The Hilbert space formulation of (non-relativistic) quantum mechanics is one of the great achievements of mathematical physics. Typically in undergraduate physics courses it is introduced as a set of postulates (e.g. the Dirac-von Neumann axioms) and hard to motivate without some knowledge of functional analysis or at least probability theory. Some of that motivation and the connection with probability theory is summarised in the notes here – in fact it can be said that quantum mechanics is essentially non-commutative probability theory [2]. Furthermore having an algebraic point of view seems to provide a unified picture of classical and quantum mechanics.

The important difference between classical and quantum mechanics is that in the latter, the order in which measurements are taken sometimes matters. This is because obtaining the value of one measurement can disturb the system of interest to the extent that a consistently precise value of the other cannot be found. A famous example is position and momentum of a quantum particle – the Heisenberg uncertainty relation states that the product of their uncertainties (variances) in measurement is strictly greater than zero.

If measurements are treated as real-valued functions of the state space of system, we will not be able to capture the fact that the measurements do not commute. Since linear operators (e.g. matrices) do not commute in general, we use algebras of operators instead. We make use of the spectral theory leading from a special class of algebras with norm and adjoint known as von Neumann algebras which in turn are a special case of C*-algebras. The **spectrum** of an operator A is the set of numbers for which does not have an inverse. Self-adjoint operators have a real spectrum and will represent the set of values that an observable (a physical variable that can be measured) can take. Hence we have this correspondence between self-adjoint operators and observables.

By the Gelfand-Naimark theorem C*-algebras can be represented as bounded operators on a Hilbert space . See Section II.6.4 of [**3**] for proof details. If the C*-algebra is commutative the representation is as continuous functions on a locally compact Hausdorff space that vanish at infinity. Furthermore we assume the C*-algebra and corresponding Hilbert space are separable, meaning the space contains a countable dense subset (analogous to how the subset of rationals are dense in the set of real numbers). This ensures that the Stone-von Neumann theorem holds which was used to show that the Heisenberg and Schrödinger pictures of quantum physics are equivalent [see pp7-8 here].

The link between C*-algebras and Hilbert spaces is made via the notion of a **state** which is a positive linear functional on the algebra of norm 1. A state evaluated on a self-adjoint operator outputs a real number that will represent the expected value of the observable corresponding to that operator. Note that it is impossible to have two different states that have the same expected values across over observables. A state is called **pure** if it is an extreme point on the boundary of the (convex) space of states. In other words, we cannot write a pure state as where are states and ). A state that is not pure is called **mixed**.

Now referring to a Hilbert space , for any mapping of bounded operators to expectation values such that

- (it makes sense that the identity should have expectation value 1),
- self-adjoint operators are mapped to real numbers with positive operators (those with positive spectrum) mapped to positive numbers and
- is continuous with respect to the strong convergence in – i.e. if for all , then ,

then there is a is a unique self-adjoint non-negative trace-one operator (known as a **density matrix**) such that for all (see [1] Proposition 19.9). (The trace of an operator is defined as where is an orthonormal basis in the separable Hilbert space – in the finite dimensional case it is the sum of the operator’s eigenvalues.) Hence states are represented by positive self-adjoint operators with trace 1. Such operators are compact and so have a countable orthonormal basis of eigenvectors.

When corresponds to a projection operator onto a one-dimensional subspace it has the form where and . In this case we can show , which recovers the alternative view that unit vectors of correspond to states (known as **vector states**) so that the expected value of an observable corresponding to the operator is . This is done by choosing the orthonormal basis where and computing

Trace-one operators can be written as a convex combination of rank one projection operators: . From this it can be shown that those density operators which cannot be written as a convex combination of other states (called pure states) are precisely those of the form . Hence vector states and pure states are equivalent notions. Mixed states can be interpreted as a probabilistic mixture (convex combination) of pure states.

Let us now look at the similarity with probability theory. A measure space is a triple where is a set, is a collection of measurable subsets of called a -algebra and is a additive measure. If is a non-negative integrable function with it is called a **density function** and then we can define a **probability measure** by

.

A **random variable** maps elements of a set to real numbers in such a way that for any Borel subset of . This enables us to compute their expectation with respect to the density function as

.

This is like the quantum formula with our density operator playing the role of and operator playing the role of random variable . Hence a probability density function is the commutative probability analogue of a quantum state (density operator).

While Borel sets are the **events** from which we define simple functions and then random variables, in the non-commutative case we define operators in terms of **projections** (equivalently closed subspaces) of a Hilbert space . A projection operator is self-adjoint, satisfies and has the discrete spectrum . Hence they are analogous to 0-1 indicator random variables, the answers to yes/no events. For any unit vector the expected value

is interpreted as the probability the observable corresponding to will have value 1 when measured in the state corresponding to . In particular this probability will be 1 if and only if is in the invariant subspace of . We define meet and join operations on these closed subspaces to create a **Hilbert lattice** :

Borel sets form a algebra in which the distributive law holds for any elements of . However in the Hilbert lattice the corresponding rule (where are projection operators) only holds some of the time (see here for an example). This failure of the distributive law is equivalent to the general non-commutativity of projections.

A **quantum probability measure** can be defined by combining projections in a -additive way, namely and where are mutually orthogonal projections (). **Gleason’s theorem** says that for Hilbert space dimension at least 3 a state is uniquely determined by the values it takes on the orthogonal projections – a quantum probability measure can be extended from projections to bounded operators to obtain , similar to how characteristic functions are extended to integrable functions. Hence this is a key result for non-commutative integration (note: the continuity conditions defining in 1-3 above are stronger). We choose von Neumann algebras over C*-algebras since the former contain all spectral projections of their self-adjoint elements while the latter may not [ref].

So far we have seen that expected values of observables are derived via the formula . To derive the distribution itself, we make of the **spectral theorem** and for self-adjoint operators with continuous spectrum this requires **projection valued measures**. A self-adjoint operator has a corresponding function mapping Borel sets to projections so that represents the event that the outcome of measuring observable is in the set : we require that and is a complex additive function (measure) for all . We use as shorthand for . Similar to the way a finite dimensional self-adjoint matrix may be eigen-decomposed in terms of its eigenvalues and normalised eigenvectors as

the spectral theorem for more general self-adjoint operators allows us to write

which means that for every ,

.

Here, the integrals are over the spectrum of . Through this formula we can work with functions of operators and in particular the distribution of the random variable corresponding to operator in state will be

.

The similarities we have seen here between classical probability and quantum mechanics are summarised in the table below, largely taken from [2] which greatly aided my understanding. Note how the pairing between trace class and bounded operators is analogous to the duality of and functions.

Classical Probability |
Quantum Mechanics(non-commutative probability) |

– measure space | – Hilbert space model of QM |

– set | – Hilbert space |

– Boolean algebra of Borel subsets of called events |
– orthomodular lattice of projections (equivalently closed subspaces) of |

disjoint events | orthogonal projections |

– additive positive measure | – functional |

– integrable functions (probability density functions) | – trace class operators (density operators) |

– probability measure mapping Borel sets to numbers in [0,1] in a sigma-additive way |
– quantum state mapping projections to numbers in [0,1] in a sigma-additive way |

– essentially bounded measurable functions (bounded random variables) |
– von Neumann algebra of bounded operators (bounded observables) |

– expectation value of with respect to |
– expectation value of in state |

In summary, the fact that measurements don’t always commute lead us to consider non-commutative operator algebras. This leads us to the Hilbert space representation of quantum mechanics where a quantum state is a trace-one density operator and an observable is a bounded linear operator. We also saw that projections can be viewed as 0-1 events. The spectral theorem is used to decompose operators into a sum or integral of projections.

The richer mathematical setting for quantum mechanics allows us to model non-classical phenomena such as quantum interference and entanglement. We have not mentioned the time evolution of states, but in short, state vectors evolve unitarily according to the Schrödinger equation, generated by an operator known as the Hamiltonian.

#### References and Further Reading

[1] Hall, B.C., *Quantum Theory for Mathematicians*, Springer, Graduate Texts in Mathematics #267, June 2013 (relevant section)

[2] Redei, M., Von Neumann’s work on Hilbert space quantum mechanics

[3] Blackadar, B., Operator Algebras: Theory of C*-Algebras and von Neumann Algebras

[4] Wilce, Alexander, “Quantum Logic and Probability Theory“, *The Stanford Encyclopedia of Philosophy * (Spring 2017 Edition), Edward N. Zalta (ed.).

[6] Planetmath.org – Lattice of Projections

[7] Planetmath.org – Spectral Measure

[8] quantum mechanics – Intuitive meaning of Hilbert Space formalism – Physics Stack Exchange

[9] This answer to: mathematical physics – Quantum mechanics in a metric space rather than in a vector space, possible? – Physics Stack Exchange

[10] functional analysis – Resolution of the identity (basic questions) – Mathematics Stack Exchange