# Is there a physically meaningful example of a spacetime scalar potential?

From Misner, Thorne and Wheeler, page 115.

### 0-Form or Scalar, $$f$$

An example in the context of 3-space and Newtonian physics is temperature $$T\left(x,y,z\right),$$ and in the context of spacetime, a scalar potential, $$\phi\left(t,x,y,z\right).$$

I'm trying to think of an example of such a scalar potential. Is there one? Electrostatic potential is the time component of the electromagnetic 4-vector potential, so it's really a vector with 0-valued space components.

Within the Standard Model, the simplest model of the Higgs field is a multiplet of Lorentz scalar fields. This multiplet does have a non-trivial transformation under an underlying gauge group of the Standard Model; but under Lorentz transformations, the Higgs field is invariant, as all good Lorentz scalars should be.

Of course, the Higgs field is not a "potential" in the sense that a "potential" is a field whose derivative is a physically observable field; so if you're strictly looking for a scalar potential this isn't what you're looking for. But to the best of my knowledge it is the only fundamental scalar field we know to date.

• This answer is not entirely in my field of expertise, so if I've garbled anything above, feel free to send me corrections, amendments, and execrations. Feb 11 at 3:15

If you have a vector or tensor field, then you can get a scalar field by contraction.

Examples:

$$J^\mu$$ = 4-flux of some quantity. Scalar field: $$\rho = \sqrt{J^\mu J_\mu}/c$$. Interpretation: proper density.

$$k^\mu$$ = 4-wave vector; $$x^\mu$$ = 4-position. Scalar field: $$\phi = k^\mu x_\mu$$. Interpretation: phase of a plane wave.

Electromagnetic field tensor $$F^{\mu\nu}$$. Scalar fields: $$F^\mu_\mu$$ and $$F^{\mu\nu} F_{\mu \nu}$$ and $$F^{\mu \nu} \tilde{F}_{\mu \nu}$$. The first of these is zero, the second is $$2(E^2 - c^2 B^2)/c^2$$ and the third is $$4 {\bf E} \cdot {\bf B}$$.

The above are scalar fields, though not normally called 'potentials' because their gradient does not relate to a force. However we can introduce a potential which is by definition a scalar invariant, and then consider the gradient to be a 4-force. We thus obtain $$f^\mu = - \partial^\mu \phi.$$ Such a 4-force is not the electromagnetic force, but it can be used to construct simple models of the strong force.

• Why should we consider those to be potentials? Feb 11 at 3:53
• @StevenThomasHatton I may have misunderstood your question, but since you mentioned temperature I thought you were just thinking of anything scalar and invariant and a continuous field. I modified the answer a bit. Feb 11 at 9:24
• I didn't ask the question well. I should have specified "classical physics". And even excluded variational dynamics. I figure there must be some way of concocting a Lagrangian with a 4-potential that is "physically meaningful". But, IMO, that is a different mathematical space, which is "less real" than the mathematical space of classical general relativity. I wasn't even considering QFT. And since I haven't looked at QFT for quite some time, I'm not in a position to assess any of the answers involving it. It may be a year or more before I am prepared to accept an answer. Feb 14 at 20:14
• @StevenThomasHatton The added remark I made about strong force but construed either as a classical force or as a contribution to a quantum field. Feb 14 at 20:36

There are two different things one can mean by a potential.

The first is in the sense of a gauge field whose derivatives (in some combination) give the field-strength tensor. For example, the electromagnetic potential $$A^\mu(t,\vec{x})$$ as you mention. One can certainly write down a theory with a scalar gauge field $$B(t,\vec{x})$$, however, such a gauge field does not appear in the standard model of particle physics. But, there is nothing wrong with writing down such a theory. See, for example, this paper where the $$U(1)$$ symmetry is gauged using a scalar gauge field.

The second is in the sense of the potential terms, i.e., the interaction/self-interaction terms in the Lagrangian density. For example, we say things like "the shape of the Higgs potential looks like the logo of a famous StackExchange site". All such potential terms are always scalar because a Lagrangian is not allowed to be charged.