nLab indivisible stochastic process interpretation of quantum mechanics

Preface

The sections to follow give a summary of the account in Barandes 2025. As a brief preview, here is the gist of what is going on:

For a finite (physical) system with $N \in \mathbb{N}$ states, a one-step random evolution is given by a stochastic matrix $(P_{i j})_{i, j =1}^N$ whose entry $P_{i j} \in [0,1]$ gives the probability that the system, if in state $j$, evolves to state $i$.

Assuming there is a (temporal) sequence of such one-step random evolutions then these stochastic matrices depend on a pair of (time) parameters $t_1 \geq t_0 \in \mathbb{R}$.

Say that the stochastic process defined thereby is divisible iff these matrices of transition probabilities over some (time) interval are the matrix product $(-) \cdot (-)$ over those associated with any decomposition of the interval:

$$\text{decomposable} \;\;\;\; \Leftrightarrow \;\;\;\; \forall_{t_1 \leq t \leq t_0} \;\; P(t_1, t_0) = P(t_1, t) \cdot P(t, t_0) \,.$$

Now, if these transition probabilities $P$ are given by the rules of coherent time-evolution of pure quantum states according to quantum mechanics, then:

1. the Born rule says that the probabilities are norm-squares

   $$P_{i j} = \vert\Theta_{i j}\vert^2$$

   of complex numbers known as probability amplitudes and traditionally denoted as on the right here:

   $$\begin{aligned} \Theta_{i j} & = \langle i \vert \Theta \vert j \rangle \\ P_{i j} & = \langle i \vert \Theta \vert j \rangle \, \langle j \vert \Theta^\dagger \vert i \rangle \end{aligned}$$

   (where traditionally one writes “$U$” for where Barandes has “$\Theta$”),

2. (what is essentially) the Schrödinger equation says that it is these probability amplitudes that compose under matrix multiplication:

   $$\forall_{t_0 \leq t \leq t_1} \;\; \Theta(t_1, t_0) = \Theta(t_1, t) \cdot \Theta(t, t_0) \,.$$

But with the probability amplitudes $\Theta$ composing according to matrix multiplication, then in general the corresponding (doubly) stochastic matrices $P$ of transition probabilities will clearly not compose by the rules of matrix multiplication.

In conclusion: The stochastic processes given by quantum processes are generically indivisible.

That seems to be the key point highlighted by Barandes 2025 (without maybe explicitly saying so).

NB: The above argument generalizes immediately to countable sets of states, with $\big\{ {\vert i \rangle} \big\}_{i \in \mathbb{N}}$ a Hilbert space basis of a Hilbert space $\mathscr{H}$, with $\Theta$ a (unitary) linear operator on $\mathscr{H}$, and $P$ an infinite stochastic matrix acting on $\ell^1(\mathbb{N})$.

Idea

The indivisible stochastic process interpretation of quantum mechanics proposes that quantum systems can be understood as stochastic processes that lack the “Markov property,” meaning they cannot be broken down into independent, sequential steps. Instead, they evolve over finite chunks of time through a more general, non-Markovian, stochastic law. This interpretation, developed by Jacob Barandes, suggests that the core features of quantum mechanics, such as quantum interference, decoherence, and entanglement are an artifact of the indivisible dynamics of the underlying indivisible stochastic processes being approximated by Markovian dynamics in the Hilbert space formalism of quantum mechanics.

Definitions and the stochastic-quantum correspondence

Let $\Omega$ be a measurable space which we interpret as the space of all possible configurations of some system, and let $T$ denote an interval of time starting at $0$. An indivisible real-time stochastic process on $\Omega$ consists of a family of morphisms in the Kleisli category of the Giry monad $G$,

$$\Omega_0 \xrightarrow{\Gamma_t} \Omega_t$$

where $\Omega_t$ is a copy of $\Omega$ at time $t \in T$. These Markov kernels $\Gamma_t$ are defined as functions

$$\begin{array}{ccc} \Omega_0 \times \Sigma_{\Omega_t} & \xrightarrow{\Gamma_t} & [0,1] \\ (\omega, U) & \mapsto & \Gamma_t(U | \omega) \end{array}$$

which we read as ‘’the probability of $\Gamma_t$ on the measurable set $U$ given the point $\omega \in \Omega_0$’’. For each fixed $\omega \in \Omega_0$ the function $\Gamma_t(\bullet | \omega)$ defines a probability measure on $\Omega_t$ ($\Gamma_t(\bullet | \omega) \in G(\Omega_0)$), and for each fixed $U \in \Sigma_{\Omega_t}$ the function $\Omega_0 \xrightarrow{\Gamma_t(U | \bullet)} [0,1]$ is a measurable function.

To say that the stochastic process is indivisible means that for any $0 \lt s \lt t$ there exists no morphism $\Omega_s \xrightarrow{ \Gamma_{t,s}} \Omega_t$ in the Kleisli category of the Giry monad $G$ such that $\Gamma_t = \Gamma_{t,s} \circ \Gamma_s$ where the composition in the Kleisli category is defined by $(\Gamma_{t,s} \circ \Gamma_s)(U | \omega) = \int_{\lambda \in \Omega_s} \Gamma_{t,s}(U | \lambda) \, d\Gamma_s(\bullet | \omega)$.

The stochastic-quantum correspondence assumes the configuration (or “state’’) space $\Omega$ is a countable measurable space with the discrete $\sigma$-algebra. Here we assume $\Omega$ is finite, say $|\Omega| = N$. It follows that every singleton set $\{i\} \subset \Omega_t$ is measurable, and hence the morphism $\Gamma_t$ is completely determined by the values $\Gamma_t(\{i\} | j)$ which represent the transition probability from the state $j$ at time $0$ to the state $i$ at time $t$. Since $\Omega$ is finite the morphisms $\Gamma_t$ in the Kleisli category of $G$ can be representated as an $N \times N$ matrix, and the composition in the Kleisli category reduces to just matrix multiplication. Let $\Gamma(t)$ denote the matrix whose component at row $i$ and column $j$ is $\Gamma_t(\{i\}|j)$, and denote that component of $\Gamma(t)$ by $\Gamma_{i,j}(t)$.

Let us now proceed, from the indivisible stochastic process specified by the laws $\Gamma(t)$, to a Hilbert space formulation of that same process. Since $\Gamma_{i,j}(t) \in [0,1]$ we can introduce an $N \times N$ potential matrix $\Theta(t)$ consisting of complex-valued matrix elements $\Theta_{i,j}(t)$ related to $\Gamma_{i,j}(t)$ by the modulus squared relation

$$\Gamma_{i,j}(t) = |\Theta_{i,j}(t)|^2.$$

The matrix $\Theta(t)$ is not unique. Let $P_i = diag(0,\ldots,1,0,\ldots,0)$ denote the $N \times N$ diagonal matrix with value one at row $i$ and column $i$. We can represent the elements of the matrix $\Gamma(t)$ as the trace of the composite of four matrices,

$$\Gamma_{i,j}(t) = tr( \Theta^{\dagger}(t) P_j \Theta(t) P_i )$$

where $\Theta^{\dagger}(t)$ is the conjugate transpose matrix of $\Theta(t)$. We call this equation the dictionary formula as it provides the translation between indivisible stochastic processes, as represented by the left-hand side, and the formalism of quantum-theoretic Hilbert spaces as represented on the right-hand side. This dictionary formula is the basic ingredient of the stochastic-quantum correspondence.

Given an initial probability measure $p(0)$ on $\Omega_0$ with components $p_j(0)$ we can define the density matrix $\rho(0) = diag(p_1(0),p_2(0),\ldots,p_N(0))$. We then define a time-dependent density matrix by

$$\rho(t) = \Theta(t) \rho(0) \Theta^{\dagger}(t)$$

which is not generally diagonal, but is self-adjoint, positive semi-definite, and has trace equal to one.

Provided the density matrix is of rank one we can write the density matrix as the outer product of an $N \times 1$ column vector $\Psi(t)$ and its complex-conjugate-transpose (adjoint)

$$\rho(t) = \Psi^{\dagger}(t) \Psi(t).$$

The state vector or wave function $\Psi(t)$ then evolves over time according to

$$\Psi(t) = \Theta(t) \Psi(0),$$

and $\Psi(t)$ satisfies the property $||\Psi(t)||=1$. Thus we have derived the Hilbert space viewpoint of the stochastic process. This procedure can be reversed using the dictionary formula.

Note that the measurement process for an indivisible stochastic process can be modeled within the framework of the Kleisli category of the Giry monad $(G, \eta, \mu)$ as follows:

If $\Omega_t \xrightarrow{f} \mathbb{R}$ is a measurable function then $G(\Omega_t) \xrightarrow{G(f)} G(\mathbb{R})$ just pushes the probability measure on $\Omega_t$ forward to a probability measure on $\mathbb{R}$. If we want to characterize the measurement of the process $\Gamma$ in terms of the expected value of the measurement $f$ then the model of the process, measurement, and computation of the expected value can be displayed within the larger category of algebras of the Giry monad as

where the standard Borel space $\mathbb{R}_{\infty}$ is the one point compactification of the real line. (There is no $G$-algebra $G(\mathbb{R}) \rightarrow \mathbb{R}$; so to model any process with measurement $\Omega \xrightarrow{f} \mathbb{R}$ we first need to embedd $\mathbb{R}$ into $\mathbb{R}_{\infty}$.) The operator $\mathbb{E}_{\bullet}(id_{\mathbb{R}_{\infty}})$ is defined at each $P \in G(\mathbb{R}_{\infty})$ by $\mathbb{E}_{P}(id_{\mathbb{R}_{\infty}}) =\int id_{\mathbb{R}_{\infty}} \, dP$, yielding the expected value

$$\mathbb{E}_{\mu(G(\Gamma(t)(P)))}(f) = \int_{\omega \in \Omega} \Big( \int_{x \in \Omega} f(x) \, d\Gamma_t(\bullet | \omega) \Big) \, dP.$$

Editorial Note: The dictionary formula only makes sense for the configuration space $\Omega$ being a countable measurable space. If $\Omega$ is a standard Borel space with cardinality of the continuum and $P$ is a probability measure on $\Omega$ such that $P(\{i\})=0$ for every $i \in \Omega$ then $\Gamma_{i,j}(t)=0$. Hence our assumption that $\Omega$ be a countable measurable set. Note however that Barandes 2025 (page 5) claims that the stochastic-quantum correspondence extends to uncountable spaces provided the dictionary formula is characterized in terms of probability density functions. He has yet to provide that correspondence and, at present, this nLab author does not understand how that construction can be carried out.

Interpretations

Being developed: How quantum interference, decoherence, entanglement arise within the framework of indivisible stochastic processes.

History

The development of the indivisible stochastic interpretation of quantum systems is due to Jacob Barandes.

Related concepts

• The Interpretation of Quantum Mechanics

• SEP: Copenhagen interpretation of qm

• SEP: qm:collapse theories,

• SEP: many-world interpretation of qm,

• SEP: modal-interpretations of qm

• SEP: Everett’s relative-state formulation of qm

• hidden variable theory

  • Bohmian mechanics

References

The development of the indivisible stochastic process interpretation is based upon the following three articles:

• Jacob Barandes, Quantum Systems as Indivisible Stochastic Processes [arXiv:2507.21192]

• Jacob Barandes, The Stochastic-Quantum Theorem [arXiv:2309.03085[quant-ph]]

• Jacob Barandes, The Stochastic-Quantum Correspondence [arXiv:2302.10778[quant-ph]]

For a discussion of the measurement problem arising in other interpretations of quantum mechanics see

• David Wallace, The Quantum Measurement Problem: State of Play (arXiv:0712.0149)