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Consider the typical Lagrangian: $$L=\frac{1}{2}(\partial_\mu\phi)(\partial^\mu\phi) - V(\phi).$$ I interpret the above (please correct me) as a theory consisting of a field which can move through spacetime with a certain potential.

However, this theory does not contain any forces (no gauge fields). Intuitively, a potential necessitates a force. I think back to examples of gravitational potential and electrical potential in class, where: $$F(x)=-\frac{\mathrm{d}V(x)}{\mathrm{d}x}.$$

How can there be a notion of potential without any forces?

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  • $\begingroup$ Can you pick the potential $V$ in a gauge-invariant way? $\endgroup$
    – Marius Ladegård Meyer
    Commented Jul 30 at 22:11

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In terms of a field theory, the term $V$ in the Lagrangian is not connected with an interaction with a different field (which is missing), but it rules the dynamics and the ground state of the "free" field $\phi$.

In a field theory, forces as spatial gradients of the potential energy do not appear because the dynamical degrees of freedom are not the positions but the field values. This should be clear if one considers the equations of motion resulting as Euler-Lagrange equations of the Lagrangian density $L$.

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    $\begingroup$ 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? $\endgroup$
    – James
    Commented Jul 31 at 5:12
  • $\begingroup$ @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). $\endgroup$
    – GiorgioP-DoomsdayClockIsAt-90
    Commented Jul 31 at 5:34
  • $\begingroup$ 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? $\endgroup$
    – James
    Commented Jul 31 at 5:54
  • $\begingroup$ 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? $\endgroup$
    – James
    Commented Jul 31 at 6:00
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    $\begingroup$ @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. $\endgroup$
    – GiorgioP-DoomsdayClockIsAt-90
    Commented Jul 31 at 7:09

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