Contents

# 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 $\sqrt{-1}$ 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 $h$; some quantum systems with spatial interpretation in the limit $h\to 0$ 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 $(A, (-)^\ast)$ over the complex numbers. The category of of quantum mechanical systems is the opposite category of $\ast$-algebras:

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

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:

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

$(a,b) := a b + b a \,.$

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

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

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

###### Definition

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

• an observable is an element $a \in A$ such that $a^\ast = a$;

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

###### Remark

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

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

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

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

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

(…details…)

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

### Spaces of states

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

$\rho(A) = \langle \sqrt\rho, a \sqrt \rho\rangle \,.$

If the star algebra $A$ 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 $|\psi\rangle$ represents a state and $\langle\psi|$ 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 $\rho$, state evolution as maps of the form $\rho \mapsto U^\dagger \rho U$ for unitary maps $U$, and measurements are positive operator-valued measures (POVM’s). There is a natural embedding of pure states into the space of density matrices: $|\psi\rangle \mapsto |\psi\rangle\langle\psi|$. So, one way to think of mixed states is a probabilistic mixture of pure states.

$\rho = \sum_i a_i |\psi_i\rangle\langle\psi_i|$

Composite systems are formed by taking the tensor product of Hilbert spaces. If a pure state $|\Psi\rangle \in H_1 \otimes H_2$ can be written as $|\psi_1\rangle \otimes |\psi_2\rangle$ for $|\psi_i\rangle \in H_i$ it is said to be separable. If no such $|\psi_i\rangle$ exist, $|\Psi\rangle$ 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 $a \in A$ by

$\frac{d}{d \lambda} b_\lambda = \frac{1}{i \hbar}[b_\lambda, a] \,.$

In a non-relativistic system one specifies an observable $H$ – 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 $X$ as being $(0+1)$-dimensional quantum field theory:

the fields of this system are maps $\Sigma \to X$ where $\Sigma \in Riem Bord_1$ 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

$U(-) : Riem Bord_1 \to Hilb$

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 $L_2$-sections (with respect to a Riemannian metric on $X$) of the background gauge field on $X$ under which the particle in question is charged

• to the cobordism of Riemannian length $t$ the operator

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

where $H$ is the Hamiltonian operator, typically of the form $H = \nabla^\dagger \circ \nabla$ 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 $X$ 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 $\mathcal{A}$ an algebra describing a quantum system, def. , a subsystem is a subalgebra (a subobject) $B \hookrightarrow \mathcal{A}$.

Two subsystems $B_1, B_2 \hookrightarrow \mathcal{A}$ are called independent subsystems if the linear map

$B_1 \otimes B_2 \to \mathcal{A}$
$(b_1, b_2) \mapsto b_1 \cdot b_2$

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

$B_1 \otimes B_2 \stackrel{\simeq}{\to} B_1 \vee B_2 \hookrightarrow \mathcal{A}$

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

See for instance (BrunettiFredenhagen, section 5.2.2).

###### Definition

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

$(\rho_1 \otimes \rho_2) : (b_1 , b_2) \mapsto \rho_1(b_1) \rho_2(b_2) \,.$
###### Definition

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

## Quantum physics in general

More generally, quantum physics is all the known physics not including classical physics in wider sense; it includes relativistic and nonrelativistic phenomena. Quantum mechanics is the standard formalism with the Hilbert space, unitary evolution etc. which explains theoretically phenomena of quantum physics: in this generality of the formalism a la von Neumann, it includes the quantum field theory. Quantum statistical mechanics in fact uses some additional assumptions not in exact quantum mechanics, which are believed to be derivable eventually (like quantum ergodicity etc.). Thus quantum statistical mechanics may or may not be included within quantum mechanics.

Remark: Another way to look at quantum processes is via quantum channels which are completely positive trace-preserving maps.

For a previous query about quantum physics (includes experimental phenomena) and quantum mechanics (formalism for such, sometimes with or without statistical principles) see here.

## Formulations and formalization

### 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

## References

Classical textbooks (on the Hilbert space description) include

• John von Neumann, Mathematische Grundlagen der Quantenmechanik.

(German) Mathematical Foundations of Quantum Mechanics. Berlin, Germany: Springer Verlag, 1932.

• George Mackey, The Mathematical Foundations of Quamtum Mechanics A

Lecture-note Volume, ser. The mathematical physics monograph series. Princeton university, 1963

• E. Prugovecki, Quantum mechanics in Hilbert Space. Academic Press, 1971.

• Paul Dirac, The Principles of Quantum Mechanics, International series of monographs on physics, Oxford University Press, 1978

• Anthony Sudbery, Quantum mechanics and the particles of nature: an outline for mathematicians

Modern textbooks:

Lecture notes include

Some standard references include

Dicussion of quantum measurement is in

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

• Jonathan Gleason, The $C^*$-algebraic formalism of quantum mechanics, 2009 (pdf, pdf)

• Jürg Fröhlich, B. Schubnel, Quantum Probability Theory and the Foundations of Quantum Mechanics. In: Blanchard P., Fröhlich J. (eds.) The Message of Quantum Science. Lecture Notes in Physics, vol 899. Springer 2015 (arXiv:1310.1484, doi:10.1007/978-3-662-46422-9_7)

• Jürg Fröhlich, The structure of quantum theory, Chapter 6 in The quest for laws and structure, EMS 2016 (doi, doi:10.4171/164-1/8)

• Klaas Landsman, Foundations of quantum theory – From classical concepts to Operator algebras, Springer Open 2017 (pdf)

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.

Last revised on May 6, 2021 at 04:26:00. See the history of this page for a list of all contributions to it.