# Kaluza Klein and non-abelian gauge transformation

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• LCSphysicist
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TL;DR Summary
Under a special type of coordinate transformation, the KK reduce metric transforms such that a non-abelian gauge vector appears. How?
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The Kaluza-Klein metric, by reduction, can be written as a ##(4+m) \times (4+m)##symmetric matrix, where ##m## is the dimension of the additional spacetime (if we decompose ##M_D = M_4 \times M_m##). It was show by Bryce de. Witt that, if the non-diagonal metric has the form

$$g_{\mu m} = B^I_{\mu}(x^{\nu}) V^I_m(y^n)$$

And we make a general coordinate transformation

$$\epsilon_m = \lambda^I (x^{\nu}) V_m^I(y^n)$$

We can show that ##\lambda## is the parameter of a non-abelian gauge transformation of ##B_{\mu}##. While every source talks about de Witt derivation, i was not able to find it, indeed, the citations used are not open (B.S. DeWitt, In Relativity, Groups and Topology, eds. C. and B.S. DeWitt (Gordon and Breach, New York, 1964). So i have tried to show it here

$$\delta g_{rs} = \epsilon^m \partial_m g_{rs} + \partial_{r} \epsilon^m g_{m s} + \partial_s \epsilon^m g_{m s}$$

$r = \mu, s = n$

$$\delta g_{\mu n} = \lambda V^{m} \partial_m g_{\mu n} + \partial_{\mu} \lambda V^m g_{mn} + \lambda \partial_n V^m g_{m \mu}$$

$g_{\mu n} = B_{\mu} V_n$, $g_{mn} = \delta_{mn}$

$$\delta g_{\mu n} = \delta B_{\mu} V_n + B_{\mu} \delta V_{n} = \lambda B_{\mu} V^m \partial_m V_n + V_n \partial_{\mu} \lambda + \lambda B_{\mu} \partial_n V^m V_m$$

But ##\delta V_n## is ##0## (honestly, i am just assuming it). Also, i assumed that ##\partial_m V_n V^n = V^n \partial_m V_n##

$$\delta B_{\mu} V_n = \lambda B_{\mu} V^m \partial_m V_n + V_n \partial_{\mu} \lambda + \lambda B_{\mu} V_m \partial_n V^m$$

This, in fact, reduces to a "looks like" non-gauge transformation

$$\delta B_{\mu} = \partial_{\mu} \lambda + (V_n^{-1} V_m \partial_m V_n + V_n^{-1} V_m \partial_n V^m ) \lambda B_{\mu}$$

But i am not really sure if it is correct. I mean, the term in parenthesis shouldn't be something like $~f$, the structure constant of some group rep?

Last edited:
Hmm I am intrigued by the initial " Here's your text with the changes you requested:" Does it mean that the IA had all this info?

Usually KK non abelian refers to a short burst of research between 1981 and 1986; the research in the sixties was more about quantised fields on curved spaces.

arivero said:
Hmm I am intrigued by the initial " Here's your text with the changes you requested:" Does it mean that the IA had all this info?

Usually KK non abelian refers to a short burst of research between 1981 and 1986; the research in the sixties was more about quantised fields on curved spaces.

hahahaha oh! I am not a native english speaker, as probably you will notice with this message i am writing to you, so I generally write a text and puts on chatgpt so that it "corrects" the bad gramatic and maybe write it a little better. I mean, of course you can understand me, but it is embarassing to make english mistakes haha

The main question in this thread is "what is the role of De Witt in the formulation of non abelian kaliza Klein theories and how does this formulation differs of Witten's 'realistic kaluza Klein' models."

Wait for the next ChatGPT update to see if it gets some sensical answer.

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