Quantum field theory 1, lecture 05

Scalar field theory in general coordinates

Action for complex scalar field in general coordinates and with external gauge field

There exists are more elegant formulation of symmetries and the associated conservation laws wich is the subject of Noethers second theorem. The idea it to promote the global symmetry to a local transformation, i. e. one that depends on spacetime coordinates. Consider the following action for a complex relativistic scalar field \[S[\phi] = \int d^4 x \sqrt{g(x)} \left\{ - g^{\mu\nu}(x) \left[\partial_\mu \phi^*(x) + i A_\mu(x) \phi^*(x)\right] \left[\partial_\nu \phi(x) - i A_\mu(x) \phi(x) \right] - V(\phi^*(x) \phi(x)) \right\}, \label{eq:actionComplexScalarGeneral}\] with \(\sqrt{g(x)} = \sqrt{- \det(g_{\mu\nu}(x))}\). We have written the theory in general (not necessarily Cartesian) coordinates which leads to the appearance of the spacetime-dependent metric \(g_{\mu\nu}(x)\) with inverse \(g^{\mu\nu}(x)\). We have also introduced an external gauge field \(A_\mu(x)\) and replaced partial derivatives \(\partial_\mu \phi(x)\) by covariant derivatives \(\partial_\mu \phi(x) - i A_\mu(x) \phi(x)\).

Local U\((1)\) symmetry

Due to the presence of the gauge field \(A_\mu(x)\) we can now consider local U\((1)\) transformations of the form \[\phi(x) \to e^{i \alpha(x)} \phi(x), \quad\quad\quad \phi^*(x) \to e^{-i \alpha(x)} \phi^*(x), \quad\quad\quad A_\mu(x) \to A_\mu(x) + \partial_\mu \alpha(x). \label{eq:localU1}\] This leaves the action \(\eqref{eq:actionComplexScalarGeneral}\) invariant. This is immediately clear for the potential term \(V(\phi^*\phi)\) when it is taken to depend only on the U\((1)\) invariant combination \(\phi^*\phi\). For the derivative terms we have \[\partial_\mu \phi(x) \to \partial_\mu \left[ e^{i\alpha(x)} \phi(x)\right] = e^{i\alpha(x)} \partial_\mu \phi(x) + e^{i\alpha(x)} \phi(x) i \partial_\mu \alpha(x).\] However, the inhomogeneous term on the right-hand side gets canceled by the transformation of the gauge field term in covariant derivatives.

Conservation law from gauge invariance

Noether’s second theorem is concerned with the conservation laws that arise from local symmetries. Let us consider the local U\((1)\) symmetry in eq. \(\eqref{eq:localU1}\) in the infinitesimal form \(\delta_\alpha \phi(x) = i \alpha(x) \phi(x)\), \(\delta_\alpha \phi^*(x) = - i \alpha(x) \phi^*(x)\) and \(\delta A_\mu(x) = \partial_\mu \alpha(x)\). The change in the action can be written in the form \[\delta S = \int d^4x \left\{ i \alpha \frac{\delta S}{\delta \phi(x)} \phi(x) - i \alpha \frac{\delta S}{\delta \phi^*(x)} \phi^*(x) + \frac{\delta S}{\delta A_\mu(x)} \partial_\mu \alpha(x)\right\} = 0.\] By the principle of stationary action the functional derivatives with respect to the fields \(\phi(x)\) and \(\phi^*(x)\) vanish. For the third term we define the current \(J^\mu(x)\) through \[\frac{\delta S}{\delta A_\mu(x)} = \sqrt{g(x)} J^\mu(x). \label{eq:currentDefVariationAction}\] Using partial integration we obtain from local U\((1)\) gauge invariance the conservation law \[\frac{1}{\sqrt{g(x)}}\partial_\mu \left[ \sqrt{g(x)} J^\mu(x) \right] = 0. \label{covariantCurrentConservation}\] This can be seen as electromagnetic current conservation. To make things concrete we give the corresponding expression for the action in eq. \(\eqref{eq:actionComplexScalarGeneral}\), as obtained by variation of \(A_\mu(x)\), \[J^\mu(x) = - i g^{\mu\nu}(x) \left[ \phi^*(x) \partial_\nu \phi(x) - \phi(x) \partial_\nu \phi^*(x) \right].\]

General coordinate invariance

The action in eq. \(\eqref{eq:actionComplexScalarGeneral}\) is also invariant under invertible general coordinate transformations, \(x^\mu \to x^{\prime\mu}(x)\). The scalar fields transform like \[\phi(x) \to \phi^\prime(x^\prime) = \phi(x(x^\prime)),\] which implies for its derivatives \[\frac{\partial}{\partial x^\mu} \phi(x) \to \frac{\partial}{\partial x^{\prime\mu}} \phi^\prime(x^\prime) = \frac{\partial x^\nu}{\partial x^{\prime\mu}} \frac{\partial}{\partial x^\nu} \phi(x(x^\prime)).\] A similar transformation behavior is needed for the external gauge field, \[A_\mu(x) \to A^\prime_\mu(x^\prime) = \frac{\partial x^\nu}{\partial x^{\prime\mu}} A_\nu(x(x^\prime)),\] and for the metric, \[g_{\mu\nu}(x) \to g^\prime_{\mu\nu}(x^\prime) = \frac{\partial x^\rho}{\partial x^{\prime\mu}} \frac{\partial x^\sigma}{\partial x^{\prime\nu}} g_{\rho\sigma}(x(x^\prime)).\] Using the Jacobi determinant one finds that \(d^4 x \sqrt{g}\) is a covariant spacetime volume element. Also, for the inverse metric this implies \[g^{\mu\nu}(x) \to g^{\prime\mu\nu}(x^\prime) = \frac{\partial x^{\prime\mu}}{\partial x^\rho} \frac{\partial x^{\prime\nu}}{\partial x^\sigma} g^{\rho\sigma}(x(x^\prime)).\] Combining terms we find that the action \(\eqref{eq:actionComplexScalarGeneral}\) is indeed invariant under general coordinate transformations.

Infinitesimal general coordinate transformations

In an action as in eq. \(\eqref{eq:actionComplexScalarGeneral}\) the coordinates \(x^\mu\) are just integration variables. One may therefore label them from \(x^{\prime\mu}\) back to \(x^\mu\) after the coordinate transformation. For the metric this leads to the transformation rule \[g_{\mu\nu}(x) \to g^\prime_{\mu\nu}(x) = \frac{\partial x^\rho}{\partial x^{\prime\mu}} \frac{\partial x^\sigma}{\partial x^{\prime\nu}} g_{\rho\sigma}(x) - \left[ g^{\prime}_{\mu\nu}(x^\prime) - g^\prime_{\mu\nu}(x) \right].\] We now also specialize to infinitesimal coordinate transformations, \(x^{\prime\mu} = x^\mu - \varepsilon^\mu(x)\). This implies for the metric \[g_{\mu\nu}(x) \to g_{\mu\nu}(x) + \mathcal{L}_\varepsilon g_{\mu\nu}(x),\] where the Lie derivative of the metric is defined as \[\mathcal{L}_\varepsilon g_{\mu\nu}(x) = \varepsilon^\rho(x) \partial_\rho g_{\mu\nu}(x) + g_{\rho\nu}(x) \partial_\mu \varepsilon^\rho(x) + g_{\mu\rho}(x) \partial_\nu \varepsilon^\rho(x).\] Similarly, the external gauge field transforms \(A_\mu(x) \to A_\mu(x) + \mathcal{L}_\varepsilon A_\mu(x)\) where the Lie derivative of a one-form field is given by \[\mathcal{L}_\varepsilon A_\mu(x) = \varepsilon^\nu(x) \partial_\nu A_\mu(x) + A_\nu(x) \partial_\mu \varepsilon^\nu(x).\] Finally, the scalar field transforms as \(\phi(x) \to \phi(x) + \mathcal{L}_\varepsilon \phi(x)\) with the Lie derivative of a scalar field defined as \[\mathcal{L}_\varepsilon \phi(x) = \varepsilon^\mu(x) \partial_\mu \phi(x).\] Formulated in this way, general coordinate transformations resemble closely other local symmetry transformations like the local U\((1)\) gauge transformations discussed above.

Energy-momentum conservation from general coordinate invariance

We also know that the action must be invariant under general coordinate transformations. For an infinitesimal transformation we can write \[\begin{split} \delta S = & \int d^4 x \left\{ \frac{\delta S}{\delta \phi(x)} \varepsilon^\mu(x) \phi(x) + \frac{\delta S}{\delta \phi^*(x)} \varepsilon^\mu(x) \phi^*(x) + \frac{\delta S}{\delta A_\mu(x)} \left[ \varepsilon^\nu(x) \partial_\nu A_\mu(x) + A_\nu(x) \partial_\mu \varepsilon^\nu(x)\right] \right. \\ & \quad\quad \quad + \left. \frac{\delta S}{\delta g_{\mu\nu}(x)} \left[ \varepsilon^\rho(x) \partial_\rho g_{\mu\nu}(x) + g_{\rho\nu}(x) \partial_\mu \varepsilon^\rho(x) + g_{\mu\rho}(x) \partial_\nu \varepsilon^\rho(x) \right] \right\} = 0. \end{split}\] Again the first two terms on the right hand side vanish when the equation of motion for \(\phi(x)\) is fulfilled. We use now the definition \(\eqref{eq:currentDefVariationAction}\) and also define the energy-momentum tensor \(T^{\mu\nu}(x)\) through \[\frac{\delta S}{\delta g_{\mu\nu}(x)} = \frac{1}{2} \sqrt{g(x)} T^{\mu\nu}(x).\] It is symmetric by definition, \(T^{\mu\nu}(x) = T^{\nu\mu}(x)\). The change in action can now be written as \[\begin{split} \delta S = & \int d^4 x \sqrt{g(x)} {\Big \{} J^\mu(x) \left[ \varepsilon^\nu(x) \partial_\nu A_\mu(x) + A_\nu(x) \partial_\mu \varepsilon^\nu(x)\right] \\ & + \frac{1}{2} T^{\mu\nu}(x) \left[ \varepsilon^\rho(x) \partial_\rho g_{\mu\nu}(x) + g_{\rho\nu}(x) \partial_\mu \varepsilon^\rho(x) + g_{\mu\rho}(x) \partial_\nu \varepsilon^\rho(x) \right] {\Big \}} = 0. \end{split}\] For the last term in the first line we perform a partial integration and use the current conservation law \(\eqref{covariantCurrentConservation}\). This leads to a term involving the field strength tensor \(F_{\mu\nu} = \partial_\mu A_\nu - \partial_\nu A_\mu\). For the second and third term in the last line we also perform partial integrations and use the symmetry of the energy-momentum tensor. This yields, up to boundary terms, \[\begin{split} \delta S = & \int d^4 x \sqrt{g(x)} \, \varepsilon^\rho(x) {\bigg \{} F_{\rho\mu}(x) J^\mu(x) - \frac{1}{\sqrt{g(x)}} \partial_\mu \left[ \sqrt{g(x)} T^\mu_{\phantom{\mu}\rho}(x) \right] + \frac{1}{2} T^{\mu\nu}(x) \partial_\rho g_{\mu\nu}(x) {\bigg \}} = 0. \end{split}\] Because \(\varepsilon^\rho(x)\) is arbitrary the term in curly brackets must vanish. For vanishing field strength of the external gauge field, \(F_{\mu\nu}(x)=0\), and constant Minkowski space metric, \(g_{\mu\nu}(x) = g_{\mu\nu}\), this reduces to the standard energy-momentum conservation law in Cartesian coordinates, \(\partial_\mu T^{\mu\nu}(x) = 0\). In summary, energy-momentum conservation can be seen as a consequence of general coordinate invariance of the action in terms of Noether’s second theorem.

Energy-momentum tensor for complex scalar field in general coordinates

Let us finally calculate the energy-momentum tensor for the action in eq. \(\eqref{eq:actionComplexScalarGeneral}\) by variation of the metric. Varying only the metric but keeping the scalar field \(\phi(x)\) fixed, and setting the gauge field \(A_\mu(x)\) to zero for simplicity, we find \[\begin{split} \delta S = & \int d^4 x \left[\delta \sqrt{g}\right] \left\{ - g^{\mu\nu} \partial_\mu \phi^* \partial_\nu \phi - V(\phi^* \phi) \right\} + \int d^4 x \sqrt{g} \left\{ - \left[\delta g^{\mu\nu} \right] \partial_\mu \phi^* \partial_\nu \phi \right\} \end{split}\] We need the formulas \[\delta \sqrt{g} = \frac{1}{2} \sqrt{g} g^{\mu\nu} \delta g_{\mu\nu}, \quad\quad\quad \delta g^{\mu\nu} = - g^{\mu\rho} g^{\nu\sigma} \delta g_{\rho\sigma}.\] Writing then \[\delta S = \int d^4 x \sqrt{g} \left\{ \frac{1}{2} T^{\mu\nu} \delta g_{\mu\nu} \right\},\] and comparing terms, leads to the energy-momentum tensor of a complex scalar field \[T^{\mu\nu} = 2 g^{\mu\rho} g^{\nu\sigma} \partial_\rho \phi^* \partial_\sigma \phi - g^{\mu\nu} \left[ g^{\rho\sigma} \partial_\rho \phi^* \partial_\sigma \phi + V(\phi^* \phi)\right].\] Specializing to the zero-zero component we find the energy density \[T^{00} = \mathscr{H} = \dot \phi^* \dot \phi + \boldsymbol{\nabla} \phi^* \boldsymbol{\nabla} \phi + V(\phi^* \phi).\]