Noncommutative probability

The traditional mathematical axiomatization of probability, due to Kolmogorov, begins with a probability space Image may be NSFW.
Clik here to view. and constructs random variables as certain functions Image may be NSFW.
Clik here to view. $P \to \mathbb{R}$ . But start doing any probability and it becomes clear that the space Image may be NSFW.
Clik here to view. is de-emphasized as much as possible; the real focus of probability theory is on the algebra of random variables. It would be nice to have an approach to probability theory that reflects this.

Moreover, in the traditional approach, random variables necessarily commute. However, in quantum mechanics, the random variables are self-adjoint operators on a Hilbert space Image may be NSFW.
Clik here to view., and these do not commute in general. For the purposes of doing quantum probability, it is therefore also natural to look for an approach to probability theory that begins with an algebra, not necessarily commutative, which encompasses both the classical and quantum cases.

Happily, noncommutative probability provides such an approach. Terence Tao’s notes on free probability develop a version of noncommutative probability approach geared towards applications to random matrices, but today I would like to take a more leisurely and somewhat scattered route geared towards getting a general feel for what this formalism is capable of talking about.

Classical and quantum probability

(Below, if the reader chooses, she can restrict herself to finite sets and finite-dimensional Hilbert spaces so as to ignore measure-theoretic and analytic difficulties.)

A classical probability space Image may be NSFW.
Clik here to view. $(S, \mathcal{E}, \mathbb{P})$ consists of the following data:

A set Image may be NSFW.
Clik here to view., the sample space. Elements of this set describe possible states of some system.
A Image may be NSFW.
Clik here to view. $\sigma$ -algebra Image may be NSFW.
Clik here to view. $\mathcal{E}$ of subsets of Image may be NSFW.
Clik here to view., the events. Events describe properties of states. The pair Image may be NSFW.
Clik here to view. $(S, \mathcal{E})$ is a measurable space
A probability measure Image may be NSFW.
Clik here to view. $\mathbb{P}$ on Image may be NSFW.
Clik here to view. $(S, \mathcal{E})$ . This measures the probability that the system has some property.

Example. Let Image may be NSFW.
Clik here to view. $S = \{ H, T \}^n$ be the set of possible outcomes of Image may be NSFW.
Clik here to view. coin flips. Letting Image may be NSFW.
Clik here to view. $\mathcal{E}$ be the set of all subsets of Image may be NSFW.
Clik here to view., we can describe various events like “no heads are flipped” or “at most three tails are flipped,” and we can compute their probabilities using the fact that every point in Image may be NSFW.
Clik here to view. has probability Image may be NSFW.
Clik here to view. $\frac{1}{2^n}$ .

Example. Let Image may be NSFW.
Clik here to view. be a Image may be NSFW.
Clik here to view.-dimensional symplectic manifold with symplectic form Image may be NSFW.
Clik here to view. $\omega$ (e.g. the cotangent bundle of some other manifold). The Image may be NSFW.
Clik here to view. $n^{th}$ exterior power of the symplectic form defines a volume form on Image may be NSFW.
Clik here to view. which defines a Borel measure on Image may be NSFW.
Clik here to view. called Liouville measure (locally just Lebesgue measure). Since Liouville measure is built from the symplectic form, it is preserved under all symplectomorphisms, and in particular under time evolution with respect to any Hamiltonian.

A random variable is a measurable function Image may be NSFW.
Clik here to view. $X : S \to \mathbb{R}$ (where Image may be NSFW.
Clik here to view. $\mathbb{R}$ is given the Borel Image may be NSFW.
Clik here to view. $\sigma$ -algebra generated by the Euclidean topology). In our coin-flipping example, “number of heads flipped” is a random variable. A random variable which only takes the values Image may be NSFW.
Clik here to view. or Image may be NSFW.
Clik here to view. encodes the same data as an event (more precisely, it is the indicator function Image may be NSFW.
Clik here to view. $\chi_E$ of a unique event, which takes the value Image may be NSFW.
Clik here to view. on Image may be NSFW.
Clik here to view. and Image may be NSFW.
Clik here to view. on its complement). More generally, we can construct events from random variables: for any Borel subset Image may be NSFW.
Clik here to view. $E \subseteq \mathbb{R}$ the preimage Image may be NSFW.
Clik here to view. $X^{-1}(E)$ is an event (the event that Image may be NSFW.
Clik here to view. lies in Image may be NSFW.
Clik here to view.), often written Image may be NSFW.
Clik here to view. $X \in E$ , and so we can consider its probability Image may be NSFW.
Clik here to view. $\mathbb{P}(X \in E)$ . (This is the pushforward measure.)

Random variables should be thought of as real-valued observables of our system (and events, as random variables which take the value Image may be NSFW.
Clik here to view. or Image may be NSFW.
Clik here to view., are the observables given by answers to yes-no questions). By repeatedly measuring an observable and averaging, we can obtain its expected value

Image may be NSFW.
Clik here to view. $\displaystyle \mathbb{E}(X) = \int_S X \, d \mu$ .

(if this integral converges). If Image may be NSFW.
Clik here to view. only takes the values Image may be NSFW.
Clik here to view. 0, 1 , then Image may be NSFW.
Clik here to view. $\mathbb{E}(X)$ reduces to the probability of the corresponding event. In general, if Image may be NSFW.
Clik here to view. is a random variable and Image may be NSFW.
Clik here to view. is a Borel subset of Image may be NSFW.
Clik here to view. $\mathbb{R}$ , then Image may be NSFW.
Clik here to view. $\chi_E(X)$ is the indicator function of Image may be NSFW.
Clik here to view. $X^{-1}(E)$ , and Image may be NSFW.
Clik here to view. $\mathbb{P}(X \in E) = \mathbb{E}(\chi_E(X))$ .

If we wanted to define a quantum probability space by analogous data, it would consist of the following (not standard):

A Hilbert space Image may be NSFW.
Clik here to view., the space of states.
An abstract Image may be NSFW.
Clik here to view. $\sigma$ -algebra Image may be NSFW.
Clik here to view. $\mathcal{H}$ of closed subspaces of Image may be NSFW.
Clik here to view., the events. The intersection of two subspaces is their set-theoretic intersection, the union is the closure of their span, and the complement is the orthogonal complement.
A unit vector Image may be NSFW.
Clik here to view. $\psi \in H$ , the state vector.

Example. Every classical probability space Image may be NSFW.
Clik here to view. $(S, \mathcal{E}, \mathbb{P})$ defines a quantum probability space as follows: Image may be NSFW.
Clik here to view. is the Hilbert space Image may be NSFW.
Clik here to view. $L^2(S, \mathbb{P})$ of (equivalence classes of) square-integrable functions Image may be NSFW.
Clik here to view. $S \to \mathbb{C}$ under the inner product

Image may be NSFW.
Clik here to view. $\displaystyle \langle f, g \rangle = \int_S \overline{f(s)} g(s) \, d \mu$ ,

Image may be NSFW.
Clik here to view. $\mathcal{H}$ consists of the closed subspaces of functions which are (a.e.) equal to zero except on a given event Image may be NSFW.
Clik here to view. $E \in \mathcal{E}$ , and Image may be NSFW.
Clik here to view. $\psi$ is the function Image may be NSFW.
Clik here to view. $S \to \mathbb{C}$ which is identically equal to Image may be NSFW.
Clik here to view..

Example. The quantum probability space describing a qubit comes from applying the above construction to a bit; thus Image may be NSFW.
Clik here to view. $H = \mathbb{C}^2$ is a Image may be NSFW.
Clik here to view.-dimensional Hilbert space with orthonormal basis Image may be NSFW.
Clik here to view. $|0\rangle, |1\rangle$ , Image may be NSFW.
Clik here to view. $\mathcal{H}$ consists of four subspaces Image may be NSFW.
Clik here to view. $0, \text{span}(|0\rangle), \text{span}(|1\rangle), H$ , and Image may be NSFW.
Clik here to view. $\psi = c_0 |0\rangle + c_1 |1\rangle$ is the state of the qubit.

A quantum probability space does not have points in the classical sense, but we can still talk about the probability of an event Image may be NSFW.
Clik here to view.: if Image may be NSFW.
Clik here to view. P_E denotes the projection onto Image may be NSFW.
Clik here to view., then it is given by

Image may be NSFW.
Clik here to view. $\displaystyle \mathbb{P}(E) = \langle \psi, P_E \psi \rangle$

and writing Image may be NSFW.
Clik here to view. $\psi$ as the sum of its components parallel and orthogonal to Image may be NSFW.
Clik here to view. we see that this is the square of the absolute value of the component of Image may be NSFW.
Clik here to view. $\psi$ parallel to Image may be NSFW.
Clik here to view.. This is a simple form of the Born rule, and it describes the probability that Image may be NSFW.
Clik here to view. $\psi$ , when measured to determine whether or not it lies in Image may be NSFW.
Clik here to view., will in fact lie in Image may be NSFW.
Clik here to view.. Applied to a qubit, we conclude that a qubit described by Image may be NSFW.
Clik here to view. $\psi = c_0 |0\rangle + c_1 |1\rangle$ , when measured, takes the value Image may be NSFW.
Clik here to view. with probability Image may be NSFW.
Clik here to view. |c_0|^2 and the value Image may be NSFW.
Clik here to view. with probability Image may be NSFW.
Clik here to view. |c_1|^2 .

Note that if Image may be NSFW.
Clik here to view. is the entire Hilbert space then the condition that the corresponding probability is Image may be NSFW.
Clik here to view. is precisely the condition that Image may be NSFW.
Clik here to view. $\psi$ is a unit vector. Note also that the probability assigned by Image may be NSFW.
Clik here to view. $\psi$ to an event does not change if Image may be NSFW.
Clik here to view. $\psi$ is multiplied by a unit complex number; for this reason, state vectors are really points in the projective space over Image may be NSFW.
Clik here to view.. Thus the possible states of a qubit are parameterized by the Riemann sphere (called in this context the Bloch sphere).

A (real-valued) quantum random variable (probably not standard) is a self-adjoint operator Image may be NSFW.
Clik here to view. on Image may be NSFW.
Clik here to view. (possibly unbounded and/or densely defined in general). The values taken by Image may be NSFW.
Clik here to view. are precisely its spectral values (the points in its spectrum Image may be NSFW.
Clik here to view. $\sigma(X)$ ). This specializes even to the classical case: the values Image may be NSFW.
Clik here to view. $\lambda$ for which a random variable Image may be NSFW.
Clik here to view. $X : S \to \mathbb{R}$ has the property that Image may be NSFW.
Clik here to view. $\lambda - X$ fails to be invertible are precisely its values (up to the subtlety that we can ignore the behavior of Image may be NSFW.
Clik here to view. on a set of measure zero, but in practice we cannot meaningfully evaluate random variables at points anyway). In particular, for Image may be NSFW.
Clik here to view. bounded, Image may be NSFW.
Clik here to view. takes only the values Image may be NSFW.
Clik here to view. 0, 1 if and only if it is idempotent by Gelfand-Naimark, hence if and only if it is a projection; thus as in the classical case, random variables generalize events.

The expected value of a quantum random variable is

Image may be NSFW.
Clik here to view. $\displaystyle \mathbb{E}(X) = \langle \psi, X \psi \rangle$

(when Image may be NSFW.
Clik here to view. $\psi$ lies in the domain of Image may be NSFW.
Clik here to view.). If Image may be NSFW.
Clik here to view. happens to have a countable orthonormal basis Image may be NSFW.
Clik here to view. $\psi_k$ of eigenvectors with eigenvalues Image may be NSFW.
Clik here to view. $\lambda_k$ , then writing Image may be NSFW.
Clik here to view. $\psi = \sum c_k \psi_k$ we compute that

Image may be NSFW.
Clik here to view. $\displaystyle \langle \psi, X \psi \rangle = \sum_k \lambda_k |c_k|^2$

so this really is the expected value of Image may be NSFW.
Clik here to view. if we think of it classically as a random variable taking the value Image may be NSFW.
Clik here to view. $\lambda_k$ with probability Image may be NSFW.
Clik here to view. $\langle \psi, P_{\psi_k} \psi \rangle = |c_k|^2$ .

As in the classical case, we can make sense of probabilities such as Image may be NSFW.
Clik here to view. $\mathbb{P}(X \in E)$ where Image may be NSFW.
Clik here to view. is a Borel subset of Image may be NSFW.
Clik here to view. $\mathbb{R}$ , but this requires more work. If Image may be NSFW.
Clik here to view. has a countable orthonormal basis this is straightforward; in general, we need the Borel functional calculus in order to define Image may be NSFW.
Clik here to view. $\chi_E(X)$ as an operator so that we can compute Image may be NSFW.
Clik here to view. $\mathbb{E}(\chi_E(X))$ (note that we do not need Image may be NSFW.
Clik here to view. $\chi_E(X)$ to be a projection whose image lies in Image may be NSFW.
Clik here to view. $\mathcal{H}$ to compute this expectation, although this would be the appropriate analogue of the random variable Image may be NSFW.
Clik here to view. being measurable). Roughly speaking we ought to be able to start from the continuous functional calculus and approximate the indicator function of Image may be NSFW.
Clik here to view. by continuous functions, then show that the corresponding limit exists as a self-adjoint operator.

Unlike the classical case, the expected value Image may be NSFW.
Clik here to view. $\langle \psi, X \psi \rangle$ can be computed independently of any measurability hypotheses on Image may be NSFW.
Clik here to view.; in particular, the probability of a particular event occurring (that is, the expected value of an arbitrary projection) is automatically well-defined.

Noncommutative probability

The classical and quantum cases above have several features in common. In both cases we saw that, although we started with a description of events and their probabilities and moved on to a description of random variables and their expected values, we could recover events through their indicator functions as the idempotent random variables and recover probabilities of events as expected values of indicator functions. This suggests that we might fruitfully approach probability in general using algebras of random variables and the expectation.

If the algebra is commutative, we might hope to recover an underlying probability space, but a random-variables-first approach will allow us to work independently of a particular representation of a family of random variables as an algebra of functions on a probability space. If the algebra is noncommutative, we might hope to recover a Hilbert space on which it acts, but again, a random-variables-first approach will allow us to work independently of a particular representation as operators on a Hilbert space. We can also think of the algebra as the algebra of functions on a noncommutative space in the spirit of noncommutative geometry. Although noncommutative spaces don’t have a good notion of point, quantum probability spaces suggest that they have a good notion of measure (which we can think of as a “smeared-out” point, the Dirac measures corresponding to ordinary points).

The following definition is morally due to von Neumann and Segal. A random algebra (not standard) is a complex Image may be NSFW.
Clik here to view. $^{\dagger}$ -algebra Image may be NSFW.
Clik here to view. together with a Image may be NSFW.
Clik here to view. $^{\dagger}$ -linear functional Image may be NSFW.
Clik here to view. $\mathbb{E} : A \to \mathbb{C}$ such that

Image may be NSFW.
Clik here to view. $\displaystyle \mathbb{E}(a^{\dagger} a) \ge 0, \mathbb{E}(1) = 1$ .

Such a functional is called a state on Image may be NSFW.
Clik here to view. (as it describes the state of some probabilistic system by describing the expected value of observables). The (real-valued) random variables in Image may be NSFW.
Clik here to view. are its self-adjoint elements (and an event is a projection; that is, a self-adjoint idempotent).

A morphism of random algebras Image may be NSFW.
Clik here to view. $(A_1, \mathbb{E}_1) \to (A_2, \mathbb{E}_2)$ is a morphism Image may be NSFW.
Clik here to view. $\phi : A_1 \to A_2$ of complex Image may be NSFW.
Clik here to view. $^{\dagger}$ -algebras such that Image may be NSFW.
Clik here to view. $\mathbb{E}_2 \circ \phi = \mathbb{E}_1$ . This defines the category Image may be NSFW.
Clik here to view. $\text{Rand}$ of random algebras, and the category of noncommutative probability spaces is Image may be NSFW.
Clik here to view. $\text{Rand}^{op}$ . (This is probably the wrong choice of morphisms, but we’ll ignore that for now.)

Example. From a classical probability space Image may be NSFW.
Clik here to view. $(S, \mathcal{E}, \mathbb{P})$ we obtain a random algebra by letting Image may be NSFW.
Clik here to view. be the von Neumann algebra Image may be NSFW.
Clik here to view. $L^{\infty}(S)$ of essentially bounded measurable functions Image may be NSFW.
Clik here to view. $S \to \mathbb{C}$ under conjugation and letting Image may be NSFW.
Clik here to view. $\mathbb{E}$ be the integral.

Example. From a quantum probability space Image may be NSFW.
Clik here to view. $(H, \mathcal{H}, \psi)$ we obtain a random algebra by letting Image may be NSFW.
Clik here to view. be the span of the space of self-adjoint operators Image may be NSFW.
Clik here to view. $a : H \to H$ such that Image may be NSFW.
Clik here to view. $\chi_E(a) \in \mathcal{H}$ for all Borel subsets Image may be NSFW.
Clik here to view. $E \subseteq \mathbb{R}$ and letting Image may be NSFW.
Clik here to view. $\mathbb{E}$ be the functional Image may be NSFW.
Clik here to view. $a \mapsto \langle \psi, a \psi \rangle$ .

(Because we have not developed the Borel functional calculus, it will be cleaner just to work with an arbitrary Image may be NSFW.
Clik here to view. $^{\dagger}$ -algebra of bounded operators Image may be NSFW.
Clik here to view. $H \to H$ from which Image may be NSFW.
Clik here to view. $\mathcal{H}$ can but need not be derived. We can do the same thing in the classical case by starting with a collection of functions Image may be NSFW.
Clik here to view. $S \to \mathbb{C}$ and taking the preimages under all of them of the Borel subsets of Image may be NSFW.
Clik here to view. $\mathbb{C}$ to define a Image may be NSFW.
Clik here to view. $\sigma$ -algebra on Image may be NSFW.
Clik here to view..)

The above examples require some analysis to define in full generality. However, the reason we do not require any analytic hypotheses on Image may be NSFW.
Clik here to view. is to have a formalism flexible enough to discuss more algebraic examples such as the following.

Example. Let Image may be NSFW.
Clik here to view. be a group. The group algebra Image may be NSFW.
Clik here to view. $\mathbb{C}[G]$ is a Image may be NSFW.
Clik here to view. $^{\dagger}$ -algebra in the usual way (with involution extending Image may be NSFW.
Clik here to view. $g^{\dagger} = g^{-1}$ , so that every element of Image may be NSFW.
Clik here to view. is unitary). There is a distinguished state given by Image may be NSFW.
Clik here to view. $\mathbb{E}(1) = 1$ and Image may be NSFW.
Clik here to view. $\mathbb{E}(g) = 0$ for every non-identity Image may be NSFW.
Clik here to view..

The axioms we have chosen require some explanation. Working in a complex Image may be NSFW.
Clik here to view. $^{\dagger}$ -algebra is both convenient and has clear ties to quantum mechanics, but I do not have a good explanation of this axiom from first principles. The condition that Image may be NSFW.
Clik here to view. $\mathbb{E}$ is Image may be NSFW.
Clik here to view. $^{\dagger}$ -linear reflects linearity of expectation, which holds both in the classical and quantum cases, and the fact that we want the expected value of a self-adjoint element to be real. The condition that Image may be NSFW.
Clik here to view. $\mathbb{E}(a^{\dagger} a) \ge 0$ (positivity) reflects the fact that we want probabilities to be non-negative in the following sense.

In any complex Image may be NSFW.
Clik here to view. $^{\dagger}$ -algebra Image may be NSFW.
Clik here to view., we may define a positive (really non-negative) element to be an element of the form Image may be NSFW.
Clik here to view. $a^{\dagger} a, a \in A$ . A positive element is in particular self-adjoint. In the case of measurable functions on a probability space, the positive elements are precisely the elements which are (a.e.) non-negative, and in the case of operators on a Hilbert space, the positive elements are precisely the self-adjoint elements which have non-negative spectrum (~~by Gelfand representation~~ this is subtle; see the comments below). Hence positivity is a natural analogue in the algebraic setting of the condition that probabilities are non-negative.

(Edit, 1/2/22: It’s been pointed out in the comments that a more natural definition of “positive” is an element which is a sum of the form Image may be NSFW.
Clik here to view. $\sum_i a_i^{\dagger} a_i$ ; this makes the positive elements form a convex cone. However, this doesn’t affect the definition of a positive linear functional, and the two are equivalent in any C*-algebra.)

Finally, the condition that Image may be NSFW.
Clik here to view. $\mathbb{E}(1) = 1$ reflects the fact that we want the total probability to be Image may be NSFW.
Clik here to view..

The semi-inner product

The state allows us to define a bilinear map Image may be NSFW.
Clik here to view. $\langle a, b \rangle = \mathbb{E}(a^{\dagger} b)$ on any random algebra Image may be NSFW.
Clik here to view. which satisfies all of the axioms of an inner product except that it is not necessarily positive-definite, but only satisfies the weaker axiom that Image may be NSFW.
Clik here to view. $\langle a, a \rangle \ge 0$ . We call such a gadget a semi-inner product (since it is positive-semidefinite).

As for classical random variables we can define the covariance

Image may be NSFW.
Clik here to view. $\displaystyle \text{Cov}(a, b) = \langle a - \mathbb{E}(a), b - \mathbb{E}(b) \rangle = \mathbb{E}(a^{\dagger} b) - \mathbb{E}(a^{\dagger}) \mathbb{E}(b)$

of two elements, and positive-semidefiniteness implies that the variance Image may be NSFW.
Clik here to view. $\text{Var}(a) = \text{Cov}(a, a)$ is non-negative, hence that Image may be NSFW.
Clik here to view. $\mathbb{E}(a^{\dagger} a) \ge \mathbb{E}(a^{\dagger}) \mathbb{E}(a)$ . More generally, the proof of the Cauchy-Schwarz inequality goes through without modification, and we conclude that

Image may be NSFW.
Clik here to view. $\displaystyle |\langle a, b \rangle|^2 \le \langle a, a \rangle \langle b, b \rangle$ .

This is already enough for us to prove the following general version of Heisenberg’s uncertainty principle.

Theorem (Robertson uncertainty): Let Image may be NSFW.
Clik here to view. a, b be self-adjoint elements of a random algebra Image may be NSFW.
Clik here to view.. Then

Image may be NSFW.
Clik here to view. $\displaystyle \text{Var}(a) \text{Var}(b) \ge \frac{1}{4} \left| \mathbb{E}([a, b]) \right|^2$ .

Proof. Since both sides are invariant under translation of either Image may be NSFW.
Clik here to view. or Image may be NSFW.
Clik here to view. by a nonzero constant, we may assume without loss of generality that Image may be NSFW.
Clik here to view. a, b have mean zero (that is, that Image may be NSFW.
Clik here to view. $\mathbb{E}(a) = \mathbb{E}(b) = 0$ ). This gives

Image may be NSFW.
Clik here to view. $\displaystyle \text{Var}(a) \text{Var}(b) \ge | \mathbb{E}(ab) |^2$

by Cauchy-Schwarz. We can write Image may be NSFW.
Clik here to view. as the sum of its real and imaginary parts

Image may be NSFW.
Clik here to view. $\displaystyle ab = \frac{ab + ba}{2} + i \frac{ab - ba}{2i}$

and computing Image may be NSFW.
Clik here to view. $| \mathbb{E}(ab) |^2$ using the above decomposition gives

Image may be NSFW.
Clik here to view. $\displaystyle | \mathbb{E}(ab) |^2 = \left| \mathbb{E} \left( \frac{a \circ b}{2} \right) \right|^2 + \left| \mathbb{E} \left( \frac{[a, b]}{2} \right) \right|^2 \ge \frac{1}{4} \left| \mathbb{E}([a, b]) \right|^2$ .

where Image may be NSFW.
Clik here to view. $a \circ b = ab + ba$ and Image may be NSFW.
Clik here to view. [a, b] = ab - ba . The conclusion follows. Image may be NSFW.
Clik here to view. $\Box$

Interpreting Robertson uncertainty will be easier once we do a little more work. By Cauchy-Schwarz, if an element satisfies Image may be NSFW.
Clik here to view. $\langle n, n \rangle = 0$ then in fact it satisfies Image may be NSFW.
Clik here to view. $\langle a, n \rangle = 0$ for all Image may be NSFW.
Clik here to view. (and the converse is clear). In the classical picture, a function satisfying either of these conditions is equal to zero almost everywhere, which motivates the following definition. An element Image may be NSFW.
Clik here to view. of a random algebra Image may be NSFW.
Clik here to view. is null or zero almost surely (abbreviated a.s.) if Image may be NSFW.
Clik here to view. $\langle a, n \rangle = 0$ for all Image may be NSFW.
Clik here to view. $a \in A$ , which as we have seen is equivalent to Image may be NSFW.
Clik here to view. $\langle n, n \rangle = 0$ . The null elements Image may be NSFW.
Clik here to view. form a subspace of Image may be NSFW.
Clik here to view.. Two elements Image may be NSFW.
Clik here to view. a, b are equal almost surely if Image may be NSFW.
Clik here to view. $a - b \in N$ , hence equality a.s. is equivalent to equality in the quotient Image may be NSFW.
Clik here to view. A/N .

An element has variance zero if and only if it is constant almost surely. Robertson uncertainty then says that if two self-adjoint elements Image may be NSFW.
Clik here to view. $a, b \in A$ have the property that their commutator Image may be NSFW.
Clik here to view. [a, b] has nonzero expectation, then neither of them can be constant almost surely in a strong sense: the product of their variances is bounded below by a positive constant, so as one increases, the other must decrease. In other words, not only are they uncertain, but a state in which Image may be NSFW.
Clik here to view. is less uncertain is a state in which Image may be NSFW.
Clik here to view. is more uncertain.

The standard application of Robertson uncertainty is to the case that Image may be NSFW.
Clik here to view. a, b are the position and momentum operators respectively acting on a quantum particle on Image may be NSFW.
Clik here to view. $\mathbb{R}$ . This application has the following purely mathematical interpretation: a function in Image may be NSFW.
Clik here to view. $L^2(\mathbb{R})$ and its Fourier transform cannot simultaneously be too localized.

Independence

A fundamental notion in classical probability theory is the notion of independence. It can be generalized to random algebras as follows: two Image may be NSFW.
Clik here to view. $^{\dagger}$ -subalgebras Image may be NSFW.
Clik here to view. A, B of a random algebra Image may be NSFW.
Clik here to view. are independent if

Image may be NSFW.
Clik here to view. $\displaystyle \mathbb{E}(ab) = \mathbb{E}(ba) = \mathbb{E}(a) \mathbb{E}(b)$

for all Image may be NSFW.
Clik here to view. $a \in A, b \in B$ .

Example. Let Image may be NSFW.
Clik here to view. be the random algebra Image may be NSFW.
Clik here to view. $L^{\infty}(S)$ associated to a classical probability space Image may be NSFW.
Clik here to view. $(S, \mathcal{E}, \mathbb{P})$ and let Image may be NSFW.
Clik here to view. A, B be the subalgebras of functions which are measurable with respect to two Image may be NSFW.
Clik here to view. $\sigma$ -subalgebras Image may be NSFW.
Clik here to view. $\mathcal{E}_1, \mathcal{E}_2$ of Image may be NSFW.
Clik here to view. $\mathcal{E}$ . Then Image may be NSFW.
Clik here to view. A, B are independent in the above sense if and only if Image may be NSFW.
Clik here to view. $\mathcal{E}_1, \mathcal{E}_2$ are independent in the sense that

Image may be NSFW.
Clik here to view. $\displaystyle \mathbb{P}(E_1 \cap E_2) = \mathbb{P}(E_1) \mathbb{P}(E_2)$

where Image may be NSFW.
Clik here to view. $E_i \in \mathcal{E}_i$ (by the monotone class theorem). Note that this condition is equivalent to Image may be NSFW.
Clik here to view. $\mathbb{E}(\chi_{E_1} \chi_{E_2}) = \mathbb{E}(\chi_{E_1}) \mathbb{E}(\chi_{E_2})$ .

Example. Let Image may be NSFW.
Clik here to view. A, B be two random algebras with expectations Image may be NSFW.
Clik here to view. $\mathbb{E}_A, \mathbb{E}_B$ . Their tensor product Image may be NSFW.
Clik here to view. $A \otimes B$ acquires a natural Image may be NSFW.
Clik here to view. $^{\dagger}$ -algebra structure given by Image may be NSFW.
Clik here to view. $(a \otimes b)^{\dagger} = a^{\dagger} \otimes b^{\dagger}$ on pure tensors (it is the universal Image may be NSFW.
Clik here to view. $^{\dagger}$ -algebra admitting morphisms from Image may be NSFW.
Clik here to view. A, B whose images commute), and moreover we can define on it a state given by

Image may be NSFW.
Clik here to view. $\displaystyle \mathbb{E}_{A \otimes B}(a \otimes b) = \mathbb{E}_A(a) \mathbb{E}_B(b)$

on pure tensors. Conversely, any state on Image may be NSFW.
Clik here to view. $A \otimes B$ such that Image may be NSFW.
Clik here to view. and Image may be NSFW.
Clik here to view. are independent is of this form. This is a noncommutative generalization of product measure; when Image may be NSFW.
Clik here to view. A, B come from classical probability spaces Image may be NSFW.
Clik here to view. S_1, S_2 , a suitable completion of Image may be NSFW.
Clik here to view. $A \otimes B$ is the corresponding algebra of functions on the product Image may be NSFW.
Clik here to view. $S_1 \times S_2$ , and the state above comes from integration against the corresponding product measure.

Example. Image may be NSFW.
Clik here to view. is independent of itself (in Image may be NSFW.
Clik here to view.) if and only if the state Image may be NSFW.
Clik here to view. $\mathbb{E}$ is actually a homomorphism Image may be NSFW.
Clik here to view. $A \to \mathbb{C}$ of Image may be NSFW.
Clik here to view. $^{\dagger}$ -algebras. Thinking of the case that Image may be NSFW.
Clik here to view. is a C*-algebra in particular, the corresponding states can be thought of as Dirac measures supported at points of Image may be NSFW.
Clik here to view. $\text{MaxSpec } A$ . In the noncommutative case, Image may be NSFW.
Clik here to view. may admit no homomorphisms to Image may be NSFW.
Clik here to view. $\mathbb{C}$ (for example if Image may be NSFW.
Clik here to view. contains the Weyl algebra), hence no Dirac measures, an expression of the general intuition that noncommutative spaces are “smeared out” and not easily expressible in terms of points.

Independence is a formalization of the intuitive idea that knowing the values of the random variables in Image may be NSFW.
Clik here to view. doesn’t allow you to deduce anything about the values of the random variables in Image may be NSFW.
Clik here to view. and vice versa. One indication of how this works in the setting of random algebras is as follows: if Image may be NSFW.
Clik here to view. $p \in B$ is a projection with Image may be NSFW.
Clik here to view. $\mathbb{E}(p) > 0$ (that is, an event that occurs with positive probability) we can define a conditional expectation

Image may be NSFW.
Clik here to view. $\displaystyle \mathbb{E}(a | p) = \frac{\mathbb{E}(pap)}{\mathbb{E}(p)}$ .

(The first factor of Image may be NSFW.
Clik here to view. is necessary in the noncommutative case to ensure that the result is still a state.) This represents the expected value of Image may be NSFW.
Clik here to view. given that the event Image may be NSFW.
Clik here to view. occurred. If Image may be NSFW.
Clik here to view. A, B are independent, it follows that Image may be NSFW.
Clik here to view. $\mathbb{E}(a | p) = \mathbb{E}(a)$ ; in other words, knowing that Image may be NSFW.
Clik here to view. occurred has no effect on the expected value of any of the elements of Image may be NSFW.
Clik here to view..

Independence is a very strong condition to impose if the subalgebras Image may be NSFW.
Clik here to view. A, B do not commute. For example, it implies that Image may be NSFW.
Clik here to view. $\mathbb{E}(ab - ba) = 0$ for all Image may be NSFW.
Clik here to view. $a \in A, b \in B$ , which is the only condition under which Robertson uncertainty cannot relate the variances of Image may be NSFW.
Clik here to view. a, b . In the particular case of the position and momentum operators, Image may be NSFW.
Clik here to view. ab - ba is a nonzero scalar, hence always has nonzero expectation; it follows that position and momentum cannot be made independent! (By contrast, in the classical setting any pair of random variables is independent with respect to a Dirac measure.)

In the noncommutative setting, a different notion of independence, free independence (replacing the tensor product with the free product), becomes more natural and useful. We will not discuss this issue further, but see Terence Tao’s notes linked above.

The Gelfand-Naimark-Segal construction

If Image may be NSFW.
Clik here to view. is any inner product space, Image may be NSFW.
Clik here to view. any Image may be NSFW.
Clik here to view. $^{\dagger}$ -algebra of linear operators on Image may be NSFW.
Clik here to view., and Image may be NSFW.
Clik here to view. $\psi \in V$ is any unit vector, then Image may be NSFW.
Clik here to view. is a concrete random algebra with expectation Image may be NSFW.
Clik here to view. $\mathbb{E}(a) = \langle \psi, a \psi \rangle$ . This subsumes the examples coming from both classical and quantum probability spaces. The goal of this section is to determine to what extent we can prove a Cayley’s theorem for random algebras to the effect that random algebras are concrete.

The above suggests the following definition. If Image may be NSFW.
Clik here to view. is a complex Image may be NSFW.
Clik here to view. $^{\dagger}$ -algebra, then a Image may be NSFW.
Clik here to view. $^{\dagger}$ -representation of Image may be NSFW.
Clik here to view. is a homomorphism Image may be NSFW.
Clik here to view. $A \to (V \Rightarrow V)$ from Image may be NSFW.
Clik here to view. to the endomorphisms of an inner product space Image may be NSFW.
Clik here to view. such that

Image may be NSFW.
Clik here to view. $\displaystyle \langle v, aw \rangle_V = \langle a^{\dagger} v, w \rangle_V$

for all Image may be NSFW.
Clik here to view. $v, w \in V$ . (Note that if Image may be NSFW.
Clik here to view. is not a Hilbert space then Image may be NSFW.
Clik here to view. $\text{End}(V)$ is not necessarily a Image may be NSFW.
Clik here to view. $^{\dagger}$ -algebra because adjoints may not exist in general.) A Hilbert Image may be NSFW.
Clik here to view. $^{\dagger}$ -representation is a Image may be NSFW.
Clik here to view. $^{\dagger}$ -representation on a Hilbert space.

The semi-inner product Image may be NSFW.
Clik here to view. $\langle a, b \rangle = \mathbb{E}(a^{\dagger} b)$ on a random algebra Image may be NSFW.
Clik here to view. descends to the quotient space Image may be NSFW.
Clik here to view. A/N , where it is becomes an inner product because we have quotiented by the elements of norm zero. Moreover, since

Image may be NSFW.
Clik here to view. $\langle a, n \rangle = 0 \forall n \Leftrightarrow \mathbb{E}(a^{\dagger} n) = 0 \Rightarrow \mathbb{E}(a^{\dagger} bn) = 0 \Leftrightarrow \langle a, bn \rangle = 0$

it follows that Image may be NSFW.
Clik here to view. is a left ideal, so the quotient map Image may be NSFW.
Clik here to view. $A \to A/N$ is a quotient of left Image may be NSFW.
Clik here to view.-modules; consequently, Image may be NSFW.
Clik here to view. acts on Image may be NSFW.
Clik here to view. A/N by linear operators. Since

Image may be NSFW.
Clik here to view. $\displaystyle \langle v, aw \rangle = \mathbb{E}(v^{\dagger} aw) = \langle a^{\dagger} v, w \rangle$

it follows that Image may be NSFW.
Clik here to view. A/N defines a Image may be NSFW.
Clik here to view. $^{\dagger}$ -representation of Image may be NSFW.
Clik here to view.. The procedure we have outlined is essentially the Gelfand-Naimark-Segal (GNS) construction: we associate to any state on a Image may be NSFW.
Clik here to view. $^{\dagger}$ -algebra a corresponding Image may be NSFW.
Clik here to view. $^{\dagger}$ -representation such that the state can be recovered from the representation as

Image may be NSFW.
Clik here to view. $\displaystyle \mathbb{E}(a) = \langle \psi, a \psi \rangle_{A/N}$

where Image may be NSFW.
Clik here to view. $\psi = 1$ . This may be regarded as a weak Cayley’s theorem: unfortunately, this Image may be NSFW.
Clik here to view. $^{\dagger}$ -representation is not faithful in general. To get a stronger statement about random algebras, we will now assume another condition, namely that the if Image may be NSFW.
Clik here to view. $a^{\dagger} a \neq 0$ , then Image may be NSFW.
Clik here to view. $\mathbb{E}(a ^{\dagger} a) \neq 0$ (the state is faithful).

The faithfulness axiom is equivalent to requiring Image may be NSFW.
Clik here to view. N = 0 and also equivalent to requiring that Image may be NSFW.
Clik here to view. is an inner product space (rather than a semi-inner product space). It implies, but is stronger than, the assumption that the action of Image may be NSFW.
Clik here to view. on Image may be NSFW.
Clik here to view. A/N is faithful. The remarks about the state above then prove the following.

“Cayley’s theorem for random algebras”: A random algebra with a faithful state is concrete.

This is still not a true analog of Cayley’s theorem because the converse is false: the state Image may be NSFW.
Clik here to view. $\mathbb{E}(a) = \langle \psi, a \psi \rangle$ of a concrete random algebra need not be faithful.

From here we will assume, in addition to faithfulness, another condition, namely that for every Image may be NSFW.
Clik here to view. $a \in A$ there exists a constant Image may be NSFW.
Clik here to view. such that Image may be NSFW.
Clik here to view. $|\langle v, av \rangle| \le C \langle v, v \rangle$ (boundedness). Boundedness is equivalent to requiring that Image may be NSFW.
Clik here to view. acts on itself by bounded linear operators. This action therefore uniquely extends to the completion of Image may be NSFW.
Clik here to view. with respect to its inner product, which we’ll denote by Image may be NSFW.
Clik here to view., and consequently it follows that in this case Image may be NSFW.
Clik here to view. admits a Hilbert Image may be NSFW.
Clik here to view. $^{\dagger}$ -representation Image may be NSFW.
Clik here to view. $A \mapsto (H \Rightarrow H)$ . The closure of the image of Image may be NSFW.
Clik here to view. in Image may be NSFW.
Clik here to view. $(H \Rightarrow H)$ is a C*-algebra Image may be NSFW.
Clik here to view. $\overline{A}$ of bounded linear operators on Image may be NSFW.
Clik here to view., and moreover since Image may be NSFW.
Clik here to view. $\mathbb{E}(a) = \langle \psi, a \psi \rangle$ where Image may be NSFW.
Clik here to view. $\psi = 1$ , the expectation uniquely extends to Image may be NSFW.
Clik here to view. $\overline{A}$ .

This motivates the following definition: a random C*-algebra is a random algebra which is also a C*-algebra. The above discussion proves the following.

Theorem: Let Image may be NSFW.
Clik here to view. be a random algebra with a faithful state satisfying boundedness. Then Image may be NSFW.
Clik here to view. canonically embeds as a dense Image may be NSFW.
Clik here to view. $^{\dagger}$ -subalgebra of a random C*-algebra Image may be NSFW.
Clik here to view. $\overline{A}$ equipped with a Hilbert Image may be NSFW.
Clik here to view. $^{\dagger}$ -representation Image may be NSFW.
Clik here to view. via the GNS construction; moreover, there is a canonical vector Image may be NSFW.
Clik here to view. $\psi \in H$ such that Image may be NSFW.
Clik here to view. $\mathbb{E}(a) = \langle \psi, a \psi \rangle$ for all Image may be NSFW.
Clik here to view. $a \in \bar{A}$ .

This is a much stronger conclusion than the conclusion that Image may be NSFW.
Clik here to view. is concrete, since it allows us to use facts from the theory of C*-algebras.

Corollary: Let Image may be NSFW.
Clik here to view. be a commutative random algebra with a faithful state satisfying boundedness. Then Image may be NSFW.
Clik here to view. canonically embeds as a dense Image may be NSFW.
Clik here to view. $^{\dagger}$ -subalgebra of the algebra of continuous functions on a compact Hausdorff space Image may be NSFW.
Clik here to view..

Proof. The closure of a commutative Image may be NSFW.
Clik here to view. $^{\dagger}$ -subalgebra of Image may be NSFW.
Clik here to view. $H \Rightarrow H$ is also commutative, since commutativity is a continuous condition. The conclusion then follows from Gelfand-Naimark. Image may be NSFW.
Clik here to view. $\Box$

Corollary (“Maschke’s theorem”): Let Image may be NSFW.
Clik here to view. be a finite-dimensional random algebra with a faithful state. Then Image may be NSFW.
Clik here to view. is semisimple.

Proof. A finite-dimensional random algebra automatically satisfies boundedness. The GNS construction equips Image may be NSFW.
Clik here to view. with a faithful Image may be NSFW.
Clik here to view. $^{\dagger}$ -representation, namely Image may be NSFW.
Clik here to view. itself. Let Image may be NSFW.
Clik here to view. be a submodule of Image may be NSFW.
Clik here to view.. Then for every Image may be NSFW.
Clik here to view. $a \in A$ ,

Image may be NSFW.
Clik here to view. $\displaystyle \forall v \in V : \langle av, w \rangle = 0 \Leftrightarrow \forall v \in V : \langle v, a^{\dagger} w \rangle = 0$

so Image may be NSFW.
Clik here to view. $V^{\perp}$ is also a submodule of Image may be NSFW.
Clik here to view.. So every submodule of Image may be NSFW.
Clik here to view. is a direct summand; consequently, Image may be NSFW.
Clik here to view. is semisimple. Image may be NSFW.
Clik here to view. $\Box$

Note that we really do recover Maschke’s theorem for complex representations of finite groups as a corollary, since Image may be NSFW.
Clik here to view. $\mathbb{C}[G]$ is a finite-dimensional random algebra with a faithful state.

Moments

The axioms for a random algebra may not seem strong enough to capture random variables. For example, it does not seem possible to directly access probabilities like Image may be NSFW.
Clik here to view. $\mathbb{P}(a \in E)$ . However, our axioms are enough to define the moments

Image may be NSFW.
Clik here to view. $\displaystyle \mathbb{E}(a), \mathbb{E}(a^2), \mathbb{E}(a^3), ...$

of a random variable, and under suitable hypotheses (discussed under the general heading of the moment problem) it is possible to recover a random variable in the classical sense from its moments. We prove a result of this type for random C*-algebras.

Proposition: Any state Image may be NSFW.
Clik here to view. $\mathbb{E} : A \to \mathbb{C}$ on a C*-algebra has norm Image may be NSFW.
Clik here to view. (and in particular is continuous).

Proof. By examining real and imaginary parts, it suffices to show that a self-adjoint element Image may be NSFW.
Clik here to view. $a \in A$ of norm Image may be NSFW.
Clik here to view. maps to an element of norm at most Image may be NSFW.
Clik here to view.. Since Image may be NSFW.
Clik here to view. 1 - a has non-negative spectrum, by the continuous functional calculus it has a square root, hence is positive, so

Image may be NSFW.
Clik here to view. $\displaystyle \mathbb{E}(1 - a) = 1 - \mathbb{E}(a) \ge 0$ .

Similarly, Image may be NSFW.
Clik here to view. 1 + a has non-negative spectrum, so by the continuous functional calculus it has a square root, hence is positive, so

Image may be NSFW.
Clik here to view. $\mathbb{E}(1 + a) = 1 + \mathbb{E}(a) \ge 0$ .

We conclude that Image may be NSFW.
Clik here to view. $|\mathbb{E}(a)| \le 1$ , with equality if Image may be NSFW.
Clik here to view. a = 1 . Image may be NSFW.
Clik here to view. $\Box$

In fact a much stronger statement is true due to the following corollary of the Riesz-Markov theorem, which we will not prove; see Terence Tao’s notes.

Theorem: Let Image may be NSFW.
Clik here to view. be a compact Hausdorff space and Image may be NSFW.
Clik here to view. $\mathbb{E} : C(X) \to \mathbb{C}$ be a positive linear functional. Then there is a unique Radon measure Image may be NSFW.
Clik here to view. $\mu$ on Image may be NSFW.
Clik here to view. such that

Image may be NSFW.
Clik here to view. $\displaystyle \mathbb{E}(f) = \int_X f \, d \mu$

for all Image may be NSFW.
Clik here to view. $f \in C(X)$ (and conversely any Radon measure defines a positive linear functional on Image may be NSFW.
Clik here to view. C(X) ).

It follows by Gelfand-Naimark that specifying a commutative random C*-algebra is equivalent to specifying a compact Hausdorff space and a Radon measure on it of total measure Image may be NSFW.
Clik here to view..

Corollary: Let Image may be NSFW.
Clik here to view. be a random C*-algebra. If Image may be NSFW.
Clik here to view. $a \in A$ is normal, then there is a unique Radon measure Image may be NSFW.
Clik here to view. $\mu$ on Image may be NSFW.
Clik here to view. $X = \text{MaxSpec } \overline{\mathbb{C}[a, a^{\dagger}]}$ such that

Image may be NSFW.
Clik here to view. $\displaystyle \mathbb{E}(f(a)) = \int_X f(a) \, d \mu$

for all continuous functions Image may be NSFW.
Clik here to view. $f : \sigma(a) \to \mathbb{C}$ (where Image may be NSFW.
Clik here to view. f(a) is defined using the continuous functional calculus).

Proof. Since Image may be NSFW.
Clik here to view. $a, a^{\dagger}$ is dense in Image may be NSFW.
Clik here to view. $\overline{\mathbb{C}[a, a^{\dagger}]} \cong C(X)$ by construction, a morphism Image may be NSFW.
Clik here to view. $C(X) \to \mathbb{C}$ is uniquely determined by what it does to Image may be NSFW.
Clik here to view., hence Image may be NSFW.
Clik here to view., regarded as a function Image may be NSFW.
Clik here to view. $X \to \sigma(a) \subset \mathbb{C}$ , is injective. Since it is a continuous map between compact Hausdorff spaces, it is also an embedding, so we may regard Image may be NSFW.
Clik here to view. as canonically embedded into Image may be NSFW.
Clik here to view. $\sigma(a)$ . (This embedding is actually a homeomorphism but we do not need this.) By Tietze extension, any continuous function Image may be NSFW.
Clik here to view. $X \to \mathbb{C}$ extends to a continuous function Image may be NSFW.
Clik here to view. $\sigma(a) \to \mathbb{C}$ , so the continuous functions Image may be NSFW.
Clik here to view. $f(a) : X \to \mathbb{C}$ given by applying the continuous functional calculus to Image may be NSFW.
Clik here to view. include all continuous functions Image may be NSFW.
Clik here to view. $X \to \mathbb{C}$ , and we reduce to the previous result. Image may be NSFW.
Clik here to view. $\Box$

Corollary: With the same hypotheses as above, the Radon measure Image may be NSFW.
Clik here to view. $\mu$ above is uniquely determined by the values

Image may be NSFW.
Clik here to view. $\displaystyle \mathbb{E}(p(a, a^{\dagger})) = \int_X p \, d \mu$

where Image may be NSFW.
Clik here to view. is a polynomial in Image may be NSFW.
Clik here to view. and Image may be NSFW.
Clik here to view. $\bar{z}$ . Consequently, Image may be NSFW.
Clik here to view. $\mu$ is uniquely determined by the Image may be NSFW.
Clik here to view. $^{\dagger}$ -moments Image may be NSFW.
Clik here to view. $\mathbb{E}(a^n (a^{\dagger})^m), n, m \ge 0$ . If Image may be NSFW.
Clik here to view. is self-adjoint, Image may be NSFW.
Clik here to view. $\mu$ is uniquely determined by the values

Image may be NSFW.
Clik here to view. $\displaystyle \mathbb{E}(p(a)) = \int_X p \, d \mu$

where Image may be NSFW.
Clik here to view. is a polynomial in one variable. Equivalently, Image may be NSFW.
Clik here to view. $\mu$ is uniquely determined by the moments Image may be NSFW.
Clik here to view. $\mathbb{E}(a^n), n \ge 0$ .

Proof. Image may be NSFW.
Clik here to view. $\sigma(a)$ is a compact subset of Image may be NSFW.
Clik here to view. $\mathbb{C}$ , so by Stone-Weierstrass the polynomial functions in Image may be NSFW.
Clik here to view. and Image may be NSFW.
Clik here to view. $\bar{z}$ are uniformly dense in the space of continuous functions Image may be NSFW.
Clik here to view. $\sigma(a) \to \mathbb{C}$ . Now recall that the continuous functional calculus and Image may be NSFW.
Clik here to view. $\mathbb{E}$ both preserve uniform limits. If Image may be NSFW.
Clik here to view. is self-adjoint, Image may be NSFW.
Clik here to view. $\sigma(a)$ is real, so we only need to take polynomial functions in Image may be NSFW.
Clik here to view.. Image may be NSFW.
Clik here to view. $\Box$

The proofs above generalize essentially unchanged to the following.

Corollary: Let Image may be NSFW.
Clik here to view. a_1, ... a_k be commuting normal elements of a random C*-algebra Image may be NSFW.
Clik here to view.. Then there exists a unique Radon measure Image may be NSFW.
Clik here to view. $\mu$ on Image may be NSFW.
Clik here to view. $X = \text{MaxSpec } \overline{\mathbb{C}[a_1, a_1^{\dagger}, ... a_k, a_k^{\dagger}]}$ such that

Image may be NSFW.
Clik here to view. $\displaystyle \mathbb{E}(f(a_1, ... a_k)) = \int_X f(a_1, ... a_k) \, d \mu$

for all continuous functions Image may be NSFW.
Clik here to view. $f : \sigma(a_1) \times ... \times \sigma(a_k) \to \mathbb{C}$ . Furthermore, Image may be NSFW.
Clik here to view. $\mu$ is uniquely determined by the joint Image may be NSFW.
Clik here to view. $^{\dagger}$ -moments

Image may be NSFW.
Clik here to view. $\displaystyle \mathbb{E}(a_1^{n_1} (a_1^{\dagger})^{m_1} a_2^{n_2} (m_2^{\dagger})^{e_2}...), n_i, m_i \ge 0$

of the Image may be NSFW.
Clik here to view. a_i . If the Image may be NSFW.
Clik here to view. a_i are self-adjoint, Image may be NSFW.
Clik here to view. $\mu$ is uniquely determined by the joint moments Image may be NSFW.
Clik here to view. $\mathbb{E}(a_1^{n_1} ... a_k^{n_k}), n_i \in \mathbb{Z}_{\ge 0}$ of the Image may be NSFW.
Clik here to view. a_i .

The hypothesis that the Image may be NSFW.
Clik here to view. a_i commute is crucial in the following sense. We restrict to the self-adjoint case for simplicity.

Proposition: Let Image may be NSFW.
Clik here to view. a, b be self-adjoint elements of a random C*-algebra Image may be NSFW.
Clik here to view. with faithful state such that there exists a measure Image may be NSFW.
Clik here to view. $\mu$ on a measure space Image may be NSFW.
Clik here to view. and two measurable functions Image may be NSFW.
Clik here to view. $x, y : X \to \mathbb{R}$ satisfying

Image may be NSFW.
Clik here to view. $\displaystyle \mathbb{E}(a^{n_1} b^{m_1} a^{n_2} b^{m_2} ... a^{n_k} b^{m_k}) = \int_X x^{\sum n_i} y^{\sum m_i} \, d \mu$

for all Image may be NSFW.
Clik here to view. $n_i, m_i \in \mathbb{Z}_{\ge 0}$ . Then Image may be NSFW.
Clik here to view. ab = ba .

Proof. If Image may be NSFW.
Clik here to view. a, b are self-adjoint then so is Image may be NSFW.
Clik here to view. $c = \frac{ab - ba}{i}$ . The hypothesis above implies that Image may be NSFW.
Clik here to view. $\mathbb{E}(c^2) = 0$ , but since Image may be NSFW.
Clik here to view. is self-adjoint Image may be NSFW.
Clik here to view. c^2 is positive, hence by faithfulness Image may be NSFW.
Clik here to view. c = 0 . Image may be NSFW.
Clik here to view. $\Box$

This result may be interpreted as saying that two noncommuting random variables do not in general have a reasonable notion of joint distribution.

Some closing remarks about quantumness

Classical mechanics is in principle deterministic: if the initial state of a system is known deterministically, then classical mechanics can in principle determine all future states. The predictions of quantum mechanics are, however, probabilistic: all that can be determined is a probability distribution on possible outcomes of a given experiment.

The two can be made to seem more similar if classical mechanics is generalized by allowing the state of the system to be probabilistic in the classical sense. Then classical and quantum mechanics can both be subsumed under the heading of random algebras, where in the classical case we do not keep track of the position and momentum of a particle but a probability distribution over all possible positions and momenta. What distinguishes the classical from the quantum cases is the noncommutativity of the random algebras in the latter case, and in particular the fact that the random algebras occurring in quantum mechanics generally do not admit any homomorphisms Image may be NSFW.
Clik here to view. $A \to \mathbb{C}$ , hence admit no Dirac measures, so we are forced to always work probabilistically.

The formal similarity between classical and quantum mechanics described here only applies to states and observables; to get time evolution back into the picture we should endow our random algebras with Poisson brackets, giving us random Poisson algebras, and Hamiltonians…

Noncommutative probability

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