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This question arises from a personal misunderstanding about a conversation with a friend of mine. He asked me a question about the "truly nature" of spinors, i.e., he asked a question to me about what is an spinor object. After a few lines of dialogue, he asked something quite alien to me:

"So, spinors are Levi-Civita connections?"

The relationship between a mathematical object which models physical entities in field theory (a Dirac spinor for example) and a purely mathematical entity like a Levi-Civita connection, still intrigues me.

Now, today I encountered this question here:

Under what representation do the Christoffel symbols transform?

and in the second answer, the user made another relationship between field theory and connections:

"The "Christoffel symbols" are now just the components of a principal connection on that bundle, where a "connection form" is better known to physicists as a gauge field"

I'm asking this question because, from the point of view of elementary general relativity, we are taught that we need a pseudo-riemanninan manifold and a (Levi-Civita) connection to, roughly speaking, make a well-defined notion of derivative of tensor fields. From this point of view a connection is nothing more than a linear map.

So, what are Connections in physics, INDEED?

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    $\begingroup$ A connection in physics is, indeed, a connection, in the strict mathematical sense. A connection has a precise definition and, in physics, we use that definition (sometimes implicitly or unbeknown to us). I'm not sure what else you want to know. $\endgroup$ – AccidentalFourierTransform Jul 10 at 1:19
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    $\begingroup$ "The relationship between a mathematical object which models physical entities [...] and a purely mathematical entity like a Levi-Civita connection, still intrigues me." What do you think is the difference between a mathematical object and a "purely" mathematical one? Why are Dirac spinors less "purely" mathematical than connections? $\endgroup$ – knzhou Jul 10 at 1:23
  • $\begingroup$ I just think of connections as covariant derivatives. In GR you want derivatives that are covariant under general coordinate transformations, and “adding in” Christoffel symbols accomplishes this. In gauge theories you want derivatives that are covariant under an internal gauge transformation, and “adding in” gauge fields accomplishes this. There is a level of abstraction at which these are the same thing, but when first learning GR and QFT I’m not convinced you have to understand it. $\endgroup$ – G. Smith Jul 10 at 2:06
  • $\begingroup$ It wasn’t obvious to the first people doing GR and QFT. If I understand correctly, it emerged after differential geometers generalized manifolds with a tangent space at each point and manifolds with “a Lie group at each point” into the concept of a fiber bundle. $\endgroup$ – G. Smith Jul 10 at 2:06
  • $\begingroup$ In any case, spinors are not connections, but there are “spin connections” for covariantly differentiating spinors. $\endgroup$ – G. Smith Jul 10 at 2:09
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Connections in physics "are" the same as they are in mathematics but are usually interpreted as field potentials, with the exception of GR.

The interpretation follows naturally from the concept of a covariant derivative: local transformations of the field being studied must not change the physics involved (i.e. the Lagrangian must be invariant) so one introduces another "gauge" field which has dynamics of its own to cancel changes from the matter field's transformation.

Take the case of quantum electrodynamics: The Lagrangian (density) is $$\mathcal{L}=\overline{\psi}\left(i\gamma^\mu\partial_\mu + m\right)\psi$$

You can check that it is invariant under the transformation $\psi\to e^{i\lambda}\psi$ when $\lambda$ is a constant. To make the transformation local we "promote" $\lambda$ to a function, but now we have an offending term $\overline{\psi}\gamma^\mu\partial_\mu\lambda e^{i\lambda}\psi$! All is well if we introduce the covariant derivative $\mathcal{D}_\mu=\partial_\mu-ie_0A_\mu$ such that $A_\mu\to A_\mu +\partial_\mu \lambda$ is the corresponding transformation.

The full Lagrangian is then $$\mathcal{L}=\overline{\psi}\left(i\gamma^\mu\partial_\mu + m\right)\psi + e_0\overline{\psi}\gamma^\mu A_\mu \psi + \frac{1}{4}F^{\mu\nu}F_{\mu\nu}$$

where $F_{\mu\nu}$ is the electromagnetic field strength tensor introduced to account for the dynamics of the potential (photon) field $A_\mu$

Connections in the context of relativity are instead the gravitational field strength since in our currently accepted theory gravity is not a "gauge field" like the photon field. The identification of gravitation with spacetime curvature makes particles travel according to the geodesic equation which can recover the usual Gauss's Law for gravity in the Newtonian limit.

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  • $\begingroup$ "our currently accepted theory gravity is not a "gauge field" like the photon field." Well, "spin connection" $\omega$ is the "gauge field" of gauge theory of gravity, where the gauge group is spin(1,3) (double cover of Lorentz group). $\endgroup$ – MadMax Jul 10 at 14:18
  • $\begingroup$ @MadMax I was not aware of that. Is the gauge theory of gravity more generally accepted than GR? What books/articles may I find more information in? $\endgroup$ – Quantumness Jul 11 at 20:59
  • $\begingroup$ Here is a reference for gauge gravity: journals.aps.org/rmp/abstract/10.1103/RevModPhys.48.393 $\endgroup$ – MadMax Jul 12 at 13:39

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