I ask a physical interpretation of the relationship between states of a quantum system introduced, in analogy with the causal relationship between events in Minkowski space-time by means of the formula xRy if and only if < x-y/ T( x-y)> > or = 0. Here x, y are vectors of a Hilbert space and T is a Hermitian operator applied in it. In the original relation x,y are Minkowski space-time four- vectors and T is the metric tensor diag ( 1,-1,-1--1). The relation is equivalent to the inequality (x) + (y) > or= 2 Re () where the first member is the sum of the mean values of the operator T in the states x and y respectively. ( Gennaro Franco, Giuseppe Marino, Possible causal relationship extensions and properties of related causal isomorphisms, Linear and nonlinear analysis, january 2020)
It is completely my own point of view, and I am afraid to say this. I speculate that there are only two varieties of mathematical operations (be that due to Hermitian operators, arithmetical operators, transformation operators (e.g. reflection, transposition, etc.), action/abstract algebra, set algebra, or anything whatsoever). These are additive or integrative (uneven number of reflections) and subtractive or differentiative/derivative (even number of reflections). What is the Hermitian operator doing in the Minkowski space? Is it capable of doing something else? This is a case of introduction of a nearly-Euclidean quantity into the expanse of 4 dimensions, which can be innovative enough to theorize patterns of 'coming from' and 'going to' between 'general levels of perception (Newtonian/classical mechanics)' and 'quantum levels of events (arising out of Heisenberg's works).' In my humble opinion, your question is something that is inventive/pioneering in nature. If you please, you can see my work that I feel little contextual in this milieu. Article Rudimentary Exploration of the Concept of Action from Abstra...
The relationship described here is not a relationship between the states of a quantum system. In fact, if it were, the implication should hold: given the vectors x, y in a Hilbert space on the complex field, if >or= 0 then for every pair a,b of complex numbers it should result < ax-by/ A (ax-by)> >or=0. This is because the states of a quantum system are defined up to arbitrary complex constants. In fact, the following counterexample applies: x= (1,0); y= (0,1); a=1, b=5; A = diag (2,-1). The relation introduced above is therefore only a reflective and symmetric (non-transitive) binary relation between the vectors of a Hilbert space (real or complex).
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