!include Skyrme hadrodynamics with vector mesons -- references ]]

[[!include Skyrme hadrodynamics with heavy mesons -- references]]

[[!include WZW term of QCD chiral perturbation theory -- references]]

[[!include baryon chiral perturbation theory -- references]]

[[!redirects hadrodynamics]]


quantum hadrodynamics -- references

### Walecka hadrodynamics with nucleon fields
 {#QuantumHadrodynamicsReferences}

On [[quantum hadrodynamics]] ([[relativistic field theory|relativivist]] [[effective field theory]] of [[nuclear physics]], [[interaction|coupling]] [[mesons]] and [[nucleons]]) in the sense of the [[Walecka model]],  hence with [[nucleons]] appearing as explicit [[field (physics)|fields]] (as opposed to being [[soliton|solitonic]] [[Skyrmions]] in the [[pion field]] as in [[chiral perturbation theory]]).

Precursor:

* [[Hans-Peter Duerr]], _Relativistic Effects in Nuclear Forces_, Phys. Rev. 103, 469 (1956) ([doi:10.1103/PhysRev.103.469](https://doi.org/10.1103/PhysRev.103.469))

The original [[Walecka model]] (*QHD-I model*), with [[nucleons]] coupled to [[sigma-mesons]] and [[omega-mesons]]:

* {#Walecka74} [[John Dirk Walecka]], _A Theory of highly condensed matter_, Annals Phys. 83 (1974) 491 ([spire:91609](http://inspirehep.net/record/91609?ln=en), dpi:10.1016/0003-4916(74)90208-5)

Inclusion into the [[Walecka model]] also of the [[pion]] and the [[rho-meson]] (the _QHD-II model_):

* {#Serot79} [[Brian Serot]], _A relativistic nuclear field theory with $\pi$ and $\rho$ mesons_, Physics Letters B Volume 86, Issue 2, 24 (1979), Pages 146-150 (doi:10.1016/0370-2693(79)90804-9)

* T Matsui, [[Brian Serot]], _The pion propagator in relativistic quantum field theories of the nuclear many-body problem_, Annals of Physics Volume 144, Issue 1, November 1982, Pages 107-167 (doi:10.1016/0003-4916(82)90106-3)

Further discussion of these models:

* S. A. Chin, _A relativistic many-body theory of high density matter_, Annals of Physics Volume 108, Issue 2, October 1977, Pages 301-367 (doi:10.1016/0003-4916(77)90016-1)

* [[Brian Serot]], [[John Dirk Walecka]], _The Relativistic Nuclear Many Body Problem_, Adv. Nucl. Phys. 16 (1986) 1-327 ([spire:207866](https://inspirehep.net/literature/207866))

* [[Brian Serot]], _Quantum hadrodynamics_, Reports on Progress in Physics, Volume 55, Number 11 (1992) ([doi:10.1088/0034-4885/55/11/001](https://doi.org/10.1088/0034-4885/55/11/001))

* {#SerotWalecka92} [[Brian Serot]], [[John Dirk Walecka]], _Chiral QHD with vector mesons_, Acta Phys. Polon. B 23 (1992) 655-679 ([spire:343513](https://inspirehep.net/literature/343513))

* [[Maciej Nowak]], [[Mannque Rho]], [[Ismail Zahed]], _[[Chiral Nuclear Dynamics]]_,  World Scientific 1996 ([doi:10.1142/1681](https://doi.org/10.1142/1681))

* {#SerotWalecka97} [[Brian Serot]], [[John Dirk Walecka]], _Recent Progress in Quantum Hadrodynamics_,  Int. J. Mod. Phys. E6:515-631, 1997 ([arXiv:nucl-th/9701058](http://arxiv.org/abs/nucl-th/9701058))

* R. V. Poberezhnyuk, V. Vovchenko, D. V. Anchishkin, M. I. Gorenstein, _Quantum van der Waals and Walecka models of nuclear matter_ ([arXiv:1708.05605](https://arxiv.org/abs/1708.05605))

Further inclusion of [[electromagnetism]] ([[photon]] [[field (physics)|field]]):

* A. Yu. Korchin, D. Van Neck, M. Waroquier, _Electromagnetic interaction in chiral quantum hadrodynamics and decay of vector and axial-vector mesons_, Phys. Rev. C67 (2003) 015207 ([arXiv:nucl-th/0302042](https://arxiv.org/abs/nucl-th/0302042))

Relation to [[quark-meson coupling model]]:

* Koichi Saito, _Relationship between Quark-Meson Coupling Model and Quantum Hadrodynamics_, Prog. Theor. Phys. 108 (2002) 609-614 ([arXiv:nucl-th/0207053](https://arxiv.org/abs/nucl-th/0207053))

### Hadrodynamics via Light-front QCD

On [[light-front QCD]] for [[quantum hadrodynamics]]:

* International Light Cone Advisory Committee, *Light-Front Quantum Chromodynamics: A framework for the analysis of hadron physics*,  Nuclear Physics B - Proceedings Supplements
Volumes 251–252, June–July 2014, Pages 165-174 ([arXiv:1309.6333](https://arxiv.org/abs/1309.6333), [doi:10.1016/j.nuclphysbps.2014.05.004](https://doi.org/10.1016/j.nuclphysbps.2014.05.004))

* [[Edward Shuryak]], [[Ismail Zahed]], *Hadronic structure on the light-front I: Instanton effects and quark-antiquark effective potentials* ([arXiv:2110.15927](https://arxiv.org/abs/2110.15927))

* [[Edward Shuryak]], [[Ismail Zahed]], *Hadronic structure on the light-front II: QCD strings, Wilson lines and potentials* ([arXiv:2111.01775](https://arxiv.org/abs/2111.01775))


quantum Hall effect

+-- {: .rightHandSide}
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### Context
#### Topological physics
+--{: .hide}
[[!include topological physics -- contents]]
=--
=--
=--

# Contents
* table of contents
{: toc}

## Idea

A [[physical system]] in [[solid state physics]] consisting of [[electrons]] confined to an essentially 2-dimensional surface and subject to a perpendicular [[magnetic field]].

In an appropriate limit of low [[temperature]] aspects of this system are described by the [[topological quantum field theory]] called _[[Chern-Simons theory]]_. In this limit the system has been proposed as constituting a possible implementation of [[topological quantum computation]].

## Properties

### As a topological insulator

The bulk/edge behaviour in a quantum Hall effect is is that of a [[topological insulator]]. (While topological insulator materials typically show this behaviour without the need of a strong magnetic field.)

(...)

## Related concepts

* [[Hall effect]]

* [[spin Hall effect]], [[quantum spin Hall effect]]

* [[Laughlin wavefunction]]

* [[braid group statistics]], [[anyon]]

* [[topological insulator]]

* [[topological quantum computation]]

## References

### General

Review:

* {#vonKlitzing86} [[Klaus von Klitzing]], _The quantized Hall effect_, Rev. Mod. Phys. **58** 519 (1986) $[$[doi:10.1103/RevModPhys.58.519](https://doi.org/10.1103/RevModPhys.58.519)$]$

* [[Steven M. Girvin]], *Introduction to the Fractional Quantum Hall Effect*, Séminaire Poincaré **2** (2004) 53–74, reprinted in *The Quantum Hall Effect*, Progress in Mathematical Physics **45**, Birkhäuser (2005)  $[$[pdf](http://www.bourbaphy.fr/girvin.pdf), [doi:10.1007/3-7643-7393-8_4](https://doi.org/10.1007/3-7643-7393-8_4)$]$

* {#Tong16} [[David Tong]], *The Quantum Hall Effect* (2016) $[$[course webpage](https://www.damtp.cam.ac.uk/user/tong/qhe.html), [pdf](http://www.damtp.cam.ac.uk/user/tong/qhe/qhe.pdf), [[Tong-QuantumHallEffect.pdf:file]]$]$

* _The quantum Hall effect_ ([pdf](http://www.tezu.ernet.in/rctp2015/SR%20On%20quantum%20Hall%20effect.pdf))

Discussion via Newton-Cartan theory:

* William Wolf, James Read, Nicholas Teh, *Edge modes and dressing fields for the Newton-Cartan quantum Hall effect* ([arXiv:2111.08052](https://arxiv.org/abs/2111.08052))

See also:

* Wikipedia, _[Quantum Hall effect](https://en.wikipedia.org/wiki/Quantum_Hall_effect)_,

* Wikipedia _[Fractional quantum Hall effect](http://en.wikipedia.org/wiki/Fractional_quantum_Hall_effect)_

On [[anyon statistics|anyon phases]] (specifically in the [[quantum Hall effect]]) as [[Berry phases]] of a [[quantum adiabatic theorem|adiabatic]] transport of anyon positions:

* {#AvrosSchriefferWilczek84} [[Daniel P. Arovas]], [[John Robert Schrieffer]], [[Frank Wilczek]], _Fractional Statistics and the Quantum Hall Effect_, Phys. Rev. Lett. 53, 722 (1984) $[$[doi:10.1103/PhysRevLett.53.722](https://doi.org/10.1103/PhysRevLett.53.722)$]$

### Integral quantum Hall effect

#### Experiment

Original experimental detection:

* [[Klaus von Klitzing]], G. Dorda, and M. Pepper, _New Method for High-Accuracy Determination of the Fine-Structure Constant Based on Quantized Hall Resistance_, Phys. Rev. Lett. 45, 494 (1980) ([doi:10.1103/PhysRevLett.45.494](https://doi.org/10.1103/PhysRevLett.45.494))

* M. A. Paalanen, D. C. Tsui, and A. C. Gossard, _Quantized Hall effect at low temperatures_, Phys. Rev. B 25, 5566(R) April 1982 ([doi:10.1103/PhysRevB.25.5566](https://doi.org/10.1103/PhysRevB.25.5566))

#### Theory

While an intuitive understanding for the quantization of the Hall conductance has been given in

* [[Robert B. Laughlin]], *Quantized Hall conductivity in two dimensions*, Phys. Rev. B **23** 5632(R) (1981) $[$[doi:10.1103/PhysRevB.23.5632](https://journals.aps.org/prb/abstract/10.1103/PhysRevB.23.5632)$]$

a theoretical derivation of the effect was obtained only much later in

* [[Matthew Hastings]], [[Spyridon Michalakis]], _Quantization of Hall conductance for interacting electrons without averaging assumptions_, Commun. Math. Phys., 334:433–471, (2015) ([arXiv:0911.4706](https://arxiv.org/abs/0911.4706), [doi:10.1007/s00220-014-2167-x](https://doi.org/10.1007/s00220-014-2167-x))

with closely related results in

* Alessandro Giuliani, Vieri Mastropietro, Marcello Porta, _Universality of the Hall conductivity in interacting electron systems_, Commun. Math. Phys. 349, 1107–1161 (2017) ([arXiv:1511.04047](https://arxiv.org/abs/1511.04047), [doi:10.1007/s00220-016-2714-8](https://doi.org/10.1007/s00220-016-2714-8))

Review of this theory behind the quantum Hall effect:

* Yosi Afron, _Why is the Hall conductance quantized?_, 2017 ([pdf](http://web.math.princeton.edu/~aizenman/OpenProblems_MathPhys/17_Avron.pdf), [[AfronQuantumHallEffect.pdf:file]])

* [[Spyridon Michalakis]], _Why is the Hall conductance quantized?_, Nature Reviews Physics 2, 392–393 (2020) ([doi:10.1038/s42254-020-0212-6](https://doi.org/10.1038/s42254-020-0212-6))

* S. Klevtsov, X. Ma, G. Marinescu, P. Wiegmann, _Quantum Hall effect and Quillen metric_ Commun. Math. Phys. 349, 819–855 (2017) [doi](https://doi.org/10.1007/s00220-016-2789-2)

### Fractional quantum Hall effect

A quick review of the description via [[Chern-Simons theory]] with further pointers is in the introduction of

* Spencer D. Stirling, _Abelian Chern-Simons theory with toral gauge group, modular tensor categories, and group categories_, [arXiv:0807.2857](http://arxiv.org/abs/0807.2857)

J. Bellissard introduced an approach via [[noncommutative geometry]] and Connes-Chern character:

* J. Bellissard, A. van Elst, H. Schulz Baldes, _The noncommutative geometry of the quantum Hall effect_, 79 pages, J. Math. Phys. __35__, 5373 (1994) [cond-mat/9411052](http://arxiv.org/abs/cond-mat/9411052) [doi](http://link.aip.org/link/doi/10.1063/1.530758)

In terms of [[Berry phase]] and Chern numbers in

* Joseph E. Avron, Daniel Osadchy, Ruedi Seiler, _A topological look at the quantum Hall effect_, Physics Today __56__:8, [doi](http://dx.doi.org/10.1063/1.1611351)

Realization via [[AdS/CFT in condensed matter physics]]:

* Mitsutoshi Fujita, Wei Li, Shinsei Ryu, Tadashi Takayanagi, *Fractional Quantum Hall Effect via Holography: Chern-Simons, Edge States, and Hierarchy*, JHEP 0906:066 (2009) ([arXiv:0901.0924](https://arxiv.org/abs/0901.0924))

[[!include anyons in the quantum Hall effect -- references]]

category: physics

[[!redirects quantum Hall effects]]

[[!redirects fractional quantum Hall effect]]
[[!redirects fractional quantum Hall effects]]

[[!redirects quantum Hall material]]
[[!redirects quantum Hall materials]]


quantum hall effect > history

< [[quantum hall effect]]


quantum harmonic oscillator

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###Context###
#### Physics
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[[!include physicscontents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

"Harmonic oscillator" is a fancy name for a rock on a spring:

* in [[classical mechanics]] it is the [[physical system]] given by a point mass in a parabolic potential, feeling [[forces]] driving it back to a specified origin that are propertional to the distance of the [[mass]] from that origin.

* in [[quantum mechanics]] and in particular [[quantum field theory]] the quantum harmonic oscillator governs not just the dynamics of idealized point masses but crucially appears in the dynamics of all free massive quantum fields.

To quote the field theorist Sidney Coleman,
> The career of a young theoretical physicist consists of treating the harmonic oscillator in ever-increasing levels of abstraction.

## Classical oscillator

First the harmonic oscillator in [[classical mechanics]].

The [[force]] exerted by a spring is proportional to how far you stretch it:
\[F = k x.\]
The [[potential energy]] stored in a stretched spring is the integral of that:
\[V_0 = \frac{1}{2}k x^2 + C,\]
and to make things work out nicely, we're going to choose $C = -1/2.$  The total [[energy]] $H_0$ is the sum of the potential and the kinetic energy:
\[H_0 = V_0 + T = \frac{1}{2}k x^2 + \frac{1}{2}m v^2 - \frac{1}{2}.\]
By choosing units so that $k = m = 1,$ we get
\[H_0 = \frac{x^2}{2} + \frac{p^2}{2} - \frac{1}{2},\]
where $p$ is momentum.

## Quantum harmonic oscillator

Now the harmonic oscillator in [[quantum mechanics]].

We [[quantization|quantize]], getting a quantum harmonic oscillator, or QHO.  We set $p = -i \frac{\partial}{\partial x},$ taking units where $\hbar = 1.$  Now
\[\array{ [x, p]x^n & = & (x p - p x)x^n \\ & = &(- x i \frac{\partial}{\partial x} + i \frac{\partial}{\partial x} x)x^n \\ & = & -i(n x^n - (n+1)x^n) \\ & = & i x^n. }\]

If we define a new [[observable]] $z = \frac{p + ix}{\sqrt{2}},$ then
\[\array{ z z^* & = & \frac{(p + i x)}{\sqrt{2}} \frac{(p - i x)}{\sqrt{2}} \\ & = & \frac{1}{2}(p^2 + i(x p - p x) + x^2) \\ & = & \frac{1}{2}(p^2 -1 + x^2) \\ & = & H_0. }\]
We can think of $z^*$ as $\frac{d}{dz}$ and write the energy [[eigenvector]]s as [[polynomial]]s in $z:$
\[H_0 z^n = z \frac{d}{dz} z^n = n z^n.\]
The creation operator $z$ adds a photon to the mix; there's only one way to do that, so $z\cdot z^n = 1 z^{n+1}.$  The annihilation operator $\frac{d}{dz}$ destroys one of the photons; in the state $z^n$, there are $n$ [[photon]]s to choose from, so $\frac{d}{dz} z^n = n z^{n-1}.$

[[Schrodinger equation|Schrödinger's equation]] says $i \frac{d}{dt} \psi = H_0 \psi,$ so
\[\psi(t) = \sum_{n=0}^{\infty} e^{-itn} a_n z^n.\]
This way of representing the state of a QHO is known as the _Fock basis_.

## Related concepts

* [[hydrogen atom]]

* [[quantum lattice system]]

## References

### General

* [[Howard Georgi]], Section 1 of: *The Physics of Waves*, Prentice Hall (1993) $[$[web](https://www.people.fas.harvard.edu/~hgeorgi/new.htm), [pdf](https://www.people.fas.harvard.edu/~hgeorgi/onenew.pdf)$]$

### Geometric quantization

Discussion of [[geometric quantization]] of the harmonic oscillator is in

* Adrian Lim, _A non-standard geometric quantization of the harmonic oscillator_ ([pdf](http://www.math.cornell.edu/~pclim/Docs/papers/oscillator6.pdf))

* G. Sherry, _Geometric quantization of the 3-dimensional harmonic oscillator_, Quaestiones Mathematicae, 8 (1986)

* Sergey V. Zuev, _Geometric quantization of generalized oscillator_ ([arXiv:math-ph/9902024](http://arxiv.org/abs/math-ph/9902024))

### Categorification

A program initiated by [[John Baez]] aims to identify a [[categorification]] of sorts of the quantum harmonic oscillator

The notes

* [[John Baez]]'s [Fall '03](http://math.ucr.edu/home/baez/qg-fall2003/) Quantum gravity seminar

relate wavefunctions expressed in the Fock basis to [[structure types]].

This originates in

* [[John Baez]], [[James Dolan]], _From finite sets to Feynman diagrams_ ([arXiv](http://arxiv.org/abs/math/0004133)).

For more along these lines see

* [[Jeffrey Morton]], _Categorified Algebra and Quantum Mechanics_ ([arXiv](http://arxiv.org/abs/math/0601458))

[[!redirects quantum harmonic oscillators]]

[[!redirects harmonic oscillator]]
[[!redirects harmonic oscillators]]


quantum heap

##Idea##

A _quantum heap_ is a dual version of a notion of a [[heap]] internalized to the [[monoidal category]] $k$-[[Vect]], with usual tensor product over $k$. A quantum heap is to a [[Hopf algebra]] what a heap is to a [[group]].

##Definition##

A __quantum heap__ is an [[associative unital algebra]] $(H,\mu,\eta)$
together with a ternary algebra cooperation
\[ \tau : H \rightarrow H \otimes H_{op} \otimes H,\]
satisfying the following properties

$$(\id  \otimes \id \otimes \tau) \tau
= (\tau \otimes \id \otimes \id) \tau $$

$$(\id \otimes \mu) \tau = \id \otimes 1_H$$

$$(\mu \otimes \id) \tau = 1_H \otimes \id$$

Moreover, $\tau$ is required to be algebra homomorphism from $H$ into
$H \otimes H_{op} \otimes H$, where  $H_{op}$ has the opposite
algebra structure and the tensor product has the usual algebra structure.
We use heap analogue of the Sweedler notation:
\[ \tau(h) = \sum h^{(1)} \otimes h^{(2)} \otimes h^{(3)},\]
and because of the first of the above identities,
it is justified to extend it to any odd number $\geq 3$ factors, e.g.
\[ (\id \otimes \id \otimes \tau) \tau(h) =
 \sum h^{(1)} \otimes h^{(2)} \otimes h^{(3)}
\otimes h^{(4)} \otimes h^{(5)}. \]

##Versions and references##

The algebra of functions on an affine variety with a structure of an algebraic [[heap]] is an example of a commutative heap. This has been used explicitly in

* M. Kontsevich, _Operads and motives in deformation quantization_, Lett. Math. Phys. 48 1 (1999) 35--72

The noncommutative case has been first introduced and studied in chapter 9 of

* Zoran Škoda, _Cosets for quantum groups_, Ph. D. thesis, University of Wisconsin, defended January 17, 2002

Choosing a character of a quantum heap one obtains a __copointed quantum heap__. According to the main theorem in that chapter, the category of copointed quantum heap is equivalent to the category of Hopf algebras.
An updated version of that chapter 9 has been published only much later in

* Z. Škoda, _Quantum heaps, cops and heapy categories_, Mathematical Communications 12, No. 1, pp. 1--9 (2007) [math.QA/0701749](http://arxiv.org/abs/math.QA/0701749)

where also a notion of "heapy category", sort of monoidal-category like categorification of a heap is proposed.

Independently, a relative version (that is over commutative $k$-algebra $A$ instead over $k$) of a quantum heap has been proposed in 2002 under the name "quantum torsor" by

* C. Grunspan, _Quantum torsors_, [math.QA/0204280](http://arXiv.org/abs/math.QA/0204280)

where the list has an additional unnecessary and superfluous axiom, later removed by

* P. Schauenburg, _Quantum torsors and Hopf-Galois objects_, [math.QA/0208047](http://arXiv.org/abs/math.QA/0208047)

An interesting notion in Grunspan's work are the left and right automorphism Hopf algebras of a quantum heap which are analogues of the automorphism group of a [[heap]]; in addition he has exhibited a counterexample showing that the left and right automorphism Hopf algebra do not need to be isomorphic.

A recent related article is

* G. Böhm, T. Brzeziński, _Pre-torsors and equivalences_,  J. Algebra 317 544--580 (2007) ([math.QA/0607529](http://arxiv.org/abs/math.QA/0607529)), Corrigendum: J Algebra 319 1339-1340 (2008)

and generalizations with emphasis on noncommutative base are studied in

* Tomasz Brzeziński, Joost Vercruysse, _Bimodule herds_, J. Algebra __321__:9, (2009) 2670-2704 [arXiv:0805.2510](http://arXiv.org/abs/0805.2510) [doi](https://doi.org/10.1016/j.jalgebra.2009.01.020)

See also

* Thomas Booker, Ross Street, _Torsors, herds and flocks_, J. Algebra 330:1 (2011) 346–374 [arxiv/0912.4551](http://arxiv.org/abs/0912.4551) [doi](http://dx.doi.org/10.1016/j.jalgebra.2010.12.009)

[[!redirects quantum heaps]]


quantum homogeneous space

## Idea

There are many variants of a generalization of the concept of [[homogeneous space]] to [[noncommutative geometry]]. Some of them are sometimes referred to as quantum homogeneous spaces. Variants of [[quantum flag manifold]]s, are here the main examples in several different frameworks.

## Homogeneous spaces for coactions of Hopf algebras

Most often here one considers the homogeneous spaces for [[Hopf algebras]] (viewed as [[quantum group]]s).

Given a Hopf algebra $H$ and a right $H$-comodule $(E,\rho)$ (where $\rho:E\to E\otimes H$ is a right coaction), a subspace $A\subset E$ is __coinvariant subspace__ for coaction $\rho$ if $\rho(A)\subset A\otimes H$. If $(E,\rho)$ is a $H$-[[comodule algebra]], viewed as a (formal dual to) noncommutative $Spec H$-space, one can consider its coinvariant subalgebras. If $H = E$ and $\rho = \Delta$ then the coinvariant subalgebras represent (the formal duals of) quantum homogenous spaces of the (formal dual of) $H$.

For example, a [[quotient Hopf algebra]] $B$ of $H$, with quotient map $\pi: H\to B$, can be considered as representing a quantum subgroup. Then $B$ coacts on $H$ by map $(id\circ\pi)\circ\Delta: H\to H\otimes B$. The space of $B$-[[coinvariant]]s gives an example of a coinvariant subalgebra in $H$.

## Literature

* [[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)
* [[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); _Some equivariant constructions in noncommutative algebraic geometry_, Georgian Mathematical Journal __16__ (2009), No. 1, 183–202, [arXiv:0811.4770](http://arxiv.org/abs/0811.4770) [MR2011b:14004](http://www.ams.org/mathscinet-getitem?mr=2527623)
* A. U. Klimyk, K. Schmuedgen, _Quantum groups and their representations_, Springer 1997.
* H-J. Schneider, _Principal homogeneous spaces for arbitrary Hopf algebras_,
Israel J. Math. __72__ (1990), no. 1-2, 167–195 [MR92a:16047](http://www.ams.org/mathscinet-getitem?mr=1098988) [doi](http://dx.doi.org/10.1007/BF02764619)
* K. De Commer, M. Yamashita, _Tannaka-Krein duality for compact quantum homogeneous spaces. I. General theory_, [arxiv/1211.6552](http://arxiv.org/abs/1211.6552)
* Pawel Kasprzak, _On a certain approach to quantum homogeneous spaces_, Comm. Math. Phys., [arxiv/1007.2438](http://arxiv.org/abs/1007.2438)

category: noncommutative geometry
[[!redirects quantum homogeneous spaces]]


quantum information

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### Context
#### Quantum systems
+--{: .hide}
[[!include quantum systems -- contents]]
=--
#### Computation
+-- {: .hide}
[[!include constructivism - contents]]
=--
=--
=--

> under construction

#Contents#
* table of contents
{:toc}

## Idea

Quantum information refers to data that can be physically stored in a [[quantum mechanics|quantum system]].

Quantum information theory is the study of how such information can be encoded, measured, and manipulated. A notable sub-field is _quantum computation_, a term often used synonymously with quantum information theory, which studies protocols and algorithms that use quantum systems to perform computations.

Categorical quantum information refers to a program in which the cogent aspects of [[Hilbert space]]-based quantum information theory are abstracted to the level of [[symmetric monoidal categories]].

## Quantum protocols and algorithms

Brief synopsis of teleportation, [[entanglement]] swapping, BB84, E91, Deutsch-Jozsa, Shor should go here...

## Category-theoretic formulation

There is a formulation of (aspects of) [[quantum mechanics in terms of dagger-compact categories]]. This lends itself to (and is in fact motivated by) a discussion of quantum information.

The linear adjoint $(-)^\dagger$ gives Hilbert spaces the structure of a [[†-category]]. The category of Hilb of Hilbert spaces forms a **†-symmetric monoidal category**, that is, a [[symmetric monoidal category]] equipped with a symmetric monoidal functor $(-)^\dagger$ from $Hilb^{op}$ to $Hilb$. Furthermore, the category FHilb of _finite dimensional_ Hilbert spaces forms a **[[dagger-compact category|†-compact closed category]]**, or a [[compact closed category]] such that $A_*$ := $(A^*)^\dagger = (A^\dagger)^*$ and $(\eta_A)^\dagger = \epsilon_{A^*}$.

### Graphical notation

Graphical notation via [[Penrose notation]]/[[string diagrams]]/[[tensor networks]]:

Morphisms in a [[monoidal category]] (and 2-categories in general) are inherently two dimensional, where $\circ$ is _vertical_ composition and $\otimes$ is _horizontal_ composition. These satisfy an [[interchange law]]:

\[ (f_1 \otimes f_2) \circ (g_1 \otimes g_2) = (f_1 \circ g_1) \otimes (f_2 \circ g_2) \]

So, if we think of these four morphisms as occupying a spot in 2 dimensional space:

+--{: .query}
[[Aleks Kissinger]]: TODO: figure
=--

we realize that the bracketing from above is essentially meaningless syntax. This notion is the guiding concept for the graphical notation of monoidal categories, or string diagrams. In this notation, we represent objects $A,B$ as directed strings and arrows $f : A \rightarrow B$ as boxes.

<svg version="1.1" id="Layer_1" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink" x="0px" y="0px"
	 width="60px" height="109px" viewBox="-39 3 60 109" enable-background="new -39 3 60 109" xml:space="preserve">
<line fill="none" stroke="#000000" stroke-width="2" x1="-9.5" y1="12" x2="-9.5" y2="102"/>
<text transform="matrix(1 0 0 1 -0.5 30)" font-family="Helvetica" font-size="12">A</text>
<text transform="matrix(1 0 0 1 -0.5 93)" font-family="Helvetica" font-size="12">B</text>
<rect x="-27.5" y="39" fill="#CCCCEE" stroke="#000000" stroke-width="2" width="36" height="36"/>
<polyline fill="none" stroke="#000000" stroke-width="2" points="-14,22.5 -9.5,27 -5,22.5 "/>
<polyline fill="none" stroke="#000000" stroke-width="2" points="-14,86 -9.5,90.5 -5,86 "/>
<text transform="matrix(1 0 0 1 -14.4805 66.0322)" font-family="Helvetica" font-style="italic" font-size="24">f</text>
</svg>

We represent the tensor product as juxtaposition:

<svg version="1.1" id="Layer_1" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink" x="0px" y="0px"
	 width="234px" height="108px" viewBox="-60 0 174 108" xml:space="preserve">
<line fill="none" stroke="#000000" stroke-width="2" x1="62.5" y1="12" x2="62.5" y2="102"/>
<text transform="matrix(1 0 0 1 71.5 30)" font-family="'Helvetica'" font-size="12">A</text>
<text transform="matrix(1 0 0 1 71.5 93)" font-family="'Helvetica'" font-size="12">B</text>
<rect x="44.5" y="39" fill="#CECBE6" stroke="#000000" stroke-width="2" width="36" height="36"/>
<polyline fill="none" stroke="#000000" stroke-width="2" points="58,22.5 62.5,27 67,22.5 "/>
<polyline fill="none" stroke="#000000" stroke-width="2" points="58,86 62.5,90.5 67,86 "/>
<text transform="matrix(1 0 0 1 57.5195 66.0322)" font-family="Helvetica" font-style="italic" font-size="24">f</text>
<line fill="none" stroke="#000000" stroke-width="2" x1="116.5" y1="12" x2="116.5" y2="102"/>
<text transform="matrix(1 0 0 1 125.5 30)" font-family="'Helvetica'" font-size="12">C</text>
<text transform="matrix(1 0 0 1 125.5 93)" font-family="'Helvetica'" font-size="12">D</text>
<rect x="98.5" y="39" fill="#99CCCC" stroke="#000000" stroke-width="2" width="36" height="36"/>
<polyline fill="none" stroke="#000000" stroke-width="2" points="112,22.5 116.5,27 121,22.5 "/>
<polyline fill="none" stroke="#000000" stroke-width="2" points="112,86 116.5,90.5 121,86 "/>
<text transform="matrix(1 0 0 1 108.5195 63.0322)" font-family="Helvetica" font-style="italic" font-size="24">g</text>
<line fill="none" stroke="#000000" stroke-width="2" x1="-63.5" y1="12" x2="-63.5" y2="102"/>
<text transform="matrix(1 0 0 1 -54.5 30)" font-family="'Helvetica'" font-size="12">A</text>
<text transform="matrix(1 0 0 1 -54.5 93)" font-family="'Helvetica'" font-size="12">B</text>
<rect x="-81.5" y="39" fill="#CECBE6" stroke="#000000" stroke-width="2" width="36" height="36"/>
<polyline fill="none" stroke="#000000" stroke-width="2" points="-68,22.5 -63.5,27 -59,22.5 "/>
<polyline fill="none" stroke="#000000" stroke-width="2" points="-68,86 -63.5,90.5 -59,86 "/>
<text transform="matrix(1 0 0 1 -68.4805 66.0322)" font-family="Helvetica" font-style="italic" font-size="24">f</text>
<line fill="none" stroke="#000000" stroke-width="2" x1="-0.5" y1="12" x2="-0.5" y2="102"/>
<text transform="matrix(1 0 0 1 8.5 30)" font-family="'Helvetica'" font-size="12">C</text>
<text transform="matrix(1 0 0 1 8.5 93)" font-family="'Helvetica'" font-size="12">D</text>
<rect x="-18.5" y="39" fill="#99CCCC" stroke="#000000" stroke-width="2" width="36" height="36"/>
<polyline fill="none" stroke="#000000" stroke-width="2" points="-5,22.5 -0.5,27 4,22.5 "/>
<polyline fill="none" stroke="#000000" stroke-width="2" points="-5,86 -0.5,90.5 4,86 "/>
<text transform="matrix(1 0 0 1 -8.4805 63.0322)" font-family="Helvetica" font-style="italic" font-size="24">g</text>
<text transform="matrix(1 0 0 1 25.1362 62.2832)" font-family="Helvetica" font-size="18">=</text>
<text transform="matrix(1 0 0 1 -39.5469 62.1729)" font-family="Helvetica" font-size="17.9327">⊗</text>
</svg>

and composition as graph composition:

<svg xmlns:svg="http://www.w3.org/2000/svg" xmlns="http://www.w3.org/2000/svg" version="1.1" width="207" height="189" viewBox="-60 0 153.92308 189" id="Layer_1" xml:space="preserve"><defs id="defs3013"/>
<line x1="12.46154" y1="45" x2="12.46154" y2="135" id="line2979" style="fill:none;stroke:#000000;stroke-width:2"/>
<text x="21.46154" y="63" id="text2981" style="font-size:12px;font-family:Helvetica">A</text>
<text x="21.46154" y="126" id="text2983" style="font-size:12px;font-family:Helvetica">B</text>
<rect width="36" height="36" x="-5.5384598" y="72" id="rect2985" style="fill:#cecbe6;stroke:#000000;stroke-width:2"/>
<polyline points="-68,22.5 -63.5,27 -59,22.5 " id="polyline2987" style="fill:none;stroke:#000000;stroke-width:2" transform="translate(75.96154,33)"/>
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<text x="7.481041" y="99.032204" id="text2991" style="font-size:24px;font-style:italic;font-family:Helvetica">f</text>
<line x1="-50.53846" y1="45" x2="-50.53846" y2="135" id="line2993" style="fill:none;stroke:#000000;stroke-width:2"/>
<text x="-41.53846" y="63" id="text2995" style="font-size:12px;font-family:Helvetica">B</text>
<text x="-41.53846" y="126" id="text2997" style="font-size:12px;font-family:Helvetica">C</text>
<rect width="36" height="36" x="-68.53846" y="72" id="rect2999" style="fill:#99cccc;stroke:#000000;stroke-width:2"/>
<polyline points="-5,22.5 -0.5,27 4,22.5 " id="polyline3001" style="fill:none;stroke:#000000;stroke-width:2" transform="translate(-50.03846,33)"/>
<polyline points="-5,86 -0.5,90.5 4,86 " id="polyline3003" style="fill:none;stroke:#000000;stroke-width:2" transform="translate(-50.03846,33)"/>
<text x="-58.518959" y="96.032204" id="text3005" style="font-size:24px;font-style:italic;font-family:Helvetica">g</text>
<text x="42" y="95.283203" id="text3007" style="font-size:18px;font-family:Helvetica">=</text>
<text x="-24.585361" y="103.1729" id="text3009" style="font-size:36px;font-family:Helvetica"><tspan id="tspan2874" style="font-size:36px">◦</tspan></text>
<text x="93.46154" y="36" id="text2897" style="font-size:12px;font-family:Helvetica">A</text>
<text x="93.46154" y="99" id="text2899" style="font-size:12px;font-family:Helvetica">B</text>
<polyline transform="translate(147.96154,6)" style="fill:none;stroke:#000000;stroke-width:2" id="polyline2903" points="-68,22.5 -63.5,27 -59,22.5 "/><polyline transform="translate(147.96154,6)" style="fill:none;stroke:#000000;stroke-width:2" id="polyline2905" points="-68,86 -63.5,90.5 -59,86 "/><line style="fill:none;stroke:#000000;stroke-width:2;stroke-miterlimit:4;stroke-dasharray:none" id="line2909" y2="171" x2="84.46154" y1="18" x1="84.46154"/><text x="93.46154" y="162" id="text2913" style="font-size:12px;font-family:Helvetica">C</text>
<rect width="36" height="36" x="66.46154" y="108" id="rect2915" style="fill:#99cccc;stroke:#000000;stroke-width:2"/><polyline transform="translate(84.96154,69)" style="fill:none;stroke:#000000;stroke-width:2" id="polyline2919" points="-5,86 -0.5,90.5 4,86 "/><text x="76.481041" y="132.0322" id="text2921" style="font-size:24px;font-style:italic;font-family:Helvetica">g</text>
<rect width="36" height="36" x="66.46154" y="45" id="rect2901" style="fill:#cecbe6;stroke:#000000;stroke-width:2"/><text x="79.481041" y="72.032204" id="text2907" style="font-size:24px;font-style:italic;font-family:Helvetica">f</text>
</svg>

That is, we perform a pushout along the common edge in the category of typed graphs with boundaries. Consider the interchange law from above, but replacing some of the arrows with identities.

\[
(f \otimes 1_D) \circ (1_A \otimes g) = (f \circ 1_A) \otimes (1_D \circ g) =
(1_B \circ f) \otimes (g \circ 1_C) = (1_B \otimes g) \circ (f \otimes 1_C)
\]

Graphically, this means we can "slide boxes" past each other.

<svg xmlns:svg="http://www.w3.org/2000/svg" xmlns="http://www.w3.org/2000/svg" version="1.1" width="243" height="189" viewBox="-60 0 180.69231 189" id="Layer_1" xml:space="preserve"><defs id="defs3013"/>
<line x1="-10.153852" y1="18" x2="-10.153852" y2="171" id="line2854" style="fill:none;stroke:#000000;stroke-width:2;stroke-miterlimit:4;stroke-dasharray:none"/><text x="-46.153854" y="36" id="text2897" style="font-size:12px;font-family:Helvetica">A</text>
<text x="-46.153854" y="99" id="text2899" style="font-size:12px;font-family:Helvetica">B</text>
<polyline transform="translate(8.3461471,6)" style="fill:none;stroke:#000000;stroke-width:2" id="polyline2903" points="-68,22.5 -63.5,27 -59,22.5 "/><polyline transform="translate(8.3461471,6)" style="fill:none;stroke:#000000;stroke-width:2" id="polyline2905" points="-68,86 -63.5,90.5 -59,86 "/><line style="fill:none;stroke:#000000;stroke-width:2;stroke-miterlimit:4;stroke-dasharray:none" id="line2909" y2="171" x2="-55.153854" y1="18" x1="-55.153854"/><text x="-1.1538526" y="162" id="text2913" style="font-size:12px;font-family:Helvetica">D</text>
<rect width="36" height="36" x="-28.153852" y="108" id="rect2915" style="fill:#99cccc;stroke:#000000;stroke-width:2"/><polyline transform="translate(-9.6538529,69)" style="fill:none;stroke:#000000;stroke-width:2" id="polyline2919" points="-5,86 -0.5,90.5 4,86 "/><text x="-18.134352" y="132.0322" id="text2921" style="font-size:24px;font-style:italic;font-family:Helvetica">g</text>
<rect width="36" height="36" x="-73.153854" y="45" id="rect2901" style="fill:#cecbe6;stroke:#000000;stroke-width:2"/><text x="-60.134354" y="72.032204" id="text2907" style="font-size:24px;font-style:italic;font-family:Helvetica">f</text>
<text x="-1.1538526" y="99" id="text2856" style="font-size:12px;font-family:Helvetica">C</text>
<polyline points="-68,86 -63.5,90.5 -59,86 " id="polyline2858" style="fill:none;stroke:#000000;stroke-width:2" transform="translate(53.346147,6)"/><text x="23.846148" y="106" id="text2860" xml:space="preserve" style="font-size:32px;font-style:normal;font-weight:normal;fill:#000000;fill-opacity:1;stroke:none;font-family:Bitstream Vera Sans"><tspan x="23.846148" y="106" id="tspan2862" style="font-size:24px">=</tspan></text>
<line style="fill:none;stroke:#000000;stroke-width:2;stroke-miterlimit:4;stroke-dasharray:none" id="line2864" y2="171" x2="115.84615" y1="18" x1="115.84615"/><text x="79.846153" y="99" id="text2866" style="font-size:12px;font-family:Helvetica">A</text>
<text x="79.846153" y="162" id="text2868" style="font-size:12px;font-family:Helvetica">B</text>
<polyline points="-68,22.5 -63.5,27 -59,22.5 " id="polyline2870" style="fill:none;stroke:#000000;stroke-width:2" transform="translate(134.34616,69)"/><polyline points="-68,86 -63.5,90.5 -59,86 " id="polyline2872" style="fill:none;stroke:#000000;stroke-width:2" transform="translate(134.34616,69)"/><line x1="70.846146" y1="18" x2="70.846146" y2="171" id="line2874" style="fill:none;stroke:#000000;stroke-width:2;stroke-miterlimit:4;stroke-dasharray:none"/><text x="124.84615" y="99" id="text2876" style="font-size:12px;font-family:Helvetica">D</text>
<rect width="36" height="36" x="97.846153" y="45" id="rect2878" style="fill:#99cccc;stroke:#000000;stroke-width:2"/><polyline points="-5,86 -0.5,90.5 4,86 " id="polyline2880" style="fill:none;stroke:#000000;stroke-width:2" transform="translate(116.34616,6)"/><text x="107.86565" y="69.032196" id="text2882" style="font-size:24px;font-style:italic;font-family:Helvetica">g</text>
<rect width="36" height="36" x="52.846153" y="108" id="rect2884" style="fill:#cecbe6;stroke:#000000;stroke-width:2"/><text x="65.865654" y="135.0322" id="text2886" style="font-size:24px;font-style:italic;font-family:Helvetica">f</text>
<text x="124.84615" y="36" id="text2888" style="font-size:12px;font-family:Helvetica">C</text>
<polyline transform="translate(179.34616,-57)" style="fill:none;stroke:#000000;stroke-width:2" id="polyline2890" points="-68,86 -63.5,90.5 -59,86 "/></svg>

## Extensions

CPM, classical structures, ...

## Related entries

* [[quantum logic]], [[quantum theory]]

* [[quantum state preparation]], [[quantum measurement]]

* [[quantum gate]], [[quantum circuit]]

* [[quantum probability]], [[quantum mechanics]]

* [[quantum communication]]

* [[quantum computation]], [[quantum cryptography]]

* [[quantum sensing]]

* [[qbit]], [[quantum circuit model]]

* [[quantum teleportation]]

* [[quantum error correcting code]]

* [[information theory]], [[information geometry]]

* [[Kochen-Specker theorem]]

* [[MIP-star=RE]]

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

## References

### General

Textbook accounts:

* [[Michael A. Nielsen]], [[Isaac L. Chuang]], *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]]]

* [[Stephen Barnett]], *Quantum Information*, Oxford University Press (2009) [[ISBN:9780198527633](https://global.oup.com/ukhe/product/quantum-information-9780198527633)]

* [[Benjamin Schumacher]], [[Michael Westmoreland]],  *Quantum Processes, Systems, and Information*, Cambridge University Press (2010) [[doi:10.1017/CBO9780511814006](https://doi.org/10.1017/CBO9780511814006)]

* [[Mark M. Wilde]], *Quantum Information Theory*, Cambridge University Press (2013) [[doi:10.1017/CBO9781139525343](https://doi.org/10.1017/CBO9781139525343), [arXiv:1106.1445](https://arxiv.org/abs/1106.1445)]

* [[Sumeet Khatri]], [[Mark M. Wilde]], _Principles of Quantum Communication Theory: A Modern Approach_ [[arXiv:2011.04672](https://arxiv.org/abs/2011.04672)]

* [[Masahito Hayashi]], _Quantum information theory - mathematical foundation_, Graduate Texts in Physics, Springer (2017) [[doi:10.1007/978-3-662-49725-8](https://doi.org/10.1007/978-3-662-49725-8)]

* [[Giuliano Benenti]], [[Giulio Casati]], [[Davide Rossini]], *Principles of Quantum Computation and Information*, World Scientific 2018  ([doi:10.1142/10909](https://doi.org/10.1142/10909), [2004 pdf](http://mmrc.amss.cas.cn/tlb/201702/W020170224608149307696.pdf))

* [[John Watrous]], *The Theory of Quantum Information*, Cambridge University Press (2018) [[doi:10.1017/9781316848142](https://doi.org/10.1017/9781316848142), [webpage](https://cs.uwaterloo.ca/~watrous/TQI/), [pdf](https://cs.uwaterloo.ca/~watrous/TQI/TQI.pdf)]

* [[Joseph M. Renes]], *Quantum Information Theory* (2015) [[pdf](https://edu.itp.phys.ethz.ch/hs15/QIT/renes_lecture_notes14.pdf)] De Gruyter (2022) [[doi:10.1515/9783110570250](https://doi.org/10.1515/9783110570250)]

Book collection:

* [[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)]

Lecture notes:

* {#Kuperberg05} [[Greg Kuperberg]], _A concise introduction to quantum probability, quantum mechanics, and quantum computation_ (2005) [[pdf](http://www.math.ucdavis.edu/~greg/intro-2005.pdf), [[Kuperberg-ConciseQuantum.pdf:file]]]

* Reinhard Werner, _Mathematical methods of quantum information theory_, 18 lecture course (2017) video playlist [yt](https://www.youtube.com/playlist?list=PLDfPUNusx1EoBAn8vXYjcF95R7mI_eR6o)

* [[Edward Witten]], *A Mini-Introduction To Information Theory*, Rivista Nuovo Cimento **43** (2020) 187–227 [[arXiv:1805.11965](https://arxiv.org/abs/1805.11965), [doi:10.1007/s40766-020-00004-5](https://doi.org/10.1007/s40766-020-00004-5)]

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

  *Introduction to Quantum Information Science II* (2022) [[pdf](https://www.scottaaronson.com/qisii.pdf)]

See also:

* [IQOQI Vienna](https://www.iqoqi-vienna.at/), *[List of Open Quantum Problems](https://oqp.iqoqi.oeaw.ac.at/open-quantum-problems)*

With continuous variables:

* [[Samuel L. Braunstein]], Peter van Loock, *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)]

In a context of [[quantum optics]]:

* [[Tim Byrnes]], [[Ebubechukwu Ilo-Okeke]], Section 11 of: *Quantum Atom Optics: Theory and Applications to Quantum Technology*, Cambridge University Press 2021 ([arXiv:2007.14601](https://arxiv.org/abs/2007.14601), [CUP](https://www.cambridge.org/core/books/quantum-atom-optics/2D867888B5C666D3A936F1C942C99568))

Status update:

* [[John Preskill]], *The Physics of Quantum Information*, talk at
*The Physics of Quantum Information*, 28th Solvay Conference on Physics (2022) ([arXiv:2208.08064](https://arxiv.org/abs/2208.08064))

See also:

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

* Wikipedia, *[Bures metric](https://en.wikipedia.org/wiki/Bures_metric)*

* [Quantiki](https://www.quantiki.org/) -- Quantum Information Portal and Wiki

* *Report of the Snowmass 2021 Theory Frontier Topical Group on Quantum Information Science* [[arXiv:2209.14839](https://arxiv.org/abs/2209.14839)]

Further original articles:

* Carmen Maria Constantin, _Sheaf-theoretic methods in quantum mechanics and quantum information theory_, PhD thesis, Oxford 2015 [arxiv/1510.02561](https://arxiv.org/abs/1510.02561)

* [[Samson Abramsky]], Adam Brandenburger, _The sheaf-theoretic structure of nonlocality and contextuality_, [arxiv/1102.0264](https://arxiv.org/abs/1102.0264)

* Dominik Šafránek, _Simple expression for the quantum Fisher information matrix), Phys. Rev. A97 (2018) [doi](https://doi.org/10.1103/PhysRevA.97.042322)

* [[Roman Orus]], _Entanglement, quantum phase transitions and quantum algorithms_ ([arXiv:quant-ph/0608013](https://arxiv.org/abs/quant-ph/0608013))

> In Chapter 1 we consider the irreversibility of renormalization group flows from a quantum information perspective by using majorization theory and conformal field theory.

Quantum information in relation to the [[representation theory of the symmetric group]]:

* [[Alonso Botero]], *Quantum Information and the Representation Theory of the Symmetric Group*, Rev. colomb. mat. vol.50 no. 2 Bogotá July/Dec. 2016 ([doi:10.15446/recolma.v50n2.62210](https://doi.org/10.15446/recolma.v50n2.62210), [pdf](http://www.scielo.org.co/pdf/rcm/v50n2/v50n2a05.pdf))

In relation to [[topological phases of matter]]:

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

  *[[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)$]$

In relation to the [[AdS-CFT correspondence]] via [[holographic entanglement entropy]]:

* [[Bowen Chen]], [[Bartlomiej Czech]], [[Zi-zhi Wang]], *Quantum Information in Holographic Duality*, Rept. Prog. Phys. **85** (2022) 4, 046001 [[arXiv:2108.09188](https://arxiv.org/abs/2108.09188), [doi:10.1088/1361-6633/ac51b5](https://doi.org/10.1088/1361-6633/ac51b5)]

### Quantum resources
 {#ReferencesQuantumResources}

On ([[quantum entanglement|entangled]]) [[quantum states]] as *resources*, not unlike the idea of resources in [[linear logic]]:

* [[Charles H. Bennett]], *A resource-based view of quantum information*, Quantum Information & Computation **4** 6 (2004) 460–466 [[doi:10.5555/2011593.2011598](https://dl.acm.org/doi/abs/10.5555/2011593.2011598)]

* Igor Devetak; Aram W. Harrow; [[Andreas J. Winter]], *A Resource Framework for Quantum Shannon Theory*, IEEE Transactions on Information Theory **54** 10 (2008) [[doi:10.1109/TIT.2008.928980](https://doi.org/10.1109/TIT.2008.928980)]

* Eric Chitambar, Gilad Gour, *Quantum Resource Theories* [[arXiv:1806.06107](https://arxiv.org/abs/1806.06107)]

[[!include quantum information theory via string diagrams -- references]]

[[!redirects quantum information theory]]

[[!redirects quantum information science]]