Bi-Trajectory Formalism in Quantum Mechanics

• Bi-Trajectory Formalism is a reconstruction of quantum mechanics where the primitive object is a complex measure on pairs of trajectories rather than a single state probability.
• It recovers standard quantum probabilities and Hilbert-space representations as emergent features from phenomenologically derived multi-time measurement data.
• The framework addresses interference, non-classical coarse-graining, and the quantum-classical transition by generalizing the Kolmogorov extension theorem to bi-probabilities.

Bi-Trajectory Formalism is a reconstruction of quantum mechanics in which the primitive object is not a state vector or density operator, but a complex measure on pairs of trajectories. It is developed from phenomenological inputs only: experimentally estimated multi-time probability distributions for sequential measurements, with no prior assumption of Hilbert spaces, operators, or Born’s rule. In this formulation, ordinary single-time and multi-time quantum probabilities are recovered as diagonal restrictions of complex-valued bi-probabilities on pairs of histories, and the usual Hilbert-space apparatus emerges as a representation rather than a starting axiom (Szańkowski et al., 2024, Szańkowski et al., 7 Jul 2025).

1. Phenomenological basis

The starting data are multi-time probability distributions for ordered sequences of measurement outcomes,

$P^{F_n,\dots,F_1}_{t_n,\dots,t_1}(f_n,\dots,f_1),$

estimated from repeated runs in which devices measuring observables $F_1,\dots,F_n$ are deployed at times $t_1<\dots<t_n$. Each sequence $f=(f_1,\dots,f_n)$ is treated as a basic event in the sample space $$\Omega(F_n,\dots,F_1)=\Omega(F_n)\times\cdots\times\Omega(F_1)$$. Devices are introduced operationally as classical macroscopic objects with finite outcome sets $\Omega(F)$; projectors and operators are not assumed initially (Szańkowski et al., 2024).

A central phenomenological distinction is made between perfectly fine-grained devices, denoted $K,L,\dots$, and coarse-grained devices, denoted $F,G,\dots$. Perfectly fine-grained devices are those for which no further refinement is possible and which yield Markovian multi-time statistics. Coarse-grained devices are obtained by grouping outcomes of a finer device through a resolution

$$\mathrm{Res}(F\mid F)=\{\,\omega(f)\subset \Omega(F)\mid f\in\Omega(F),\ \bigsqcup_f \omega(f)=\Omega(F)\,\},$$

so that a coarse-grained outcome corresponds to a set of fine outcomes (Szańkowski et al., 2024).

The empirical observations driving the formalism are highly specific. Marginalizing over the last outcome yields the shorter-sequence distribution, but marginalizing over intermediate outcomes does not, in general, reproduce a valid shorter-sequence distribution. This failure of Kolmogorov consistency rules out an underlying single-trajectory probability measure $P[f(\cdot)]$. Coarse-graining is also nonclassical: replacing a fine-grained device by a coarse-grained one in the middle of a sequence changes probabilities in a way that cannot be reproduced by classical post-processing, although ordinary terminal marginalization is recovered at the last time step. Interference is observed to be pairwise rather than genuinely higher-order, and independent subsystems exhibit factorized probabilities while still allowing interference effects for coarse devices that do not perfectly resolve “which subsystem” (Szańkowski et al., 2024, Szańkowski et al., 7 Jul 2025).

Further phenomenological constraints fix the temporal structure. Perfectly fine-grained measurements of a single observable $F_1,\dots,F_n$0 yield Markovian chains,

$F_1,\dots,F_n$1

initialization is defined operationally as a first $F_1,\dots,F_n$2-measurement at $F_1,\dots,F_n$3, rapid repeated fine-grained measurements exhibit the quantum Zeno effect with quadratic survival probability,

$F_1,\dots,F_n$4

and inequivalent fine-grained observables $F_1,\dots,F_n$5 satisfy short-time transition limits

$F_1,\dots,F_n$6

These inputs are treated as phenomenological constraints from which the formal structure is deduced (Szańkowski et al., 2024).

2. Bi-probabilities and the doubling of trajectories

Because observed multi-time probabilities violate classical consistency while interference remains pairwise, the formalism replaces ordinary probabilities on single sequences by complex-valued functions on pairs of sequences,

$F_1,\dots,F_n$7

These objects are called bi-probability distributions or decoherence functionals. Their defining operational requirement is that the observed probabilities appear on the diagonal: $F_1,\dots,F_n$8 For coarse-grained outcomes, the relevant quantity is obtained by summing over the appropriate fine outcomes in both arguments of $F_1,\dots,F_n$9 (Szańkowski et al., 2024).

The bi-probabilities satisfy a set of structural conditions. They are normalized, Hermitian, bi-consistent under summation over intermediate times in both arguments, causal in the sense that the last outcomes on the two branches must coincide for the kernel to be nonzero, factorized for independent systems, and positive semi-definite as kernels: $t_1<\dots<t_n$0 In matrix form, the kernel $t_1<\dots<t_n$1 is positive semi-definite (Szańkowski et al., 2024).

This doubling is not an auxiliary convenience but the minimal structure compatible with the phenomenology. The real part,

$t_1<\dots<t_n$2

is the interference term used in pairwise decompositions of probabilities, while the imaginary part is required by interference between independent subsystems under coarse-grained measurements. A common misconception is to regard the formalism as a hidden-variable model on ordinary trajectories. The construction does not support that interpretation: the relevant master object is a complex measure on pairs of trajectories, not a positive probability measure on single trajectories (Szańkowski et al., 2024, Lonigro et al., 2024).

Bi-consistency suggests an analogue of the Kolmogorov extension picture. Instead of a measure on paths $t_1<\dots<t_n$3, one obtains a measure on pairs of paths

$t_1<\dots<t_n$4

written formally as

$t_1<\dots<t_n$5

Finite-time bi-probabilities are then cylinder restrictions of this master bi-measure, and empirical probabilities are recovered by restricting to the diagonal $t_1<\dots<t_n$6 at the measured times (Szańkowski et al., 2024).

3. Generalized extension theorem

The mathematical backbone of the formalism is a generalization of the Kolmogorov extension theorem to complex-valued bi-probabilities. In the classical case, consistency of finite-time distributions,

$t_1<\dots<t_n$7

guarantees the existence of a probability measure on trajectory space. Quantum sequential-measurement probabilities do not satisfy this condition in general, so no single-trajectory Kolmogorov representation exists (Lonigro et al., 2024).

The bi-probability family does satisfy a replacement condition: bi-consistency. For an abstract family $t_1<\dots<t_n$8 of complex-valued bi-probabilities on a finite outcome set, the extension theorem requires two hypotheses: bi-consistency and uniform boundedness in $t_1<\dots<t_n$9-norm,

$f=(f_1,\dots,f_n)$0

Under these assumptions, there exists a unique complex measure $f=(f_1,\dots,f_n)$1 on the space of trajectory pairs such that every finite-time bi-probability is recovered as a cylinder functional (Lonigro et al., 2024).

In the quantum case, for a finite-dimensional Hilbert space, density operator $f=(f_1,\dots,f_n)$2, PVM $f=(f_1,\dots,f_n)$3, and continuous Hamiltonian $f=(f_1,\dots,f_n)$4 on a compact interval $f=(f_1,\dots,f_n)$5, the bi-probabilities

$f=(f_1,\dots,f_n)$6

meet those hypotheses. The nontrivial step is the uniform bound

$f=(f_1,\dots,f_n)$7

which yields the existence of a master bi-measure $f=(f_1,\dots,f_n)$8 on the trajectory-pair space (Lonigro et al., 2024).

Within the broader phenomenological reconstruction, this theorem underwrites the claim that quantum multi-time statistics arise from one pair of trajectories rather than from one single trajectory. The slogan “double down on trajectories” has a precise technical meaning: the inconsistent ordinary probabilities are the diagonal slice of a consistent bi-probability family on doubled path space (Lonigro et al., 2024). In the phenomenological program, this supports the conclusion that the formalism is unique up to unitary isomorphism at the level of Hilbert-space representation, while the bi-measure itself is the more primitive object (Szańkowski et al., 2024).

4. Hilbert-space representation

Positive semi-definiteness of the bi-probability kernel allows a Hilbert-space representation through Gudder’s theorem. Given a finite outcome set, one constructs a complex vector space with basis vectors labelled by outcome sequences and defines an inner product using the kernel $f=(f_1,\dots,f_n)$9. After quotienting by the null space, this yields a Hilbert space in which

$$\Omega(F_n,\dots,F_1)=\Omega(F_n)\times\cdots\times\Omega(F_1)$$0

Equivalently, relative to a reference inner product, there exists a positive operator $$\Omega(F_n,\dots,F_1)=\Omega(F_n)\times\cdots\times\Omega(F_1)$$1 such that $$\Omega(F_n,\dots,F_1)=\Omega(F_n)\times\cdots\times\Omega(F_1)$$2 (Szańkowski et al., 2024, Szańkowski et al., 7 Jul 2025).

For single-time measurements, this representation yields the measurement–projector link. A measuring device at time $$\Omega(F_n,\dots,F_1)=\Omega(F_n)\times\cdots\times\Omega(F_1)$$3 corresponds to a complete family of orthogonal projectors $$\Omega(F_n,\dots,F_1)=\Omega(F_n)\times\cdots\times\Omega(F_1)$$4 on a system Hilbert space $$\Omega(F_n,\dots,F_1)=\Omega(F_n)\times\cdots\times\Omega(F_1)$$5,

$$\Omega(F_n,\dots,F_1)=\Omega(F_n)\times\cdots\times\Omega(F_1)$$6

and coarse-graining corresponds to summing projectors,

$$\Omega(F_n,\dots,F_1)=\Omega(F_n)\times\cdots\times\Omega(F_1)$$7

Initialization is represented by a density operator $$\Omega(F_n,\dots,F_1)=\Omega(F_n)\times\cdots\times\Omega(F_1)$$8, so the single-time probabilities become

$$\Omega(F_n,\dots,F_1)=\Omega(F_n)\times\cdots\times\Omega(F_1)$$9

with

$\Omega(F)$0

This is the Born rule, but here it is deduced rather than assumed (Szańkowski et al., 2024).

The multi-time structure is more delicate. For a sequence of measurements $\Omega(F)$1 at times $\Omega(F)$2, the full bi-probability is represented by

$\Omega(F)$3

and the experimentally accessible probabilities are its diagonal,

$\Omega(F)$4

The associated effect operator

$\Omega(F)$5

is generally not a projector (Szańkowski et al., 2024).

This last point marks one of the decisive differences from standard textbook presentations. The correspondence “measurement $\Omega(F)$6 projector” holds cleanly for a single device, but it does not extend literally to an entire sequence of measurements. Sequential measurement statistics require the full double-chain trace expression, or equivalently the underlying bi-trajectory description (Szańkowski et al., 2024, Szańkowski et al., 7 Jul 2025).

5. Dynamics, composition, and observables

The dynamical law is inferred from the phenomenology of fine-grained measurements. For a fine-grained observable $\Omega(F)$7, the projectors at different times are related by a unitary,

$\Omega(F)$8

Comparison of different observables $\Omega(F)$9 and $K,L,\dots$0, together with time-independence of the correlation matrix $K,L,\dots$1, yields a single unitary $K,L,\dots$2 common to all observables,

$K,L,\dots$3

Using the quadratic short-time behavior implied by the Zeno effect, one shows that $K,L,\dots$4 is differentiable and unitary, with generator

$K,L,\dots$5

Imposing stationarity under absolute time shifts gives a time-independent Hamiltonian,

$K,L,\dots$6

and therefore the usual unitary evolution of states,

$K,L,\dots$7

(Szańkowski et al., 2024, Szańkowski et al., 7 Jul 2025).

For composite systems, factorization of bi-probabilities for independent subsystems yields the standard tensor-product structure,

$K,L,\dots$8

For independent subsystems, the Hamiltonian is

$K,L,\dots$9

while coupling introduces interaction terms,

$F,G,\dots$0

Initialization of independent subsystems is represented by product states, and projectors for local observables take the product form $F,G,\dots$1 (Szańkowski et al., 2024, Szańkowski et al., 7 Jul 2025).

The operator notion of observable is then recovered in standard spectral form. Any Hermitian operator

$F,G,\dots$2

is interpreted as the Hilbert-space representative of a measurable observable whose device projectors are $F,G,\dots$3. In this sense, states, observables, measurements, and dynamics are exactly those of standard quantum mechanics at the empirical level. The difference is not predictive but structural: in the reconstructed formalism they are representational elements derived from the more basic bi-trajectory measure (Szańkowski et al., 2024).

6. System-level formulation, relations to other frameworks, and limitations

Beyond observable-specific bi-probabilities, the formalism introduces a system-level construction. Fixing a reference basis $F,G,\dots$4 and parameterizing basis changes by $F,G,\dots$5, one defines coordinate-labelled projectors

$F,G,\dots$6

System bi-probabilities are then defined on tuples $F,G,\dots$7 by

$F,G,\dots$8

Observable-level bi-probabilities are recovered from these system bi-probabilities by projecting onto eigenspaces and mixing over initial $F,G,\dots$9 according to the initialization distribution. This leads to the proposed master system bi-trajectory measure on maps

$$\mathrm{Res}(F\mid F)=\{\,\omega(f)\subset \Omega(F)\mid f\in\Omega(F),\ \bigsqcup_f \omega(f)=\Omega(F)\,\},$$0

(Szańkowski et al., 7 Jul 2025).

The formalism has explicit affinities with several existing approaches. It is close to consistent histories and quantum measure theory in that the bi-probability kernel is essentially a decoherence functional, and Gudder’s theorem is used directly in the reconstruction. It also resembles forward–backward path-integral and Schwinger–Keldysh structures, since the doubled trajectories play the role of two branches. Relative to quantum combs or process tensors, the relation is structural but not identical: combs encode multi-time processes through functionals on sequences of completely positive maps, whereas the bi-trajectory formalism keeps the trajectory concept tied to definite outcomes of observables and doubles it, allowing non-CP “bi-instruments”

$$\mathrm{Res}(F\mid F)=\{\,\omega(f)\subset \Omega(F)\mid f\in\Omega(F),\ \bigsqcup_f \omega(f)=\Omega(F)\,\},$$1

to reproduce bi-probabilities (Lonigro et al., 2024, Szańkowski et al., 2024).

Its conceptual claims are correspondingly restrained. The formalism is presented as empirically equivalent to standard quantum mechanics, not as a new physical theory. It does not invoke a collapse postulate or a Heisenberg cut, because all sequential experiments are treated within a single descriptive scheme. The classical limit is identified with diagonalization of the bi-measure,

$$\mathrm{Res}(F\mid F)=\{\,\omega(f)\subset \Omega(F)\mid f\in\Omega(F),\ \bigsqcup_f \omega(f)=\Omega(F)\,\},$$2

or, equivalently, collapse of the bi-trajectory measure onto $$\mathrm{Res}(F\mid F)=\{\,\omega(f)\subset \Omega(F)\mid f\in\Omega(F),\ \bigsqcup_f \omega(f)=\Omega(F)\,\},$$3, at which point full Kolmogorov consistency and an ordinary stochastic process are recovered. This suggests a reformulation of the quantum–classical transition in terms of suppression of off-diagonal bi-trajectory structure rather than state collapse (Szańkowski et al., 2024, Szańkowski et al., 7 Jul 2025).

The present scope remains limited. The reconstruction is carried out for finite-dimensional Hilbert spaces, finite outcome sets, and projective measurements. Extensions to continuous outcome spaces, infinite-dimensional systems, quantum field theory, and relativistic settings are left open. For the single-observable case, the existence of the bi-trajectory measure is proved rigorously; for the system-level master measure, existence is argued but not established with the same degree of rigor. On infinite time intervals, the required uniform bound generally fails, so the extension theorem is formulated on finite time windows. In the multi-observable case, a uniform bound analogous to the single-observable bound is not proved without extra structure (Lonigro et al., 2024, Szańkowski et al., 7 Jul 2025).

In sum, the Bi-Trajectory Formalism reconstructs quantum mechanics as a theory of complex measures on pairs of trajectories. Its distinctive claim is not that standard Hilbert-space quantum mechanics is replaced, but that it can be deduced from operational multi-time probabilities once one recognizes that the correct extension object is a bi-measure on doubled trajectory space rather than a probability measure on single paths (Szańkowski et al., 2024).