Which part is the Hamiltonian is a convention – a census of the Lindblad equation.
The Lindblad equation writes the evolution of an open system as a unitary part plus a dissipative part. This paper writes that the split is not unique, what fence makes it unique, and that the word decoherence names two different destinations. NO NEW MATHEMATICAL THEOREM AND NO NEW PHYSICAL LAW IS CLAIMED. Lindblad (1976), Gorini-Kossakowski-Sudarshan (1976), Stinespring’s dilation theorem, the Peres-Horodecki criterion and the relation between the relaxation times are all standard. Paper 118 said that the small quantity of time and the small quantity of entropy are of different logical type, and the Lindblad equation looks like the two of them set side by side in one line; this paper asks how far that arrangement is unique.
First, one of the two cores. The split is not unique. Add a multiple of the identity to a jump operator, and at the same time move the Hamiltonian by half the imaginary unit times the difference between that constant times the adjoint of the jump operator and its conjugate times the jump operator. The difference of the two generators is then exactly the zero matrix (confirmed here symbolically, along with the Hermiticity of the added term). The same evolution is thus written with different Hamiltonians and different jump operators, so the statement that the Hamiltonian of this system is such-and-such has no content by itself: which decomposition was taken must be said. The reason is visible too. Adding the identity to a jump operator produces a commutator inside the dissipative term, and a commutator has the same shape as the unitary term, so it can be moved from one side to the other. One word, Hamiltonian, is naming a quantity that depends on our choice (Paper 95).
Second, there is a fence that makes it unique. This paper computes the trace of the original jump operator as zero and the trace after the shift as twice the constant, so requiring traceless jump operators forces the constant to vanish. Imposing tracelessness makes the decomposition unique up to a constant in the Hamiltonian, and fixing the Hamiltonian to be traceless as well makes it completely unique. This is the same shape as Papers 127 and 133: uniqueness lives only inside a fence and must not be accepted without counting it, and here the fence is tracelessness. There is a practical consequence: what an experiment measures is the generator, not its decomposition, so a quantity such as the size of the Lamb shift becomes a number only once the traceless convention is adopted.
Third, the other core. Complete positivity is a demand about something one is not touching. A map is positive if it sends density matrices to density matrices, and completely positive if the map extended by the identity on a spectator of any size is still positive. The transpose map is positive: over twenty thousand random single-qubit states this paper finds the smallest eigenvalue after transposition to be positive, the minimum being 3.7 times ten to the minus five. Nothing breaks so long as one looks at the system alone. But transposing only one half of a Bell state gives eigenvalues of minus one half and three times one half, so a negative eigenvalue appears. Complete positivity is therefore a demand about something one is not touching, and it cannot be checked by looking inside the system. The Lindblad form has its present shape as a consequence of that demand (Stinespring’s dilation theorem). This is also an instance of the corpus’s distinct roots: positive and completely positive look like one family of words, and neither implies the other automatically. The separator is whether a spectator is present, and the transpose map demonstrates it.
Fourth, the coherence time cannot exceed twice the relaxation time. From the rate of amplitude damping and the rate of pure dephasing, the reciprocal of the coherence time is the sum of the reciprocal of twice the relaxation time and the reciprocal of the pure dephasing time. In the table here, with no pure dephasing the coherence time sits exactly at the bound of twice the relaxation time; with a dephasing time of ten it is 1.6667, with one it is 0.6667, and with a tenth it is 0.0952. So the one word decay names two rates, energy relaxation and phase randomisation. Only their combination is observed, and the coherence time alone does not reveal the breakdown. But if it sits at the bound, one learns that there is no pure dephasing: touching the bound is itself information.
Fifth, the statement that decoherence destroys purity is false in general. Computing the purity from an equal superposition, amplitude damping takes it down to 0.875 at time 0.693 and then back up, reaching 0.999977 at time ten. Pure dephasing takes it down monotonically, reaching 0.500000 at time ten. The reason lies in the fixed point: amplitude damping at zero temperature has a pure state, the ground state, as its fixed point, so the purity comes back. Being dissipative therefore does not by itself decide where the system ends; the fixed point of the map does, and within the one frame of an open system there are processes ending pure and processes ending mixed. A contrast with Paper 135 can also be drawn: there a symplectic map had determinant exactly one and the volume was preserved exactly, whereas here it is not, and that is not a defect of an integrator but the physics being described. Two things called not preserved, numerical error and physical dissipation, are distinct roots.
IMPORTANT (scope): a census of established standard material (Layer 2). Every work cited is standard. What is not claimed: any new mathematical theorem (there is none) / any new physical law (there is none) / that the Lindblad equation is in error (it is not) / that the Hamiltonian is physically meaningless (it is not; the point is one of standing, namely that it becomes a number only once a decomposition is specified). What is added: exhibiting the gauge freedom and showing the difference of generators to be zero, writing the fence that makes the split unique, demonstrating complete positivity as a demand about a spectator, and separating the two destinations of decoherence. What remains: in writing the equation of an open system, the line between unitary and dissipative is drawn by us, and the shape of the equation itself comes from the demand that nothing break in a system we are not touching. Care for what is not being looked at is what fixes the form of the equation for what is.
On the making of this work: The ideas and content of this work stem from the author’s own considerations. Assistance from an AI (a large language model) was used for structuring, English translation, and checking the algebra. Any remaining errors or misinterpretations are solely the author’s. Feedback and corrections are sincerely appreciated.
Reference
Yamagishi, Y. (2026). Which Part Is the Hamiltonian Is a Convention [Paper 138]. Zenodo.
DOI: 10.5281/zenodo.22081512
ORCID: 0009-0006-8587-3450