This is the second part of a wide-ranging conversation between Curt Jaimungal and Jacob Barandes, a theoretical physicist at Harvard and co-director of graduate studies there. They explore the foundations of quantum mechanics, focusing on Barandes’s reformulation of quantum theory through “indivisible stochastic processes,” a framework that replaces Hilbert spaces and wave functions with ordinary probability theory and non-Markovian dynamics. The discussion covers wave-particle duality, entanglement, Bell’s theorem, quantum field theory, quantum gravity, the many-worlds interpretation, and open research directions.
Wave-Particle Duality and the Nature of Quantum Waves
Schrödinger introduced the wave function in 1926 as a tool for computing energy levels in atoms, but he initially treated it as a physically real wave in a high-dimensional configuration space (3N dimensions for N particles), a view called wave function realism.
Einstein and Heisenberg both objected to this metaphysical picture. Einstein’s famous “God plays dice” letter to Max Born in December 1926 was immediately followed by a criticism of “waves in 3N-dimensional space,” a detail often lost in translation where the “N” was dropped.
Schrödinger later recanted wave function realism around 1928, moving closer to Born’s view that the wave function is a tool for computing measurement probabilities, though he briefly entertained an embryonic many-worlds-like idea.
Wave-particle duality is the idea that quantum systems display wave-like behavior (interference patterns in the double-slit experiment) and particle-like behavior (definite detection events), depending on what is measured.
The wave is never directly observed; only discrete detection events (“dots”) are seen, and the wave pattern is inferred statistically over many repetitions.
This is often confused with classical waves (like electromagnetic waves), but Schrödinger waves are abstract, complex-valued functions in configuration space, not physical waves in 3D space.
In Barandes’s indivisible stochastic approach, Schrödinger waves are not physically real. The theory predicts interference-like patterns from stochastic dynamics alone, without invoking actual waves.
Classical field waves (e.g., electromagnetic waves) are still valid and physically real; the claim is specifically that Schrödinger wave functions are not ontological.
Electrons have associated quantum fields (Dirac fields), but these are not the same as Schrödinger wave functions, and the Dirac field is not a classical field.
Quantum Field Theory and Gravity
Bohmian mechanics works well for systems of fixed numbers of non-relativistic particles but struggles with quantum fields, especially fermionic fields (like the Dirac field for electrons), and requires a preferred foliation of spacetime, making it difficult to reconcile with special relativity.
The indivisible stochastic approach has no pilot wave, no guiding equation, and no preferred foliation, making it more naturally generalizable to quantum fields.
The generalization is conceptually straightforward: replace particle positions with localized field intensities as degrees of freedom and apply the stochastic laws. However, quantum field theories (QFTs) are mathematically difficult because they have infinitely many degrees of freedom, and no empirically adequate QFT (including the Standard Model) has been rigorously defined.
Most practical QFT calculations use perturbation theory and renormalization, which work well up to certain energy cutoffs but break down at arbitrarily high energies. The Standard Model is expected to fail above roughly 10 TeV.
This suggests that nature might be described not by a single fundamental QFT but by a tower of “effective field theories,” each valid within certain energy bounds, raising deep questions about ultimate ontology.
Regarding whether spacetime itself fluctuates due to quantum uncertainty: in standard QFT on a fixed Minkowski background, spacetime coordinates are not fluctuating; only field values are uncertain. The Dirac-von Neumann axioms do not describe anything happening between measurements, so talk of “fluctuating fields” between measurements is not formally justified without an interpretive extension.
In the indivisible stochastic approach, fields really do fluctuate stochastically, but formulating a probabilistic theory for dynamical (curving) spacetime is obstructed because the metric tensor—which defines which directions are spacelike or timelike—would itself be fluctuating, making it unclear how to even define conditional probabilities.
Barandes suggests that a “probabilistic general relativity” (not quantum, just stochastic) is a missing intermediate step that should have been developed before attempting quantum gravity. He conjectures that a fully probabilistic version of general relativity might already be equivalent to quantum gravity within the indivisible stochastic framework.
Semi-classical quantum gravity, which plugs quantum expectation values into Einstein’s field equations, is problematic because expectation values in the Dirac-von Neumann formalism refer to measurement averages, not to “what is happening” in the absence of measurement. Using them as classical sources conflates two different categories.
Bell’s Theorem, Non-Locality, and Causation
Entanglement in textbook quantum mechanics arises when a composite system is in a superposition state that cannot be factorized into separate states for each subsystem (e.g., the singlet state |↑↓⟩ + |↓↑⟩). This leads to correlations stronger than any classical correlation.
In the EPR (Einstein-Podolsky-Rosen) thought experiment, two entangled particles are separated. Alice can choose to measure one of two incompatible observables on her particle, and depending on her choice, she can “steer” Bob’s distant particle into definite states for the corresponding observable. This is called quantum steering.
EPR argued that either Alice’s choice instantaneously influences Bob’s system (superluminal causation) or Bob’s system already “knew” what answers it would give for all possible measurements (hidden variables, meaning quantum mechanics is incomplete).
Bell’s 1964 theorem was intended to close the hidden variable escape route. Bell showed that any local hidden variable theory must satisfy an inequality that quantum mechanics violates, and experiments confirm the violation.
However, Bell’s theorem has been widely misinterpreted as proving that hidden variables are impossible. In fact, Bell explicitly acknowledged Bohmian mechanics as a viable (though “grossly non-local”) hidden variable theory. His result was that hidden variables do not restore locality, not that they are ruled out entirely.
Bell’s theorem is a mathematical theorem whose physical relevance depends on whether its premises (formulated in mathematical terms) adequately represent physical reality—a “connection problem.” The premises involve notions of “local beables” and “causation” that are not rigorously defined at the microphysical level.
The EPR argument and Bell’s theorem rely on an interventionist theory of causation (causal influences are defined by what happens when agents intervene), which is problematic in a fundamental theory where there are no agents or interventions at the microphysical level.
If causation is not a fundamental feature of nature (as Bertrand Russell argued), then the entire framework of “non-local causation” in Bell’s theorem becomes moot.
Bell’s 1975 generalization of his theorem introduced the principle of “local causality,” which requires Reichenbachian common cause factorization: if two distant beables are correlated and do not directly influence each other, there must exist common causes in their causal past that, when conditioned on, make the correlations factorize.
Barandes argues that Reichenbachian factorization is too strong a requirement. Quantum interactions that create entanglement are not the kind of “beables” that can be conditioned on in this way, so the factorization condition fails without implying non-local causation.
In the indivisible stochastic approach, the laws are formulated as directed conditional probabilities, which are the same mathematical objects used in causal modeling. Barandes has proposed a specific theory of microphysical causation based on these conditional probabilities and shown that, under this theory, EPR experiments do not involve non-local causal influences. The correlations arise from a common cause (the past interaction), but this common cause does not satisfy Reichenbachian factorization.
Entanglement in the Indivisible Stochastic Picture
In this framework, entanglement is not a state vector or superposition but a property of the stochastic dynamics. Two initially independent systems have their own factorizable stochastic laws. When they interact locally, their joint stochastic dynamics becomes non-factorizable.
Because the stochastic map is indivisible (non-Markovian), it cumulatively encodes the interaction’s effects. Even after the systems separate, their joint dynamics remains non-factorizable—they do not regain independent laws.
Entanglement is “broken” when a division event (an interaction with a measuring device or environment) occurs while the systems are far apart. This allows the stochastic dynamics to restart in a factorized form, giving each system its own independent law again.
This picture uses only ordinary probability theory (albeit non-Markovian) and does not require Hilbert spaces, wave functions, or superposition as ontological features.
The “memory” in the stochastic dynamics is not information stored in a physical sense (like qubits in space that could form a black hole via the Bekenstein bound); it is a feature of the laws themselves being non-Markovian.
Phase information in the Hilbert space representation is not lost in the stochastic picture; it is encoded in the indivisibility of the stochastic dynamics. When a measuring device is explicitly included in the model, phases are unnecessary—the overall stochastic process produces Born-rule probabilities directly. Phases become necessary only when the measuring device is excised from the description and replaced with an instantaneous collapse (the textbook Dirac-von Neumann approach), which is a valid approximation in the measurement regime.
Contrast with the Copenhagen Interpretation
Heisenberg’s version of the Copenhagen interpretation holds that the quantum world is fundamentally beyond human comprehension (which is limited to classical categories like 3D geometry and cause-effect). The mathematical formalism of quantum mechanics is merely an instrumentalist tool for predicting macroscopic classical outcomes; it does not describe what is really happening at the micro level.
This view practices “quietism” about the micro world and does not posit any substrate from which classical reality emerges. It also relies on an ill-defined “Heisenberg cut” between the quantum and classical domains.
The indivisible stochastic approach is not Copenhagen because it asserts that the micro world has an ontology (particles or fields with real configurations), that things really happen between measurements according to stochastic laws, and that classical reality emerges from this substrate in a principled way, analogous to how fluid water emerges from water molecules.
Problems with the Many-Worlds Interpretation
In the Everettian (many-worlds) interpretation, the universal wave function evolves deterministically, and every quantum measurement causes the universe to branch into multiple copies. This avoids wave function collapse but faces severe problems:
The preferred basis problem: There are infinitely many ways to decompose the universal wave function into branches. Which decomposition corresponds to the “real” branches? Decoherence dynamically picks out a preferred basis in which branches no longer interfere, but this makes branches emergent rather than fundamental.
The probability problem: If branches are emergent and not fundamental, the axioms of the theory cannot assign them probabilities. If branches are fundamental, it is unclear what it means to assign probabilities to outcomes that all occur. Branch-counting arguments (counting branches to derive the Born rule) do not work in general.
Decision-theoretic approaches (David Wallace, David Deutsch) attempt to derive the Born rule by arguing that rational agents using decision theory will assign probabilities according to the Born rule. These arguments require many extra-empirical assumptions (e.g., how to relate to copies of oneself, compatibilist free will, rationality axioms developed in a one-world context) and have been criticized by philosophers like John Norton for circularity (smuggling probability into the premises).
Barandes argues that the many-worlds interpretation is an extravagant metaphysical hypothesis not compelled by logic or experiment. Extraordinary claims require extraordinary evidence, and the interpretation only gets off the ground by adding a tower of speculative metaphysical hypotheses (SMHs). The more conservative approach is to take nature at face value: experiments yield single, probabilistic outcomes, and the theory should be built around that.
Open Research Directions
Mathematics of indivisible stochastic processes: This is a relatively new class of stochastic processes (appearing in the literature around 2021) with simple definitions but potentially profound implications. There is room for pure mathematical investigation.
Application to quantum field theory and the standard model: Formulating QFTs in the indivisible stochastic language is straightforward in principle and could reveal new features or simplify existing mathematical difficulties. This is an open and accessible research problem.
Dynamical symmetries: These play an interesting role in the new framework and deserve further study.
Statistical mechanics and the origin of probabilities: Classical statistical mechanics struggles to explain where its probabilities come from (the ergodic hypothesis fails for many systems, and information-theoretic approaches face deep problems). If the fundamental laws are stochastic, probabilities come from the laws themselves rather than from initial conditions, offering a new approach to foundational problems like the past hypothesis and the arrow of time.
Thermal fluctuations in statistical mechanics are distinct from quantum fluctuations; the indivisible stochastic approach provides a foundation for Boltzmannian statistical mechanics, from which thermal phenomena (temperature, thermal equilibrium) can be derived.
Quantum simulation and computing: Since Hilbert space descriptions are dual to stochastic descriptions, quantum hardware might efficiently simulate a wide range of non-Markovian stochastic systems that are otherwise hard to simulate.
C-star algebraic formulation: Reformulating the theory in the language of C-star algebras (an alternative to Hilbert space formulations) could be useful for certain physical systems and might suggest generalizations of quantum theory that would be natural from a probability-first starting point.
Quantum gravity: The framework suggests new approaches, such as asking what a fully probabilistic (not quantum) version of general relativity would look like, potentially as a stepping stone to or even an equivalent of quantum gravity.
On Being a Researcher and Building a Reputation
Barandes reflects on how his lack of “common sense” (which he considers a weakness in everyday life) has become an advantage in foundations of physics, where common sense often leads people astray and where the ability to disassemble arguments without relying on intuition is valuable.
He encourages people to put themselves in different contexts to discover whether features they consider weaknesses might actually be strengths in the right setting.
He advises students and researchers that the most important reputation to cultivate is not one of intimidating intelligence but one of being collaborative, helpful, and someone who makes others feel smarter after talking with them. This kind of reputation leads to more collaborations, support, and long-term success.
He emphasizes treating everyone with basic respect and as a potential future collaborator or supporter, because you never know who will be important in your life.
