I don't know enough quantum theory to be able to tell you what techniques are used! It's called *Perturbative formulation of general continuous-time

Markov model of sequence evolution via insertions/deletions, Part I: Theoretical basis*, its 99 pages long (and that's just part I). They call it theoretical, but I think it might be too applied for you. Bits are reminiscent of things you've said. A snippet with garbled formulas:

> 1.3. Differences from the quantum mechanics

> Although we borrowed the bra-ket notation and the concept of operators from the

quantum mechanics (e.g., Dirac 1958; Messiah 1961a), there are some differences

between quantum mechanics and the Markov model. For example, in the Markov

model, we made the bra-probability vector ( ?? p(t) ) evolve, as in Eq.(1.1.2’), in order

to clarify its correspondence with the traditional matrix equation for the conditional

probabilities, Eq.(1.1.2). In contrast, in quantum mechanics, it is the ket-vector, ? (t) ,

that is usually made evolve. This is simply by convention and, if desired, we could

reformulate the quantum mechanics so that the bra-vector will evolve. Another

difference, which is conceptually more important, is that, in quantum mechanics, it is

the squared absolute values of the scalar products, i ? (t) 2 ( i =1, 2,..., N ), that are

interpreted as the probabilities (and thus satisfy i ? (t) 2

i=1

N S =1 ). In the Markov

model, in contrast, it is the scalar products themselves, ?? p(t) i ( i =1, 2,..., N ), that

give the probabilities (and thus satisfy ?? p(t) i =1

i=1

N S ). This should be related to

another big difference that the time evolution in the quantum mechanics is in the pure-imaginary direction ( i?? ?

?t

? (t) = ˆH ? (t) , where ?? is the Planck constant and ˆH is

the instantaneous time-evolution operator called the Hamiltonian), whereas the time

evolution in the Markov model is in the real direction

Markov model of sequence evolution via insertions/deletions, Part I: Theoretical basis*, its 99 pages long (and that's just part I). They call it theoretical, but I think it might be too applied for you. Bits are reminiscent of things you've said. A snippet with garbled formulas:

> 1.3. Differences from the quantum mechanics

> Although we borrowed the bra-ket notation and the concept of operators from the

quantum mechanics (e.g., Dirac 1958; Messiah 1961a), there are some differences

between quantum mechanics and the Markov model. For example, in the Markov

model, we made the bra-probability vector ( ?? p(t) ) evolve, as in Eq.(1.1.2’), in order

to clarify its correspondence with the traditional matrix equation for the conditional

probabilities, Eq.(1.1.2). In contrast, in quantum mechanics, it is the ket-vector, ? (t) ,

that is usually made evolve. This is simply by convention and, if desired, we could

reformulate the quantum mechanics so that the bra-vector will evolve. Another

difference, which is conceptually more important, is that, in quantum mechanics, it is

the squared absolute values of the scalar products, i ? (t) 2 ( i =1, 2,..., N ), that are

interpreted as the probabilities (and thus satisfy i ? (t) 2

i=1

N S =1 ). In the Markov

model, in contrast, it is the scalar products themselves, ?? p(t) i ( i =1, 2,..., N ), that

give the probabilities (and thus satisfy ?? p(t) i =1

i=1

N S ). This should be related to

another big difference that the time evolution in the quantum mechanics is in the pure-imaginary direction ( i?? ?

?t

? (t) = ˆH ? (t) , where ?? is the Planck constant and ˆH is

the instantaneous time-evolution operator called the Hamiltonian), whereas the time

evolution in the Markov model is in the real direction