nLab quantum mechanics



Quantum systems

quantum logic

quantum physics

quantum probability theoryobservables and states

quantum information

quantum computation


quantum algorithms:

quantum sensing

quantum communication



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 h0h\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.


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.


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.


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

It makes sense to think of this as a deformed version of a real Poisson algebra as follows:

  • the Poisson-Lie bracket of a Poisson algebra corresponds to the commutator of the *\ast-algebra:

    [a,b]:=abba, [a,b] := a b - b a \,,
  • the commutative algebra structure of the Poisson algebra coresponds to the Jordan algebra structure of the *\ast-algebra, with commutative (but non-associative!) product

    (a,b):=ab+ba. (a,b) := a b + b a \,.

With this interpretation the derivation-property of the Poisson bracket over the other product is preserved: for all a,b,cAa,b,c \in A we have

[a,(b,c)]=([a,b],c)+(b,[a,c]). [a,(b,c)] = ([a,b],c) + (b,[a,c]) \,.

We thus may regard a non-commutative star-algebra as a non-associative Poisson algebra : a Jordan-Lie algebra. See there for more details.

Observables and states


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

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

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


One can formalize the idea that a quantum mechanical system is like a deformed classical mechanical system as follows:

To every *{}^\ast-algebra AA is associated its poset of commutative subalgebras Com(A)Com(A). Then the corresponding quantum mechanical system is a classical mechanical system internal to the sheaf topos Sh(Com(A))Sh(Com(A)):

  • The *\ast-algebra canonically induces a commuative algebra A̲Sh(Com(A))\underline A \in Sh(Com(A));

  • the (classical) states of A̲\underline{A} in Sh(Com(A))Sh(Com(A)) are in natural bijection with the quantum states externally on AA;

  • the (classical) observables of A̲\underline{A} in Sh(Com(A))Sh(Com(A)) correspond to the external quantum observables on AA.


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 aAa \in A the value of the state ρ:A\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 ρU 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|ψψ||\psi\rangle \mapsto |\psi\rangle\langle\psi|. So, one way to think of mixed states is a probabilistic mixture of pure states.

ρ= ia i|ψ iψ 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 |ΨH 1H 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 iH 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 aAa \in A by

ddλb λ=1i[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 ΣX\Sigma \to X where ΣRiemBord 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 1Hilb 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 \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(tiH):, 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


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

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

B 1B 2𝒜 B_1 \otimes B_2 \to \mathcal{A}
(b 1,b 2)b 1b 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 1B 2B 1B 2𝒜 B_1 \otimes B_2 \stackrel{\simeq}{\to} B_1 \vee B_2 \hookrightarrow \mathcal{A}

through the algebra B 1B 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).


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

(ρ 1ρ 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) \,.

There exist states on B 1B 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



Historical origins

The seed of quantum mechanics is sown in

  • Max Planck (transl. M. Martius) The Theory of Heat Radiation (1914) [pdf]

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

Quantum mechanics as such originates with:

Formulating the Born rule:

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

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 [doi:10.1016/S1355-2198(96)00017-2]

Equivalence of the Heisenberg picture and the Schrödinger picture:

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

Early discussion of composite quantum systems and their quantum entanglement:

On the historical orogin of the canonical commutation relations:


Classical textbook accounts:

More recent textbook accounts:

On the interpretation of quantum mechanics:

Lecture notes:

Further references:

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 information theory via String diagrams


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:


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 March 5, 2024 at 13:14:26. See the history of this page for a list of all contributions to it.