How Nested Quantum Observers in Infinite Spaces Create Irreversibility

A new theoretical study is turning one of physics’ oldest headaches on its head by suggesting that the quantum world’s apparent switch from reversible to irreversible behavior may emerge naturally when systems grow to infinite complexity. In a paper published in the open-access journal Foundations of Physics, Karl Svozil of the Institute for Theoretical Physics at TU Wien argues that the notorious quantum measurement problem—the puzzle of how definite outcomes arise from quantum superpositions—could dissolve once infinite tensor products are taken seriously as physical objects, with an endless chain of “Wigner’s friends” serving as the conceptual vehicle for the transition.

At the heart of the argument lies a distinction that has troubled physicists since John von Neumann formalized quantum measurement in the 1930s. The Schrödinger equation, which governs the evolution of isolated quantum systems, is unitary: it is reversible, deterministic in its way, and preserves all information about the past. Yet measurement, as actually experienced in laboratories, is manifestly irreversible. Von Neumann himself divided quantum theory into two processes, labeling the smooth, unitary evolution “process 2” and the abrupt, non-unitary collapse of the wavefunction “process 1.” The question that has defied nearly a century of interpretation-building is how process 1 can possibly emerge from process 2, given that a group of unitary transformations is, as Svozil puts it, “hermetic”—closed under its own operations and seemingly incapable of generating anything irreversible from within.

Mainstream responses to this conundrum fall into well-known camps. Decoherence theory explains the apparent collapse as the leaking of quantum information into an environment so large that retrieving it is practically impossible, though the global evolution remains strictly unitary. Objective collapse models, such as GRW-type theories, modify the dynamics to include a genuine physical collapse mechanism. Everettian many-worlds interpretations simply deny collapse, asserting that all outcomes occur in branching universes. Svozil’s approach stands apart from all three: rather than altering the postulates of quantum mechanics for finite systems, he explores what happens in the mathematical limit of infinite complexity, where the very structure of the Hilbert space changes character.

The technical vehicle for this exploration is the infinite tensor product, a construction that von Neumann himself pioneered in 1939 under the name “incomplete infinite direct products.” When infinitely many quantum systems are stitched together into a single tensor product, familiar properties of finite-dimensional quantum mechanics begin to fail in dramatic ways. Consider the inner product between two such infinite product states. If the factors are only slightly mismatched—each overlap being, say, a value slightly less than one—the infinite product of these overlaps behaves approximately like the exponential of a divergent sum and converges to zero. Two states that differ “slightly” in each of infinitely many components therefore become exactly orthogonal. As Svozil demonstrates with explicit examples, vectors differing in just a single subfactor out of infinitely many also end up with a vanishing inner product, and an infinite tensor product of elementary projection operators, which intuition suggests should be a well-behaved bounded operator, turns out to annihilate almost every vector it acts upon while leaving a measure-zero set untouched.

These pathologies are not mere curiosities; they reorganize the Hilbert space itself. Von Neumann recognized that the infinite tensor product “splits up” into what are now called superselection sectors—equivalence classes of states that differ from one another in at most finitely many subfactors, or are otherwise unitarily close. Within a sector, the infinite product of overlaps converges to a nonzero value and quantum mechanics functions normally. Between sectors, however, the inner product vanishes identically. Svozil illustrates the idea with an infinite array of spin-1/2 particles: the state in which every spin points up belongs to one sector, the all-down state belongs to another, and an alternating up-down pattern to a third. These sectors correspond to distinct macroscopic configurations—different average magnetizations, in this case—and no finite unitary operation, which can only modify finitely many components, can rotate one sector into another. Superpositions across sectors become physically meaningless; coherence between them is, in a precise mathematical sense, lost.

Svozil then connects this machinery to the celebrated Wigner’s friend scenario. In Eugene Wigner’s original thought experiment, a friend measures a quantum system inside a sealed laboratory while Wigner, outside, treats the entire laboratory—including the friend—as a single quantum system in superposition. By nesting this scenario recursively—friend observing friend observing friend, ad infinitum—one generates precisely the infinite tensor products discussed above. A single measurement interaction is standard unitary evolution of the combined system. But an infinite sequence of such interactions, the formalism suggests, forces the total state into one of the newly formed sectors, and this transition to an orthogonal sector is what constitutes a definite, macroscopic measurement outcome. Each subsequent measurement in an incompatible basis drives the system into yet another sector, a process Svozil describes as the “reshuffling” or “scrambling” of contexts.

Crucially, this decoherence mechanism is graded rather than abrupt. If each friend in the chain introduces only a minuscule mismatch—a per-link overlap close to but not exactly equal to one—the accumulated product still converges to zero. The loss of information about the originally prepared state is smooth and continuous, with no sharply localized “Heisenberg cut” at any particular scale, yet it culminates in complete orthogonality in the infinite limit. Svozil interprets each small mismatch as a form of stochastic input contributed by the friend or the environment, information unrelated to the original quantum state that cumulatively destroys coherence. He is careful to note, following John Bell, that any finite number of Wigner’s friends does not violate unitary equivalence—the breakdown is strictly a transfinite phenomenon.

To motivate the idea that infinite limits can generate genuinely new physics, Svozil marshals analogies from mathematics and statistical physics. In number theory and analysis, finite operations on rational numbers can never escape the rationals, yet infinite processes—Cantor’s diagonalization, or the construction of Specker sequences of computable numbers converging to uncomputable limits such as Chaitin’s halting probability—produce irrational, incomputable, and algorithmically random numbers. In statistical physics, the same pattern appears in the debate over irreversibility that Loschmidt raised against Boltzmann. If microscopic laws are time-reversible, how can macroscopic entropy increase be irreversible? The answer is that irreversibility is a limit phenomenon: once the limit is reached and no record of the path is kept, different microscopic routes to the same macrostate become indistinguishable. Svozil illustrates this with two different infinite expansions of the square root of two—the continued fraction and the binomial series—which share no common terms yet converge to the same limit; once arrived, no memory remains of which route produced it.

The paper also engages the deeper algebraic structure of infinite quantum systems through von Neumann algebras and their classification into factors of types I, II, and III. Unitary transformations preserve the type of a factor, so no sequence of unitary operations can convert, for instance, a type I factor built from finite-dimensional matrix algebras into the diffuse, trace-structured type II factors or the type III factors that admit no conventional density-operator description at all. Transitioning between types requires more powerful machinery, such as inductive limits. Svozil conjectures that the infinite limit of nested Wigner’s friends, mediated by entanglement, could enable such transitions, meaning that both sectorization and factorization might contribute, through different mechanisms, to the loss of unitary equivalence and the emergence of classical behavior.

What distinguishes this framework from standard decoherence theory is the nature of the resulting irreversibility. In conventional decoherence, global evolution remains unitary and coherence is merely leaked into environmental degrees of freedom—locally inaccessible, but never truly destroyed; classicality is an appearance sustained for all practical purposes. In the infinite tensor product limit, by contrast, the breakdown of unitary equivalence between macroscopic outcomes is formal and absolute: states in different sectors are structurally non-interconvertible by any finite means, not merely difficult to reconnect. The sectors play the role of decoherence’s pointer basis, but the transition between them is not a matter of practical inaccessibility—it is a feature of the geometry of infinite-dimensional Hilbert space itself.

Svozil acknowledges the epistemological weight of these commitments. Objections to transfinite recursion in physics, notably Bell’s critique of the Coleman-Hepp model, remain valid against infinite friend chains, and one can always retreat to a pragmatic stance in which the infinite is replaced by “FAPP unboundedness”—too large to handle. Yet he argues that refusing infinite limits altogether has its own costs: without transfinite capacities, even classical motion through a continuous spacetime succumbs to Zeno’s paradoxes, as Hermann Weyl observed. His concluding position is pointedly symmetrical: whatever stance one takes toward Loschmidt’s reversal paradox in classical statistical mechanics, the same stance should apply to quantum measurement. In both domains, irreversibility in the limit may simply be what “more is different” ultimately means.