Timeline for How can you have a potential in a theory without any forces?
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| Jul 31 at 10:22 | comment | added | Peter Petrov | @GiorgioP-DoomsdayClockIsAt-90. Thank you for your answer. Regarding your point of the field being the dynamical degree of freedom, would the rate fo change of the potential in configuration space correspond to some notion of force? $$F(\phi)=-\frac{\partial(V(\phi))}{\partial(\phi)}$$ If so, how can we interpret this force? | |
| Jul 31 at 7:09 | comment | added | GiorgioP-DoomsdayClockIsAt-90 | @James. Your question deserves more space than what is allowed in a comment. Moreover, it is related but not directly connected to the present question. | |
| Jul 31 at 6:05 | comment | added | James | By internal configuration space, i meant this. A wave packet with momentum $p$ should be moving in real space (because it has momentum), right? But plug this wave packet into the time-dependent Schrodinger\Dirac equation and we get $\frac{\partial \Psi}{\partial t} = -i\frac{\partial^2 \Psi}{\partial x^2}$ and voila, this wave is not moving correctly with momentum $p$ according to the TDSE. This seems to me to show that the Dirac/Schodinger equation cannot be operating in our real physical space... | |
| Jul 31 at 6:00 | comment | added | GiorgioP-DoomsdayClockIsAt-90 | @James, I think your question is more general than just the case of the Dirac field. Maybe, in a clearer form, it could be a good question to propose. However, you should clarify what you mean by internal configuration space. | |
| Jul 31 at 6:00 | comment | added | James | Experimentally, the very distinctive shapes of those higher orbital $\Psi$ of hydrogen have never actually been experimentally shown in any great detail, right? How certain are we that the distinctive shapes of those higher orbital probability clouds are exactly the same shapes in real physical space, as the solutions show in configuration space? | |
| Jul 31 at 5:54 | comment | added | James | I was thinking about the classical Dirac (or Schrodinger) field pre-quantization. When we say $\Psi_{electron}$ in the classical field has such and such distribution ("electron cloud") this is not the true distribution of the electron's probability in real physical space, but rather only in its internal configuration space, do you think? But this electron exists too in real physical space, so like EM and GR fields, there must be some mapping between its internal space and our real physical space? | |
| Jul 31 at 5:34 | comment | added | GiorgioP-DoomsdayClockIsAt-90 | @James, I am not sure I fully understood what you mean by $\Psi$ operates. In the first quantization scheme of the Dirac field, it does not operate, but it is a function of the electronic coordinates that, in a position representation, are the positions of electrons in the physical space. In a second quantization scheme, $\Psi$ becomes an operator acting on a state of the Fock's space of states, states which may be described in terms of physical space coordinates of electrons (and other particles). | |
| Jul 31 at 5:12 | comment | added | James | Nice answer! (+1) As you said "the dynamical degrees of freedom are not positions but field values". The Dirac $\Psi$ field in particular has been bothering me.. When talking about EM field and the GR field, we generally assume that "of course" these fields operate in real physical space, correct? But it doesn't seem clear to me what space the Dirac field is operating on.. If $\Psi$ operates in its own configuration space, how do you think we can translate what is happening to $\Psi_{electron}$ in this internal configuration space to what is happening to this electron in real physical space? | |
| Jul 31 at 4:47 | history | answered | GiorgioP-DoomsdayClockIsAt-90 | CC BY-SA 4.0 |