swansont Posted December 16, 2023 Share Posted December 16, 2023 3 hours ago, sethoflagos said: Not the foggiest. Hence points 1) to 5) of my earlier post that you seem to have no interest in. “The evolution of the individual quantum states in a superposition are accurately described by an appropriate wave equation such as (eg for Dirac fermions) the Dirac equation.” I have no interest in this because my experience with superposition is with the expression of the eigenstates, and not the wave equation or how the wave function was determined. There’s no common ground for me, and so I have no comment. IOW aψ1+bψ2 doesn’t really rely on the wave equation, so I’m not sure where you were going with your post Link to comment Share on other sites More sharing options...

sethoflagos Posted December 16, 2023 Share Posted December 16, 2023 33 minutes ago, swansont said: “The evolution of the individual quantum states in a superposition are accurately described by an appropriate wave equation such as (eg for Dirac fermions) the Dirac equation.” I have no interest in this because my experience with superposition is with the expression of the eigenstates, and not the wave equation or how the wave function was determined. There’s no common ground for me, and so I have no comment. IOW aψ1+bψ2 doesn’t really rely on the wave equation, so I’m not sure where you were going with your post That's fine. I have no difficulty in seeing the issues with my later inferences. Which is why I asked in the first place. So in a sense, you've been asking me to defend a position after I've announced that I've got my own problems with it. But I'd be quite interested to know where the train of thought actually went astray. If indeed it did. Link to comment Share on other sites More sharing options...

Genady Posted December 16, 2023 Share Posted December 16, 2023 (edited) On 12/14/2023 at 4:20 PM, sethoflagos said: 1) The evolution of the individual quantum states in a superposition are accurately described by an appropriate wave equation such as (eg for Dirac fermions) the Dirac equation. Corrections: 1a) The evolution of any quantum state is described by an appropriate equation of motion, such as Schrödinger equation, Dirac equation, and Klein-Gordon equation. The equations of motion are not restricted to "the individual quantum states in a superposition." 1b) Any state is a superposition of some states. Being an "individual quantum state in a superposition" is not property of a state but property of a coordinate basis in which the state is expressed. Edited December 16, 2023 by Genady Link to comment Share on other sites More sharing options...

sethoflagos Posted December 16, 2023 Share Posted December 16, 2023 6 minutes ago, Genady said: 1a) The evolution of any quantum state is described by an appropriate equation of motion, such as Schrödinger equation, Dirac equation, and Klein-Gordon equation. The equations of motion are not restricted to "the individual quantum states in a superposition." This is consistent with what I posted I think, though you have made it more general in application. 8 minutes ago, Genady said: 1b) Any state is a superposition of some states. Being an "individual quantum state in a superposition" is not property of a state but property of a coordinate basis in which the state is expressed. In the sense that any arbitrary vector can be expressed as the sum of 3 vectors in your coordinate system of choice? Or are you making some deeper point that I'm missing? Link to comment Share on other sites More sharing options...

Genady Posted December 16, 2023 Share Posted December 16, 2023 1 minute ago, sethoflagos said: In the sense that any arbitrary vector can be expressed as the sum of 3 vectors in your coordinate system of choice? Yes, but I refer to a coordinate basis of the space of states rather than the geometric 3D space. Link to comment Share on other sites More sharing options...

sethoflagos Posted December 16, 2023 Share Posted December 16, 2023 33 minutes ago, Genady said: Yes, but I refer to a coordinate basis of the space of states rather than the geometric 3D space. If you're moving onto eigenstates of Hamiltonians again then you're losing me. How does this invalidate my point 1), in easy steps. Link to comment Share on other sites More sharing options...

Genady Posted December 16, 2023 Share Posted December 16, 2023 10 minutes ago, sethoflagos said: If you're moving onto eigenstates of Hamiltonians again then you're losing me. How does this invalidate my point 1), in easy steps. I refer to eigenstates of any observable. We can expand any state in any basis. This makes the notion of "states in a superposition" arbitrary. Link to comment Share on other sites More sharing options...

sethoflagos Posted December 16, 2023 Share Posted December 16, 2023 13 minutes ago, Genady said: I refer to eigenstates of any observable. We can expand any state in any basis. This makes the notion of "states in a superposition" arbitrary. Okay... so I've arbitrarily picked something. 1 hour ago, Genady said: Yes, but I refer to a coordinate basis of the space of states rather than the geometric 3D space. Could such a space be: On 12/14/2023 at 9:20 PM, sethoflagos said: 2) Although the Dirac equation maps onto our R^{3+1} spacetime, it is expressed in terms of a 4 dimensional complex vector space that is not defined within our R^{3+1} spacetime. ... or not. Link to comment Share on other sites More sharing options...

Genady Posted December 16, 2023 Share Posted December 16, 2023 16 minutes ago, sethoflagos said: Okay... so I've arbitrarily picked something. Could such a space be: ... or not. In the Dirac equation, the evolving objects are spinors. They evolve in spacetime. They constitute states in a spinor space. The equation does not pick any specific basis in this space to expand the spinors as superposition. We are free to choose such a basis and thus such expansion is arbitrary. Link to comment Share on other sites More sharing options...

sethoflagos Posted December 16, 2023 Share Posted December 16, 2023 12 minutes ago, Genady said: In the Dirac equation, the evolving objects are spinors. They evolve in spacetime. They constitute states in a spinor space. You seem to be saying that spacetime and a complex 4-D spinor space are the same thing. To me, any object in a complex space does not have an energy content though its projection in real (non-complex) spacetime will. So they aren't the same thing,,, or are they? Link to comment Share on other sites More sharing options...

Genady Posted December 17, 2023 Share Posted December 17, 2023 (edited) 24 minutes ago, sethoflagos said: You seem to be saying that spacetime and a complex 4-D spinor space are the same thing. To me, any object in a complex space does not have an energy content though its projection in real (non-complex) spacetime will. So they aren't the same thing,,, or are they? I don't say or imply anything like that and don't have any idea how it seems so. I say that the states, which are spinors, evolve in spacetime. Just like a scalar such as temperature can evolve in spacetime but is not the same as spacetime. But you don't need to go to spinors to express your concern. Long before Dirac, in the good old Schrödinger equation, the wave functions are complex-valued functions and they represent particle states in a complex vector space. In case of the Hamiltonian observable basis, the states, complex functions, are eigenstates, while the energies, real scalars, are eigenvalues. I don't think there is any problem in this distinction. Edited December 17, 2023 by Genady 1 Link to comment Share on other sites More sharing options...

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