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quantum mechanics (changes) in nLab

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Context

Quantum systems

quantum logic

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quantum physics

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quantum probability theory – observables and states

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quantum information

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quantum technology

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quantum computing

Contents

Idea

While classical mechanics considers deterministic evolution of particles and fields, quantum physics follows nondeterministic evolution where the probability of various outcomes of measurement may be predicted from the state in a Hilbert space representing the possible reality: that state undergoes a unitary evolution, what means that the generator of the evolution is −1\sqrt{-1} times a Hermitean operator called the quantum Hamiltonian or the Hamiltonian operator of the system. The theoretical framework for describing this precisely is the quantum mechanics. It involves a constant of nature, Planck constant hh; some quantum systems with spatial interpretation in the limit h→0h\to 0 lead to classical mechanical systems (not all: some phenomena including non-integer spin are purely quantum mechanical, but the properties depending on their existence survive in the “classical” limit); in limited generality, one can motivate and find the nonfunctorial procedure to single out a right inverse to taking this classical limit under the name quantization.

While quantum mechanics may be formulated for a wide range of physical systems, interpreted as particles, extended particles and fields, the quantum mechanics of fields is often called the quantum field theory and the quantum mechanics of systems of a fixed finite number of particles is often viewed as the quantum mechanics in a narrow sense.

nPOV

Mathematically, despite the basic formalism of quantum mechanics which is sound and clear, there are two big areas which are yet not clear. One is to understand quantization, in all cases – of particles, fields, strings and so on. The second and possibly more central to nLab is a problem how to define rigorously a wide range of quantum field theories and some related quantum mechanical systems like the hypothetical superstring theory. Regarding that this is a central goal, we also put emphasis on the interpretation of quantum mechanics via the picture which is a special case of a FQFT, and where the time evolution functorially leads to evolution operators.

Definition

We discuss some basic notions of quantum mechanics.

Quantum mechanical systems

Recall the notion of a classical mechanical system: the formal dual of a real commutative Poisson algebra.

Definition

A quantum mechanical system is a star algebra (A,(−) *)(A, (-)^\ast) over the complex numbers. The category of of quantum mechanical systems is the opposite category of *\ast-algebras:

QuantMechSys:=*Alg ℂ op. QuantMechSys := {\ast}Alg_{\mathbb{C}}^{op} \,.

Observables and states

Definition

Given a quantum mechanical system in terms of a star algebra AA, we say

  • an observable is an element a∈Aa \in A such that a *=aa^\ast = a;

  • a state is a linear function ρ:A→ℂ\rho : A \to \mathbb{C} which is positive in the sense that for all a∈Aa \in A we have ρ(aa *)≥0∈ℝ↪ℂ\rho(a a^\ast) \geq 0 \in \mathbb{R} \hookrightarrow \mathbb{C}.

One also says that the internal classical mechanical system (Sh(Com(A)),A̲)(Sh(Com(A)), \underline{A}) is the “Bohrification” of the external quantum system AA. See there for more details.

Spaces of states

Given a *\ast-algebra AA together with a state ρ\rho on it, the GNS construction provides an inner product space H ρH_\rho together with an action of AA on H ρH_\rho and a vector Ω=(ρ)\Omega = \sqrt(\rho) – the vacuum vector? – such that for all a∈Aa \in A the value of the state ρ:A→ℂ\rho : A \to \mathbb{C} is obtained by applying aa to ρ\sqrt{\rho} and then taking the inner product with ρ\sqrt \rho:

ρ(A)=⟨ρ,aρ⟩. \rho(A) = \langle \sqrt\rho, a \sqrt \rho\rangle \,.

If the star algebra AA happens to be a C-star algebra, then this inner product space is naturally a Hilbert space.

Historically and still often in the literature, such a Hilbert space is taken as a fundamental input of the definition of quantum systems.

Traditionally, Dirac’s “bra-ket” notation is used to represent vectors in such Hilbert spaces of states, where |ψ⟩|\psi\rangle represents a state and ⟨ψ|\langle\psi| represents its linear adjoint. State evolutions are expressed as unitary maps. Self-adjoint operators represent physical quantities such as position and momentum and are called observables. Measurements are expressed as sets of projectors onto the eigenvectors of an observable.

In mixed state quantum mechanics, physical states are represented as density operators ρ\rho, state evolution as maps of the form ρ↦U †ρU\rho \mapsto U^\dagger \rho U for unitary maps UU, and measurements are positive operator-valued measures (POVM’s). There is a natural embedding of pure states into the space of density matrices: |ψ⟩↦|ψ⟩⟨ψ||\psi\rangle \mapsto |\psi\rangle\langle\psi|. So, one way to think of mixed states is a probabilistic mixture of pure states.

(1)ρ=∑ ia i|ψ i⟩⟨ψ i| \rho = \sum_i a_i |\psi_i\rangle\langle\psi_i|

Composite systems are formed by taking the tensor product of Hilbert spaces. If a pure state |Ψ⟩∈H 1⊗H 2|\Psi\rangle \in H_1 \otimes H_2 can be written as |ψ 1⟩⊗|ψ 2⟩|\psi_1\rangle \otimes |\psi_2\rangle for |ψ i⟩∈H i|\psi_i\rangle \in H_i it is said to be separable. If no such |ψ i⟩|\psi_i\rangle exist, |Ψ⟩|\Psi\rangle is said to be entangled. If a mixed state is separable if it is the sum of separable pure states. Otherwise, it is entangled.

Flows and time evolution

As for classical mechanics, 1-parameter families of flows in a quantum mechanical system are induced from observables a∈Aa \in A by

ddλb λ=1iℏ[b λ,a]. \frac{d}{d \lambda} b_\lambda = \frac{1}{i \hbar}[b_\lambda, a] \,.

In a non-relativistic system one specifies an observable HH – called the Hamiltonian – whose flow represents the time evolution of the system. (This is the Heisenberg picture.)

We comment on how to interpret this from the point of view of FQFT:

Quantum mechanics of point particles may be understood as a special case of the formalism of quantum field theory. It is interpreted as the quantum analog of the classical mechanics of point particles. Of course, we can take a configuration space of a system of particles looking like the configuration space of a single particle in a higher dimensional manifold.

Remark: related query on the relation between QFT and quantum mechanics (of particles and in general) can be found here.

One may usefully think of the quantum mechanics of a point particle propagating on a manifold XX as being (0+1)(0+1)-dimensional quantum field theory:

the fields of this system are maps Σ→X\Sigma \to X where Σ∈RiemBord 1\Sigma \in Riem Bord_1 are 1-dimensional Riemannian manifold cobordisms. These are the trajectories of the particle.

After quantization this yields a 1-dimensional FQFT given by a functor

U(−):RiemBord 1→Hilb U(-) : Riem Bord_1 \to Hilb

from cobordisms to Hilbert spaces (or some other flavor of vector spaces) that assigns

  • to the point the space of states ℋ\mathcal{H}, typically the space of L 2L_2-sections (with respect to a Riemannian metric on XX) of the background gauge field on XX under which the particle in question is charged

  • to the cobordism of Riemannian length tt the operator

    U(t):=exp(tiℏH):ℋ→ℋ, U(t) := \exp\left(\frac{t}{i \hbar } H \right) : \mathcal{H} \to \mathcal{H} \,,

    where HH is the Hamiltonian operator, typically of the form H=∇ †∘∇H = \nabla^\dagger \circ \nabla for ∇\nabla the covariant derivative of the given background gauge field.

Such a setup describes the quantum mechanics of a particle that feels forces of backgound gravity encoded in the Riemannian metric on XX and forces of background gauge fields (such as the electromagnetic field) encoded in the covariant derivative ∇\nabla.

(This is the Schrödinger picture.)

Quantum subsystems

Definition

For 𝒜\mathcal{A} an algebra describing a quantum system, def. 1, a subsystem is a subalgebra (a subobject) B↪𝒜B \hookrightarrow \mathcal{A}.

Two subsystems B 1,B 2↪𝒜B_1, B_2 \hookrightarrow \mathcal{A} are called independent subsystems if the linear map

B 1⊗B 2→𝒜 B_1 \otimes B_2 \to \mathcal{A}

(b 1,b 2)↦b 1⋅b 2 (b_1, b_2) \mapsto b_1 \cdot b_2

from the tensor product of algebras (the composite system) factors as an isomorphism

B 1⊗B 2→≃B 1∨B 2↪𝒜 B_1 \otimes B_2 \stackrel{\simeq}{\to} B_1 \vee B_2 \hookrightarrow \mathcal{A}

through the algebra B 1∨B 2B_1 \vee B_2 that is generated by B 1B_1 and B 2B_2 inside 𝒜\mathcal{A} (the smallest subalgebra containing both).

See for instance (BrunettiFredenhagen, section 5.2.2).

Definition

Given two independent subsystems B 1,B 2↪𝒜B_1, B_2 \hookrightarrow \mathcal{A}, and two states ρ 1:B 1→ℂ\rho_1 : B_1 \to \mathbb{C} and ρ 2:B 2→ℂ\rho_2 : B_2 \to \mathbb{C}, then the corresponding product state ρ 1⊗ρ 2\rho_1 \otimes \rho_2 on B 1∨B 2B_1 \vee B_2 is defined to be

(ρ 1⊗ρ 2):(b 1,b 2)↦ρ 1(b 1)ρ 2(b 2). (\rho_1 \otimes \rho_2) : (b_1 , b_2) \mapsto \rho_1(b_1) \rho_2(b_2) \,.

Definition

There exist states on B 1∨B 2B_1 \vee B_2 that are not (convex combinations of) product states. This phenomenon is called entanglement.

Formulations and formalization

Order-theoretic structure in quantum mechanics

See order-theoretic structure in quantum mechanics.

Quantum mechanics in terms of †\dagger-compact categories

Many aspects of quantum mechanics and quantum computation depend only on the abstract properties of Hilb characterized by the fact that it is a †-compact category.

For more on this see

Foundational theorems of quantum mechanics

The following circle of theorems

all revolve around the phenomenon that the “phase space” in quantum mechanics and hence the space of quantum states are all determined by the Jordan algebra structure on the algebra of observables, which in turn is determined by the poset of commutative subalgebras of the algebra of observables. See at order-theoretic structure in quantum mechanics for more on this.

There is also

which says roughly that linear maps between spaces of quantum states are unitary operators (or anti-unitary) already when they preserve norm, hence preserve probability.

Applications of quantum mechanics

Quantum mechanics, as opposed to classical mechanics, is necessary for an accurate description of reality whenever the characteristic scale is sufficiently small. For instance

Examples

References

Historical origins

The seed of quantum mechanics is sown in

  • Max Planck (transl. M. Martius) The Theory of Heat Radiation (1914) [[pdf](https://www.gutenberg.org/files/40030/40030-pdf.pdf)]

with the recognition of a quantum of “action”: Planck's constant (p. 164)

Quantum mechanics as such originates with:

  • Werner Heisenberg, Über quantentheoretische Umdeutung kinematischer und mechanischer Beziehungen, Zeitschrift für Physik 33 (1925) 879–893 [[doi:10.1007/BF01328377]( https://doi.org/10.1007/BF01328377), Engl. pdf]

  • Max Born, Pascual Jordan, Zur Quantenmechanik, Zeitschrift für Physik 34 (1925) 858–888 [[doi:10.1007/BF01328531](https://doi.org/10.1007/BF01328531)]

  • Paul A. M. Dirac, On the theory of quantum mechanics, Proceedings of the Royal Society 112 762 (1926) [[doi:10.1098/rspa.1926.0133](https://doi.org/10.1098/rspa.1926.0133)]

  • Paul A. M. Dirac, The physical interpretation of the quantum dynamics, Proceedings of the Royal Society of London 113 765 (1927) [[doi:10.1098/rspa.1927.0012](https://doi.org/10.1098/rspa.1927.0012)]

  • David Hilbert, John von Neumann, Lothar W. Nordheim, Über die Grundlagen der Quantenmechanik, Math. Ann. 98 (1928) 1–30 [[doi:10.1007/BF01451579](https://doi.org/10.1007/BF01451579)]

Formulating the Born rule:

  • Max Born, Zur Quantenmechanik der Stoßvorgänge, Zeitschrift für Physik 37 (1926) 863–867 [[doi:10.1007/BF01397477](https://doi.org/10.1007/BF01397477)]

  • Max Born, Quantenmechanik der Stoßvorgänge, Zeitschrift für Physik 38 (1926) 803–827 [[doi:10.1007/BF01397184](https://doi.org/10.1007/BF01397184)]

  • Max Born, Das Adiabatenprinzip in der Quantenmechanik, Zeitschrift für Physik 40 (1927) 167–192 [[doi:10.1007/BF01400360](https://doi.org/10.1007/BF01400360)]

  • Pascual Jordan, Über eine neue Begründung der Quantenmechanik, Zeitschrift für Physik 40 (1927) 809–838 [[doi:10.1007/BF01390903](https://doi.org/10.1007/BF01390903)]

Introducing the Hilbert space-formulation (and the projection postulate):

  • Von Neumann’s 1927 Trilogy on the Foundations of Quantum Mechanics (annotated translations by Anthony Duncan) [[arXiv:2406.02149](https://arxiv.org/abs/2406.02149)]

  • John von Neumann:

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

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

but see (on von Neumann’s further reasoning regarding quantum logic and then of von Neumann algebra factors):

  • Miklos Rédei, Why John von Neumann did not Like the Hilbert Space formalism of quantum mechanics (and what he liked instead), Studies in History and Philosophy of Modern Physics 27 4 (1996) 493-510 []

Equivalence of the Heisenberg picture and the Schrödinger picture:

  • Erwin Schrödinger, Über das Verhältnis der Heisenberg-Born-Jordanschen Quantenmechanik zu der meinen, Annalen der Physil 384 8 (1926) 734-756 [[doi:10.1002/andp.19263840804]( https://doi.org/10.1002/andp.19263840804)]

  • Carl Eckart, Operator Calculus and the Solution of the Equations of Quantum Dynamics, Phys. Rev. 28 4 (1926) 711-726 [[doi:10.1103/PhysRev.28.711](https://doi.org/10.1103/PhysRev.28.711)]

Introducing the tool of group theory to quantum physics (cf. Gruppenpest):

  • Hermann Weyl, Quantenmechanik und Gruppentheorie, Zeitschrift für Physik 46 (1927) 1–46 [[doi:10.1007/BF02055756](https://doi.org/10.1007/BF02055756)]

  • Hermann Weyl, Gruppentheorie und Quantenmechanik, S. Hirzel, Leipzig, (1931), translated by H. P. Robertson: The Theory of Groups and Quantum Mechanics Dover (1950) [[ISBN:0486602699](https://store.doverpublications.com/0486602699.html), ark:/13960/t1kh1w36w]

  • Eugene P. Wigner: Gruppentheorie und ihre Anwendung auf die Quantenmechanik der Atomspektren, Springer (1931) [[doi:10.1007/978-3-663-02555-9](https://doi.org/10.1007/978-3-663-02555-9), pdf]

  • Eugene P. Wigner: Group theory: And its application to the quantum mechanics of atomic spectra, 5, Academic Press (1959) [[doi:978-0-12-750550-3](https://www.elsevier.com/books/group-theory/wigner/978-0-12-750550-3)]

Early discussion of composite quantum systems and their quantum entanglement:

  • Erwin Schrödinger, Discussion of Probability Relations between Separated Systems, Mathematical Proceedings of the Cambridge Philosophical Society, 31 4 (1935) 555-563 [[doi:10.1017/S0305004100013554](https://doi.org/10.1017/S0305004100013554)]

On the historical orogin of the canonical commutation relations:

  • Severino C. Coutinho, The Many Avatars of a Simple Algebra, The American Mathematical Monthly 104 7 (1997) 593-604 [[doi:10.2307/2975052](https://doi.org/10.2307/2975052), jstor:2975052]

General

Classical textbook accounts:

More recent textbook accounts:

  • Paul Busch, Marian Grabowski, Pekka J. Lahti, Operational Quantum Physics, Lecture Notes in Physics Monographs 31, Springer (1995) [[doi:10.1007/978-3-540-49239-9](https://doi.org/10.1007/978-3-540-49239-9)]

    (perspective of quantum probability theory via POVMs)

  • Chris Isham, Lectures on Quantum Theory – Mathematical and Structural Foundations, World Scientific (1995) [[doi:10.1142/p001](https://doi.org/10.1142/p001), ark:/13960/t4xh7cs99]

  • Klaas Landsman, Mathematical Topics Between Classical and Quantum Mechanics, Springer (1998) [[doi:10.1007/978-1-4612-1680-3](https://doi.org/10.1007/978-1-4612-1680-3)]

  • Robert B. Griffiths, Consistent Quantum Theory, Cambridge University Press (2002) [[doi:10.1017/CBO9780511606052](https://doi.org/10.1017/CBO9780511606052), webpage]

  • Mikio Nakahara, Chapter 1 of: Geometry, Topology and Physics, IOP (2003) [[doi:10.1201/9781315275826](https://doi.org/10.1201/9781315275826), pdf]

  • Serge Haroche, Jean-Michel Raimond, Exploring the Quantum: Atoms, Cavities, and Photons, Oxford University Press (2006) [[doi:10.1093/acprof:oso/9780198509141.001.0001](https://doi.org/10.1093/acprof:oso/9780198509141.001.0001)]

  • Ingemar Bengtsson, Karol Życzkowski, Geometry of Quantum States — An Introduction to Quantum Entanglement, Cambridge University Press (2006) [[doi:10.1017/CBO9780511535048](https://doi.org/10.1017/CBO9780511535048)]

    (focus on the geometry of quantum state spaces and culminating in a chapter on quantum entanglement)

  • Heinz-Peter Breuer, Francesco Petruccione, The Theory of Open Quantum Systems, Oxford University Press (2007) [[doi:10.1093/acprof:oso/9780199213900.001.0001](https://doi.org/10.1093/acprof:oso/9780199213900.001.0001)]

    (focus on open quantum systems)

  • Teiko Heinosaari, Mário Ziman, The Mathematical Language of Quantum Theory – From Uncertainty to Entanglement, Cambridge University Press (2011) [[doi:10.1017/CBO9781139031103]( https://doi.org/10.1017/CBO9781139031103)]

  • Nik Weaver, Mathematical Quantization, Routledge (2011) [[ISBN 9781584880011](https://www.routledge.com/Mathematical-Quantization/Weaver/p/book/9781584880011)]

  • Thomas L. Curtright, David B. Fairlie, Cosmas K. Zachos, A Concise Treatise on Quantum Mechanics in Phase Space, World Scientific (2014) [[doi:10.1142/8870](https://doi.org/10.1142/8870)]

    (in Weyl quantization)

  • Paul Busch, Pekka J. Lahti, Juha-Pekka Pellonpää, Kari Ylinen, Quantum Measurement, Springer (2016) [[doi:10.1007/978-3-319-43389-9](https://doi.org/10.1007/978-3-319-43389-9)]

    (perspective of quantum probability via POVMs)

  • Klaas Landsman, Foundations of quantum theory – From classical concepts to Operator algebras, Springer Open (2017) [[doi:10.1007/978-3-319-51777-3](https://link.springer.com/book/10.1007/978-3-319-51777-3), pdf]

    With a focus on the relationship between quantum mechanics and representation theory

  • Peter Woit, Quantum Theory, Groups and Representations: An Introduction, Springer (2017) [[doi:10.1007/978-3-319-64612-1](https://doi.org/10.1007/978-3-319-64612-1), ISBN:978-3-319-64610-7]

On the interpretation of quantum mechanics:

Lecture notes:

Further references:

  • Sean Bates, Alan Weinstein, Lectures on the geometry of quantization AMS (1997) [[pdf](http://www.math.berkeley.edu/~alanw/GofQ.pdf)]

    (on geometric quantization)

  • Pierre Cartier, Cécile DeWitt-Morette, Functional integration: action and symmetries, Cambridge Monographs on Mathematical Physics (2006) [[ISBN:9780521143578](https://www.cambridge.org/ae/academic/subjects/physics/theoretical-physics-and-mathematical-physics/functional-integration-action-and-symmetries?format=PB#contentsTabAnchor)]

    (on rigorous path integrals)

  • Leon A. Takhtajan: Quantum mechanics for mathematicians, Amer. Math. Soc. (2008) [[ISBN:978-0-8218-4630-8](https://bookstore.ams.org/gsm-95)]

  • Michael Movshev, Concepts of Quantum Mechanics (2008) [[web](http://www.math.sunysb.edu/~mmovshev/MAT570Spring2008/syllabusfinal.html)]

  • Franco Strocchi, An introduction to the mathematical structure of quantum mechanics, Advanced Series in Mathematical Physics 28, World Scientific (2008) [[doi:10.1142/7038](https://doi.org/10.1142/7038)]

  • Steven Weinberg, Lectures on Quantum Mechanics, Cambridge University Press (2015) [[doi:10.1017/CBO9781316276105](https://doi.org/10.1017/CBO9781316276105)]

  • Valter Moretti, Spectral Theory and Quantum Mechanics – Mathematical Foundations of Quantum Theories, Symmetries and Introduction to the Algebraic Formulation, Springer (2017) [[doi:10.1007/978-3-319-70706-8](https://doi.org/10.1007/978-3-319-70706-8)]

  • Valter Moretti, Fundamental Mathematical Structures of Quantum Theory – Spectral Theory, Foundational Issues, Symmetries, Algebraic Formulation, Springer (2020) [[doi:10.1007/978-3-030-18346-2](https://doi.org/10.1007/978-3-030-18346-2)]

  • Jan Perina, Z. Hradil, Branislav Jurčo, Quantum optics and fundamentals of physics, Kluwer 1994

Introduction to mathematical foundations of quantum physics in quantum probability, operator algebra:

see also

Generalization of the algebraic perspective to quantum field theory is discussed in

for more on this see at AQFT and at perturbative AQFT

Different incarnations of this C*-algebraic locality condition are discussed in section 3 of

  • Sander Wolters, Quantum toposophy,

relating it to the topos-theoretic formulation in

Aspects of quantum mechanics in category theory and topos theory are discussed in

  • Hans Halvorson (ed.) Deep Beauty – Understanding the quantum world through mathematical innovation Cambridge (2011) (pdf)

This discusses for instance higher category theory and physics and the Bohr topos of a quantum system.

Quantum observable algebras as Groupoid algebras

The original derivation of the Heisenberg picture of quantum mechanics, introducing N×NN \times N matrix algebra for transitions between NN measurable states of atomic spectra, by

was argued by

to be fruitfully understood as the groupoid convolution algebra of the pair groupoid of transitions between NN elements.

Later but more generally, the “algebra of (selective) measurement” originally envision by

  • Julian Schwinger: Quantum Kinematics and Dynamics, CRC Press (1969, 1991) [[ISBN:9780738203034](https://www.routledge.com/Quantum-Kinematics-And-Dynamic/Schwinger/p/book/9780738203034?srsltid=AfmBOopc1mHvjXy-9P4osTWSNmgMcshwipyqnBSRWr9wbFOAe380eo8h), pdf]

  • Julian Schwinger (ed.: Berthold-Georg Englert): Quantum Mechanics – Symbolism of Atomic Measurements, Springer (2001) [[doi:10.1007/978-3-662-04589-3](https://doi.org/10.1007/978-3-662-04589-3)]

is argued to be, in modern language, the groupoid convolution algebras of groupoids whose morphisms reflect transitions between possible quantum measurement-outcomes – by:

and as such further developed in:

Proof that non-perturbative quantum observables on Yang-Mill fluxes through a surface form a convolution algebra:

Quantum information theory via String diagrams

General

The observation that a natural language for quantum information theory and quantum computation, specifically for quantum circuit diagrams, is that of string diagrams in †-compact categories (see quantum information theory via dagger-compact categories):

On the relation to quantum logic/linear logic:

Early exposition with introduction to monoidal category theory:

Review in contrast to quantum logic:

and with emphasis on quantum computation:

Generalization to quantum operations on mixed states (completely positive maps of density matrices):

Textbook accounts (with background on relevant monoidal category theory):

Measurement & Classical structures

Formalization of quantum measurement via Frobenius algebra-structures (“classical structures”):

and the evolution of the “classical structures”-monad into the “spider”-diagrams (terminology for special Frobenius normal form, originating in Coecke & Paquette 2008, p. 6, Coecke & Duncan 2008, Thm. 1) of the ZX-calculus:

ZX-Calculus

Evolution of the “classical structures”-Frobenius algebra (above) into the “spider”-ingredient of the ZX-calculus for specific control of quantum circuit-diagrams:

Relating the ZX-calculus to braided fusion categories for anyon braiding:

Last revised on February 28, 2025 at 15:14:01. See the history of this page for a list of all contributions to it.