Transformations of free fields

[bartadam]

Hi, New here...Can't seem to do latex on here so this post is incomplete until I can work it out.

This is maybe quite abstract and generic, but here goes. This problem has niggled me for a while and I need some input please.

I have an action $$S=\int d^4 x \sqrt(g(x))\overline\Phi (x)\Omega(x)\Phi(x)$$

where $$\Omega(x)$$ is a differential operator and $$\Phi(x)$$ and $$\overline\Phi(x)$$ are the free field and it's dual respectively. For example, they may well be scalars $$\phi(x)$$ and $$\overline\phi(x)$$ or vectors, or spinnors superfields tensors etc...Any generic spacetime object.

Say for the moment they are scalars and define a transformation of the fields as follows

$$\delta\phi(x)=\epsilon\phi(x_G)$$ and $$\delta\overline\phi(x)=-\epsilon\overline\phi(x_{G^{-1}})$$

where $$x \rightarrow x_{G}$$ is a finite isometry. Also omega is the operator $$g^{\mu\nu}\partial_{\mu}\partial_{nu}$$ and is invariant under isometriesThen the change in the action is

$$\delta S=\epsilon\int d^4 x \sqrt(g(x))\overline\Phi (x)\Omega(x)\Phi(x_G)-\epsilon\int d^4 x \sqrt(g(x))\overline\Phi (x_{G^{-1}})\Omega(x)\Phi(x)$$

which is zero by changing variables in the second integral and using the fact that x goes to xG is an isometry.

Now, $$\delta\phi(x)=\epsilon exp(X^u \partial_{\mu})\phi(x).$$

where X is a killing vector. I know this for a fact because it's just parameterising the isometry and using the chain rule. The each term in the taylor expansion is also a symmetry. I.e, it is the exponential of the directional derivative, defined as it's taylor expansion. It also happens to be the Lie derivative of a scalar field which pertains to something I will say in a while.

Now...I have tried to do this for a vector say $$A^{\mu}$$ and it's dual $$\overline A_{\mu}$$

I am having difficulty deciding what the transformation will be at the moment I have

$$\delta A(x)=\epsilon A^{\mu}(x_G) =\epsilon exp(X\partial)A(x)$$ and $$\delta \overline A(x)=-\epsilon A_{\mu}(x_{G^-1}}) dx^{\mu}=-\epsilon exp(-X\partial)\overline A$$

and then the change in the action is$$\delta S=\epsilon\int d^4 x \sqrt(g(x))\overline A_{\mu}(x) dx^{\mu} e_{\nu}(x)\Omega^{\nu}_{\lambda}(x) dx^{\lambda} e_{\sigma}(x) A^{\sigma}(x_G)-\epsilon\int d^4 x \sqrt(g(x))\overline A_{\mu}(x_{G^-1}}) dx^{\mu} e_{\nu}(x)\Omega^{\nu}_{\lambda}(x) dx^{\lambda} e_{\sigma}(x) A^{\sigma}(x)$$

Then perform $$x \rightarrow x_G$$ on the second integral and $$dx_G^{\mu} e_{\nu}(x_G)=dx^{\mu} e_{\nu}(x)$$ since it is simply the inner product of basis vectors so gives the delta function. Omega is invariant since it is an isometry, and $$\sqrt{g} d^4 x$$ is also invariant and so the change in the action is zero. So, again, each term in the expansion of $$exp(X\partial)$$ is a symmetry.

Now I don't believe this argument for a vector. I think the transformations should include some matrices, and should be the exponential of the Lie derivative of the vector. However, I can't get the argument with matrices to work. I.e. something like this

$$\delta A= U^{-1}\epsilon A(x_G)$$ where $$U^{-1}$$ is the matrix that maps basis vectors at $$x_G$$ back to x. I was hoping this gives the exponential of the Lie derivative, and it doesn't seem to and also the matrices don't all work out when I do the change of variables in the second integral

Is my argument for the vector field reasonable? Have I made a conceptual error? Is it simply the exponential of the normal directional derivative. I don't believe it because I would expect the indices to change.

And how on Earth will this work for more arbitrary space time objects $$\Phi$$

Sorry for the long post. I don't think I have made any latex typos

 

[Fredrik]

It's [tex]at this site, not [math]. I'm a regular at another vBulletin forum where it's $, so it seems to be up to the administrator rather than a vBulletin standard.<br /> <br /> Edit: I see you figured that out. <img src="https://cdn.jsdelivr.net/joypixels/assets/8.0/png/unicode/64/1f642.png" class="smilie smilie--emoji" loading="lazy" width="64" height="64" alt=":smile:" title="Smile :smile:" data-smilie="1"data-shortname=":smile:" />$[/tex]

 

[bartadam]

Hi, yeah thanks worked it out, looked on another thread and clicked on some maths to find the code.

Any help much appreciated.

Is my expression for the scalar correct?

$$\delta \phi(x)=\epsilon \phi(x_G)=\epsilon exp (X^{\mu}\partial_{\mu})\phi(x)$$

I proved this by having the Killing vector field over the manifold, then parameterising the integral curve x goes to xG, taylor expanding and using the chain rule.

Although I have been reading some riemannian geometry books and this exponential map seems to pertain only to geodesics, and the flows generated by isometries are not geodesics. I don't really understand them though, rather too abstract.

 

[samalkhaiat]

The following might help!

Cosider a spacetime symmetry whose infinitesimal action on the coordinates is given by

$$x^{a} \rightarrow \bar{x}^{a} = x^{a} + \delta_{f}x^{a}$$

where

$$\delta_{f}x^{a} = -f^{a}(x)$$

contains the infinitesimal parameters. Under such symmetry, a multi-component field transforms as

$$\bar{u}_{A}(\bar{x}) = D_{A}{}^{B}(f) u_{B}(x)$$

where D is a finite dimensional (matrix) representation of the group. To 1st order in the parameters, we write

$$\bar{u}_{A}(x - f) = \left(\delta_{A}{}^{B} + X_{A}{}^{B}(f) \right) u_{B}(x)$$

or,

$$\bar{u}_{A}(x) - f^{a}\partial_{a}u_{A} = u_{A}(x) + X_{A}{}^{B}(f)u_{B}(x)$$

The X's are set of matrices, appropriate for the field, satisfying the Lie algebra of the symmetry group. For scalar field X = 0, and for covariant vector field it is $\partial_{a}f^{a}$.

Thus, the infinitesimal change in the form of the field function is

$$\delta_{f}u_{A}(x) \equiv \bar{u}_{A}(x) - u_{A}(x) = f^{a}\partial_{a}u_{A}(x) + X_{A}{}^{B}u_{B}(x)$$

Except for spinor field, $\delta_{f}$ is nothing but the Lie derivative.

regards

sam