Field Transformations: Connections to Symmetries

[maxverywell]

What are the differences in (scalar) field transformations:

1) $$\phi(x)\to \phi'(x)$$

2) $$\phi(x)\to \phi'(x')$$

3) $$\phi(x)\to \phi(x')$$

How this transformations are connected to internal and external symmetries?

For example, if we take spacetime global translations $$x^{\mu}\to x'^{\mu}=x^{\mu}+\epsilon^{\mu}$$ which one of the 3 is the corresponding transformation of the field?

 

[nrqed]

What are the differences in (scalar) field transformations:

1) $$\phi(x)\to \phi'(x)$$

2) $$\phi(x)\to \phi'(x')$$

3) $$\phi(x)\to \phi(x')$$

How this transformations are connected to internal and external symmetries?

For example, if we take spacetime global translations $$x^{\mu}\to x'^{\mu}=x^{\mu}+\epsilon^{\mu}$$ which one of the 3 is the corresponding transformation of the field?

A scalar field is invariant under Lorentz transformations. What this means is that

$$\phi(x) = \phi'(x')~~~~(1)$$

What this implies is that the field must transform,$\phi(x)\to \phi'(x)$ in such a way that the field transformation compensates for the transformation of the coordinate.
To find the explicit form of $\phi'(x)$ all you must do is to plug x' into Eq. (1) and Taylor expand.

 

[maxverywell]

A scalar field is invariant under Lorentz transformations. What this means is that

$$\phi(x) = \phi'(x')~~~~(1)$$

What this implies is that the field must transform,$\phi(x)\to \phi'(x)$ in such a way that the field transformation compensates for the transformation of the coordinate.
To find the explicit form of $\phi'(x)$ all you must do is to plug x' into Eq. (1) and Taylor expand.

Edit: thnx, I get it.

Is this valid only for real scalar fields?

I'm trying to prove energy-momentum conservation for space-time translations but this isn't proof for general case of energy-momentum conservation, only for scalar fields, but they don't exists in nature so how this can be useful?

 

[The_Duck]

Energy-momentum conservation comes from the invariance of the Lagrangian under translations. To express a field phi in a translated coordinate system, you used instead of the field phi(x) the field phi(x+a). This does not depend on whether phi is a scalar field or some higher spin field. On making this replacement this you find that your Lagrangian is unchanged, which leads to energy-momentum conservation.

A scalar field is defined by its behavior under Lorentz transformations: to express phi in a Lorentz-transformed (boosted or rotated) frame you replace
$\phi(x) \to \phi(\Lambda^{-1} x)$
Contrast a higher-spin field, which will have several components that mix under Lorentz transformations:
$\psi_a(x) \to {D(\Lambda)_a}^b\psi_b(\Lambda^{-1}x)$
However you will only need to start thinking about these more complicated transformations when you ask about the conservation laws that come from Lorentz symmetry, namely angular momentum conservation.