% File: qft_symm.tex
% Section: QFT
% Title: QFT: Symmetries
% Last modified: 30.03.2006
%
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\begin{document}

\begin{center}
\large\textbf{QFT: SYMMETRIES}
\end{center}

\vspace{0cm}

\begin{center}
\large\textbf{N\"other's theorem}
\end{center}

\vspace{.1cm}

Let a Lagrangian depend on fields and their first derivatives,
\begin{equation}
\label{}
S = \int\!dx\,\mathcal{L}(x)\,, \ \ \
\mathcal{L}(x) = \mathcal{L}(\varphi^a(x), \varphi^a_{,\mu}(x))\,, \ \ \
\varphi_{,\mu}(x) = \partial_\mu\varphi(x)\,,
\end{equation}
and let infinitesimal transformations
\begin{align}
x^\mu &\rightarrow x'{}^\mu = x^\mu + \delta x^\mu\,, \label{tra1} \\
\varphi^a(x) &\rightarrow {\varphi^a}'(x') = \varphi^a(x)
+ \delta\varphi^a(x) \label{tra2}
\end{align}
be parametrized by a finite number of constant parameters $\omega^k$\,.
The `form variation'
\begin{equation}
\label{}
\bar{\delta}\varphi(x) \doteq \varphi'(x)-\varphi(x)
 = \delta\varphi(x) - \varphi_{,\mu}(x)\delta x^\mu
\end{equation}
is designed to commute with space-time derivatives, \
$ \bar{\delta}\varphi_{,\mu}^{a} = \partial_\mu\bar{\delta}\varphi^a $\,.

We say that a theory defined by $\mathcal{L}$ is invariant under
transformations (\ref{tra1}), (\ref{tra2}) if
\begin{equation}
\label{}
\mathcal{L}'(x')\,dx' = \mathcal{L}(x)\,dx\,.
\end{equation}
In that case, from
\begin{gather}
\mathcal{L}'(x')-\mathcal{L}(x)
= \frac{\partial\mathcal{L}}{\partial\varphi^a}\,\bar{\delta}\varphi^a
+ \frac{\partial\mathcal{L}}{\partial\varphi^a_{,\mu}}
\,\bar{\delta}\varphi^a_{,\mu}
+ \partial_\mu\mathcal{L}\,\delta x^\mu\,, \\
dx' = \det\Bigl(\frac{\partial x'}{\partial x}\Bigr)\,dx
 = (1+\partial_\mu\delta x^\mu)\,dx
\end{gather}
we derive
\begin{multline}
\label{main}
0 = \mathcal{L}'(x')\,\frac{dx'}{dx} - \mathcal{L}(x) =
\frac{\partial\mathcal{L}}{\partial\varphi^a}\,\bar{\delta}\varphi^a
+ \frac{\partial\mathcal{L}}{\partial\varphi^a_{,\mu}}
\,\partial_\mu\bar{\delta}\varphi^a
+ \partial_\mu(\mathcal{L}\,\delta x^\mu) \\
= \Bigl(\frac{\partial\mathcal{L}}{\partial\varphi^a}
 - \partial_\mu\frac{\partial\mathcal{L}}{\partial\varphi^a_{,\mu}}\Bigr)
 \bar{\delta}\varphi^a + \partial_\mu\Bigl(
 \frac{\partial\mathcal{L}}{\partial\varphi^a_{,\mu}}
 \bar{\delta}\varphi^a + \mathcal{L}\delta x^\mu\Bigr)
= \Bigl[\,\Bigl(\frac{\partial\mathcal{L}}{\partial\varphi^a}
 - \partial_\mu\frac{\partial\mathcal{L}}{\partial\varphi^a_{,\mu}}\Bigr)
 \psi_{k}^{a} - \partial_\mu J_{k}^{\mu}\Bigr]\omega^k
\end{multline}
where
\begin{equation}
\label{curr}
J_{k}^{\mu}
= -\frac{\partial\mathcal{L}}{\partial\varphi^a_{,\mu}}\psi_{k}^{a}
 - \mathcal{L}\xi_{k}^{\mu}\,, \ \ \ \ \
\delta x^\mu = \xi_{k}^{\mu}\omega^k\,, \ \ \ \ \
\bar{\delta}\varphi^a = \psi_{k}^{a}\omega^k\,.
\end{equation}
We conclude that on shell all N\"other currents are conserved,
\begin{equation}
\label{conserv}
\frac{\partial\mathcal{L}}{\partial\varphi^a}
- \partial_\mu\frac{\partial\mathcal{L}}{\partial\varphi^a_{,\mu}} = 0
\ \ \Rightarrow \ \ \partial_\mu J_{k}^{\mu} = 0\,.
\end{equation}
From the calculations in (\ref{main}) it is also seen that off shell,
for $x$-dependent $\omega^k$, the relations
\begin{equation}
\label{off}
\partial_\mu J_{k}^{\mu}
= \Bigl(\frac{\partial\mathcal{L}}{\partial\varphi^a}
-\partial_\mu\frac{\partial\mathcal{L}}{\partial\varphi^a_{,\mu}}\Bigr)
 \psi_{k}^{a}\,, \ \ \ \ \
\delta S = \int\!\mathcal{L}'dx' - \int\!\mathcal{L}dx
= -\!\int\!dx\,J_k^\mu\,\partial_\mu\omega^k
\end{equation}
hold in a theory which is invariant for constant $\omega$'s.

\newpage

\begin{center}
\large\textbf{Energy-momentum tensor}
\end{center}

\vspace{.1cm}

As an important example, we consider a shift of coordinates by
a constant 4-vector $\varepsilon^\mu$ (playing here the role of
$\omega^k$)\,, without additional independent field transformations.
Any fields are scalars under such shift, so
\begin{equation}
\delta x^\mu = \varepsilon^\mu\,, \ \ \ \ \delta\varphi^a = 0 \ \ \
\Rightarrow \ \ \
\bar{\delta}\varphi^a = -\varphi_{,\mu}^{a}\,\delta x^\mu\,, \ \ \
\xi_{\nu}^{\mu} = \delta_{\nu}^{\mu}\,, \ \ \
\psi_{\nu}^{a} = - \varphi_{,\nu}^{a}\,.
\end{equation}
The corresponding N\"other current is known as the energy-momentum
(or stress) tensor,
\begin{equation}
\label{stress}
T_{\nu}^{\mu} = \frac{\partial\mathcal{L}}{\partial\varphi^a_{,\mu}}
\,\varphi^a_{,\nu} - \mathcal{L}\delta_{\nu}^{\mu}\,.
\end{equation}
From (\ref{off}) we see that it describes (off shell) the change of the
action under general coordinate transformations
$\delta x^\mu=\varepsilon^\mu(x)$\,:
\begin{equation}
\label{dels}
\delta S = -\int\!dx\,T_\nu^\mu\,\partial_\mu\varepsilon^\nu(x)\,.
\end{equation}

However, it should be noted that for $x$-dependent $\omega^k$ we, in many
cases, actually deal with a more general `form variation'
$\bar{\delta}\varphi^a$ than (\ref{curr})\,:
\begin{equation}
\label{chi}
\bar{\delta}\varphi^a
 = \psi_{k}^{a}\omega^k + \chi_{k}^{a\mu}\partial_\mu \omega^k\,.
\end{equation}
In this situation
\begin{multline}
\label{off2}
\delta S = \int\!dx\,
\Bigl\{\Bigl[\,\Bigl(\frac{\partial\mathcal{L}}{\partial\varphi^a}
 - \partial_\mu\frac{\partial\mathcal{L}}{\partial\varphi^a_{,\mu}}\Bigr)
 \psi_{k}^{a} - \partial_\mu J_{k}^{\mu}\Bigr]\omega^k \\
 + \Bigl[-J_{k}^{\mu}
 + \frac{\partial\mathcal{L}}{\partial\varphi^a}\chi_{k}^{a\mu}
 + \frac{\partial\mathcal{L}}{\partial\varphi^a_{,\nu}}
  \partial_\nu \chi_{k}^{a\mu}\Bigr]\,\partial_\mu \omega^k
 + \Bigl[\frac{\partial\mathcal{L}}{\partial\varphi^a_{,\nu}}\chi_{k}^{a\mu}
 \Bigr]\,\partial_\nu \partial_\mu \omega^k \Bigr\}\,,
\end{multline}
and the relations (\ref{off})\,, (\ref{dels}) are generalized by
$\chi$-terms.

In the most interesting case, when the action $S$ is invariant under such
($x$-dependent) transformations, the expressions inside the square brackets
in (\ref{off2}) must vanish identically. This fact (the second N\"other
theorem) necessitates that some identical relations between the
equations of motion take place. In other words, a local symmetry of a given
field theory causes its equations of motion to be mutually dependent (and,
as a result, singular).

For a theory (with gravity, of course) that is invariant under general
coordinate transformations $\delta x^\mu=\varepsilon^\mu(x)$\,,
the second square bracket in (\ref{off2}) yields the following off-shell
identity:
\begin{equation}
\label{tchi}
T_{\nu}^{\mu} \ = \
\frac{\partial\mathcal{L}}{\partial\varphi^a}\chi_{\nu}^{a\mu}
 + \frac{\partial\mathcal{L}}{\partial\varphi^a_{,\lambda}}
  \partial_\lambda \chi_{\nu}^{a\mu}\,.
\end{equation}
This relation, by the way, allows to compare the definition (\ref{stress})
with another, so-called metrical, definition of the energy-momentum tensor,
\begin{equation}
\label{metrt}
\Theta_{\mu\nu} \doteq \frac{2}{\sqrt g}\,
\frac{\partial\mathcal{L}}{\partial g^{\mu\nu}}\,.
\end{equation}
Let us suppose that our theory involves the metrics $g^{\mu\nu}$ (without
derivatives) and some \emph{scalar} fields $\varphi^a$\,. For the latter
$\chi_{\nu}^{a\mu}=0$\,, whereas for $g^{\mu\nu}$ they are easily extracted
from
\begin{equation}
\label{}
\bar{\delta}g^{\mu\nu} = -g^{\mu\nu}_{,\alpha}\varepsilon^\alpha
 + g^{\alpha\nu}\partial_\alpha\varepsilon^\mu
 + g^{\mu\alpha}\partial_\alpha\varepsilon^\nu\,.
\end{equation}
We find from (\ref{tchi}) that
\begin{equation}
\label{}
T_{\nu}^{\mu} \
 = \ 2\,g^{\mu\alpha}\frac{\partial\mathcal{L}}{\partial g^{\alpha\nu}}\,,
\end{equation}
which, after lowering the index, coincides with $\Theta_{\mu\nu}$ up to a
scalar factor. However, in a more general theory $T$ and $\Theta$ may differ.


\end{document}